TPTP Problem File: SLH0110^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Fishers_Inequality/0034_Incidence_Matrices/prob_00404_017402__27984880_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1500 ( 535 unt; 232 typ; 0 def)
% Number of atoms : 3651 (1272 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9762 ( 426 ~; 84 |; 158 &;7453 @)
% ( 0 <=>;1641 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 588 ( 588 >; 0 *; 0 +; 0 <<)
% Number of symbols : 211 ( 208 usr; 19 con; 0-3 aty)
% Number of variables : 3226 ( 215 ^;2873 !; 138 ?;3226 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:47:58.854
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
produc8199716216217303280at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Matrix__Ovec_Itf__b_J_J,type,
vec_vec_b: $tType ).
thf(ty_n_t__Matrix__Omat_It__Matrix__Ovec_Itf__b_J_J,type,
mat_vec_b: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Ovec_Itf__b_J_J,type,
list_vec_b: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_Itf__b_J_J,type,
set_vec_b: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Set__Oset_Itf__b_J_J,type,
vec_set_b: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Real__Oreal_J,type,
vec_real: $tType ).
thf(ty_n_t__Matrix__Omat_It__Real__Oreal_J,type,
mat_real: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Nat__Onat_J,type,
vec_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Int__Oint_J,type,
vec_int: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Int__Oint_J,type,
mat_int: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Matrix__Ovec_Itf__b_J,type,
vec_b: $tType ).
thf(ty_n_t__Matrix__Omat_Itf__b_J,type,
mat_b: $tType ).
thf(ty_n_t__List__Olist_Itf__b_J,type,
list_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
% Explicit typings (208)
thf(sy_c_Finite__Set_Ocard_001t__Matrix__Ovec_Itf__b_J,type,
finite_card_vec_b: set_vec_b > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__b,type,
finite_card_b: set_b > nat ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Ovec_It__Int__Oint_J,type,
minus_minus_vec_int: vec_int > vec_int > vec_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Ovec_It__Nat__Onat_J,type,
minus_minus_vec_nat: vec_nat > vec_nat > vec_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Ovec_It__Real__Oreal_J,type,
minus_minus_vec_real: vec_real > vec_real > vec_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oone__class_Oone_001tf__b,type,
one_one_b: b ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__b,type,
zero_zero_b: b ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Incidence__Matrices_Omat__block__size_001t__Int__Oint,type,
incide7086334917823153261ze_int: mat_int > nat > nat ).
thf(sy_c_Incidence__Matrices_Omat__block__size_001t__Real__Oreal,type,
incide8656787765868073069e_real: mat_real > nat > nat ).
thf(sy_c_Incidence__Matrices_Omat__block__size_001tf__b,type,
incide1360831186763368190size_b: mat_b > nat > nat ).
thf(sy_c_Incidence__Matrices_Omat__rep__num_001t__Int__Oint,type,
incide7000514267430604580um_int: mat_int > nat > nat ).
thf(sy_c_Incidence__Matrices_Omat__rep__num_001t__Real__Oreal,type,
incide5781393841671188388m_real: mat_real > nat > nat ).
thf(sy_c_Incidence__Matrices_Omat__rep__num_001tf__b,type,
incide1817117124879203335_num_b: mat_b > nat > nat ).
thf(sy_c_Incidence__Matrices_Onon__empty__col_001t__Int__Oint,type,
incide6851923868969248411ol_int: mat_int > nat > $o ).
thf(sy_c_Incidence__Matrices_Onon__empty__col_001t__Nat__Onat,type,
incide6854414339478298687ol_nat: mat_nat > nat > $o ).
thf(sy_c_Incidence__Matrices_Onon__empty__col_001t__Real__Oreal,type,
incide8049862060206209947l_real: mat_real > nat > $o ).
thf(sy_c_Incidence__Matrices_Onon__empty__col_001tf__b,type,
incide3034858701194040400_col_b: mat_b > nat > $o ).
thf(sy_c_Incidence__Matrices_Oproper__inc__mat_001tf__b,type,
incide2997380824311827482_mat_b: mat_b > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001t__Int__Oint,type,
incide4964164200581851450ix_int: mat_int > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001t__Nat__Onat,type,
incide4966654671090901726ix_nat: mat_nat > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001t__Real__Oreal,type,
incide4475037519619858106x_real: mat_real > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_001tf__b,type,
incide7367983062745021297trix_b: mat_b > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_Omap__col__to__block_001t__Int__Oint,type,
incide3973235006681262014ck_int: vec_int > set_nat ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_Omap__col__to__block_001t__Nat__Onat,type,
incide3975725477190312290ck_nat: vec_nat > set_nat ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix_Omap__col__to__block_001tf__b,type,
incide5355957740755015149lock_b: vec_b > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Ovec_Itf__b_J,type,
set_vec_b2: list_vec_b > set_vec_b ).
thf(sy_c_Matrix_Ocol_001t__Int__Oint,type,
col_int: mat_int > nat > vec_int ).
thf(sy_c_Matrix_Ocol_001t__Matrix__Ovec_Itf__b_J,type,
col_vec_b: mat_vec_b > nat > vec_vec_b ).
thf(sy_c_Matrix_Ocol_001t__Nat__Onat,type,
col_nat: mat_nat > nat > vec_nat ).
thf(sy_c_Matrix_Ocol_001t__Real__Oreal,type,
col_real: mat_real > nat > vec_real ).
thf(sy_c_Matrix_Ocol_001tf__b,type,
col_b: mat_b > nat > vec_b ).
thf(sy_c_Matrix_Ocols_001tf__b,type,
cols_b: mat_b > list_vec_b ).
thf(sy_c_Matrix_Odiag__mat_001tf__b,type,
diag_mat_b: mat_b > list_b ).
thf(sy_c_Matrix_Odim__col_001t__Int__Oint,type,
dim_col_int: mat_int > nat ).
thf(sy_c_Matrix_Odim__col_001t__Matrix__Ovec_Itf__b_J,type,
dim_col_vec_b: mat_vec_b > nat ).
thf(sy_c_Matrix_Odim__col_001t__Nat__Onat,type,
dim_col_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__col_001t__Real__Oreal,type,
dim_col_real: mat_real > nat ).
thf(sy_c_Matrix_Odim__col_001tf__b,type,
dim_col_b: mat_b > nat ).
thf(sy_c_Matrix_Odim__row_001t__Int__Oint,type,
dim_row_int: mat_int > nat ).
thf(sy_c_Matrix_Odim__row_001t__Matrix__Ovec_Itf__b_J,type,
dim_row_vec_b: mat_vec_b > nat ).
thf(sy_c_Matrix_Odim__row_001t__Nat__Onat,type,
dim_row_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__row_001t__Real__Oreal,type,
dim_row_real: mat_real > nat ).
thf(sy_c_Matrix_Odim__row_001tf__b,type,
dim_row_b: mat_b > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Int__Oint,type,
dim_vec_int: vec_int > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Matrix__Ovec_Itf__b_J,type,
dim_vec_vec_b: vec_vec_b > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Nat__Onat,type,
dim_vec_nat: vec_nat > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Real__Oreal,type,
dim_vec_real: vec_real > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Set__Oset_Itf__b_J,type,
dim_vec_set_b: vec_set_b > nat ).
thf(sy_c_Matrix_Odim__vec_001tf__b,type,
dim_vec_b: vec_b > nat ).
thf(sy_c_Matrix_Oelements__mat_001t__Int__Oint,type,
elements_mat_int: mat_int > set_int ).
thf(sy_c_Matrix_Oelements__mat_001t__Matrix__Ovec_Itf__b_J,type,
elements_mat_vec_b: mat_vec_b > set_vec_b ).
thf(sy_c_Matrix_Oelements__mat_001t__Nat__Onat,type,
elements_mat_nat: mat_nat > set_nat ).
thf(sy_c_Matrix_Oelements__mat_001t__Real__Oreal,type,
elements_mat_real: mat_real > set_real ).
thf(sy_c_Matrix_Oelements__mat_001tf__b,type,
elements_mat_b: mat_b > set_b ).
thf(sy_c_Matrix_Oindex__mat_001tf__b,type,
index_mat_b: mat_b > product_prod_nat_nat > b ).
thf(sy_c_Matrix_Omat__of__row__fun_001t__Int__Oint,type,
mat_of_row_fun_int: nat > nat > ( nat > vec_int ) > mat_int ).
thf(sy_c_Matrix_Omat__of__row__fun_001tf__b,type,
mat_of_row_fun_b: nat > nat > ( nat > vec_b ) > mat_b ).
thf(sy_c_Matrix_Orow_001t__Int__Oint,type,
row_int: mat_int > nat > vec_int ).
thf(sy_c_Matrix_Orow_001t__Matrix__Ovec_Itf__b_J,type,
row_vec_b: mat_vec_b > nat > vec_vec_b ).
thf(sy_c_Matrix_Orow_001t__Nat__Onat,type,
row_nat: mat_nat > nat > vec_nat ).
thf(sy_c_Matrix_Orow_001t__Real__Oreal,type,
row_real: mat_real > nat > vec_real ).
thf(sy_c_Matrix_Orow_001tf__b,type,
row_b: mat_b > nat > vec_b ).
thf(sy_c_Matrix_Otranspose__mat_001t__Int__Oint,type,
transpose_mat_int: mat_int > mat_int ).
thf(sy_c_Matrix_Otranspose__mat_001t__Nat__Onat,type,
transpose_mat_nat: mat_nat > mat_nat ).
thf(sy_c_Matrix_Otranspose__mat_001t__Real__Oreal,type,
transpose_mat_real: mat_real > mat_real ).
thf(sy_c_Matrix_Otranspose__mat_001tf__b,type,
transpose_mat_b: mat_b > mat_b ).
thf(sy_c_Matrix_Ounit__vec_001t__Int__Oint,type,
unit_vec_int: nat > nat > vec_int ).
thf(sy_c_Matrix_Ounit__vec_001t__Nat__Onat,type,
unit_vec_nat: nat > nat > vec_nat ).
thf(sy_c_Matrix_Ounit__vec_001t__Real__Oreal,type,
unit_vec_real: nat > nat > vec_real ).
thf(sy_c_Matrix_Ounit__vec_001tf__b,type,
unit_vec_b: nat > nat > vec_b ).
thf(sy_c_Matrix_Oupdate__vec_001t__Int__Oint,type,
update_vec_int: vec_int > nat > int > vec_int ).
thf(sy_c_Matrix_Oupdate__vec_001tf__b,type,
update_vec_b: vec_b > nat > b > vec_b ).
thf(sy_c_Matrix_Ovec__first_001t__Int__Oint,type,
vec_first_int: vec_int > nat > vec_int ).
thf(sy_c_Matrix_Ovec__first_001tf__b,type,
vec_first_b: vec_b > nat > vec_b ).
thf(sy_c_Matrix_Ovec__index_001t__Int__Oint,type,
vec_index_int: vec_int > nat > int ).
thf(sy_c_Matrix_Ovec__index_001t__Matrix__Ovec_Itf__b_J,type,
vec_index_vec_b: vec_vec_b > nat > vec_b ).
thf(sy_c_Matrix_Ovec__index_001t__Nat__Onat,type,
vec_index_nat: vec_nat > nat > nat ).
thf(sy_c_Matrix_Ovec__index_001t__Real__Oreal,type,
vec_index_real: vec_real > nat > real ).
thf(sy_c_Matrix_Ovec__index_001t__Set__Oset_Itf__b_J,type,
vec_index_set_b: vec_set_b > nat > set_b ).
thf(sy_c_Matrix_Ovec__index_001tf__b,type,
vec_index_b: vec_b > nat > b ).
thf(sy_c_Matrix_Ovec__last_001t__Int__Oint,type,
vec_last_int: vec_int > nat > vec_int ).
thf(sy_c_Matrix_Ovec__last_001tf__b,type,
vec_last_b: vec_b > nat > vec_b ).
thf(sy_c_Matrix_Ovec__set_001t__Int__Oint,type,
vec_set_int: vec_int > set_int ).
thf(sy_c_Matrix_Ovec__set_001t__Matrix__Ovec_Itf__b_J,type,
vec_set_vec_b: vec_vec_b > set_vec_b ).
thf(sy_c_Matrix_Ovec__set_001t__Nat__Onat,type,
vec_set_nat: vec_nat > set_nat ).
thf(sy_c_Matrix_Ovec__set_001t__Real__Oreal,type,
vec_set_real: vec_real > set_real ).
thf(sy_c_Matrix_Ovec__set_001tf__b,type,
vec_set_b2: vec_b > set_b ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__vec_001t__Int__Oint,type,
matrix2748772424961467270ec_int: nat > vec_int ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__vec_001t__Nat__Onat,type,
matrix2751262895470517546ec_nat: nat > vec_nat ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__vec_001tf__b,type,
matrix8789069900454870053_vec_b: nat > vec_b ).
thf(sy_c_Matrix__Vector__Extras_Ocomm__monoid__add__class_Osum__vec_001t__Int__Oint,type,
matrix3634415343793898042ec_int: vec_int > int ).
thf(sy_c_Matrix__Vector__Extras_Ocomm__monoid__add__class_Osum__vec_001t__Real__Oreal,type,
matrix1363837090280519610c_real: vec_real > real ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001t__Int__Oint_001t__Int__Oint,type,
matrix4192408864422109198nt_int: ( int > int ) > $o ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001t__Int__Oint_001t__Nat__Onat,type,
matrix4194899334931159474nt_nat: ( int > nat ) > $o ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001t__Int__Oint_001t__Real__Oreal,type,
matrix8987650032624085902t_real: ( int > real ) > $o ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001t__Int__Oint_001tf__b,type,
matrix6466546049528123805_int_b: ( int > b ) > $o ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001t__Nat__Onat_001t__Int__Oint,type,
matrix3193055152521054642at_int: ( nat > int ) > $o ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001t__Nat__Onat_001t__Nat__Onat,type,
matrix3195545623030104918at_nat: ( nat > nat ) > $o ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001t__Nat__Onat_001t__Real__Oreal,type,
matrix6800728457931064114t_real: ( nat > real ) > $o ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001t__Nat__Onat_001tf__b,type,
matrix8711691017198569721_nat_b: ( nat > b ) > $o ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001t__Real__Oreal_001t__Int__Oint,type,
matrix2142174523696136718al_int: ( real > int ) > $o ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001t__Real__Oreal_001t__Nat__Onat,type,
matrix2144664994205186994al_nat: ( real > nat ) > $o ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001t__Real__Oreal_001tf__b,type,
matrix2454726049922035613real_b: ( real > b ) > $o ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001tf__b_001tf__b,type,
matrix6790486239265177588om_b_b: ( b > b ) > $o ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__vec_001t__Int__Oint_001t__Int__Oint,type,
matrix8301520909418075407nt_int: vec_int > vec_int ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__vec_001t__Int__Oint_001tf__b,type,
matrix1865072738833226460_int_b: vec_int > vec_b ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__vec_001tf__b_001t__Int__Oint,type,
matrix1311240063772730166_b_int: vec_b > vec_int ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__vec_001tf__b_001tf__b,type,
matrix7059812428859951221ec_b_b: vec_b > vec_b ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Int__Oint_001t__Int__Oint,type,
matrix1697308990001484774nt_int: int > int ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Int__Oint_001t__Nat__Onat,type,
matrix1699799460510535050nt_nat: int > nat ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Int__Oint_001t__Real__Oreal,type,
matrix1706393078865277798t_real: int > real ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Int__Oint_001tf__b,type,
matrix6038540757728371653_int_b: int > b ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Nat__Onat_001t__Int__Oint,type,
matrix697955278100430218at_int: nat > int ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Nat__Onat_001t__Nat__Onat,type,
matrix700445748609480494at_nat: nat > nat ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Nat__Onat_001t__Real__Oreal,type,
matrix8742843541027031818t_real: nat > real ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Nat__Onat_001tf__b,type,
matrix8283685725398817569_nat_b: nat > b ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Real__Oreal_001t__Int__Oint,type,
matrix4084289606792104422al_int: real > int ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Real__Oreal_001t__Nat__Onat,type,
matrix4086780077301154698al_nat: real > nat ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Real__Oreal_001t__Real__Oreal,type,
matrix3070681271257819494l_real: real > real ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001t__Real__Oreal_001tf__b,type,
matrix6537263852557659589real_b: real > b ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001tf__b_001t__Int__Oint,type,
matrix5484708082667875359_b_int: b > int ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001tf__b_001t__Nat__Onat,type,
matrix5487198553176925635_b_nat: b > nat ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001tf__b_001t__Real__Oreal,type,
matrix2280091663418064671b_real: b > real ).
thf(sy_c_Matrix__Vector__Extras_Ozero__neq__one__class_Oof__zero__neq__one_001tf__b_001tf__b,type,
matrix4781043112069605324ne_b_b: b > b ).
thf(sy_c_Missing__List_Oadjust__idx__rev,type,
missin3815256168798769645dx_rev: nat > nat > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Matrix__Ovec_Itf__b_J_M_Eo_J,type,
bot_bot_vec_b_o: vec_b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__b_M_Eo_J,type,
bot_bot_b_o: b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Matrix__Ovec_Itf__b_J_J,type,
bot_bot_set_vec_b: set_vec_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Matrix__Ovec_Itf__b_J_J,type,
ord_less_set_vec_b: set_vec_b > set_vec_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__b_J,type,
ord_less_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Int__Oint_J,type,
ord_less_eq_vec_int: vec_int > vec_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Nat__Onat_J,type,
ord_less_eq_vec_nat: vec_nat > vec_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Real__Oreal_J,type,
ord_less_eq_vec_real: vec_real > vec_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Set__Oset_Itf__b_J_J,type,
ord_le7003571194052948520_set_b: vec_set_b > vec_set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_Itf__b_J_J,type,
ord_le4862985661309304830_vec_b: set_vec_b > set_vec_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc72220940542539688at_nat: ( nat > nat ) > nat > produc8199716216217303280at_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Ring__Hom_Oone__hom__1__axioms_001t__Int__Oint_001t__Int__Oint,type,
ring_o8179449149706882820nt_int: ( int > int ) > $o ).
thf(sy_c_Ring__Hom_Oone__hom__1__axioms_001t__Int__Oint_001t__Nat__Onat,type,
ring_o8181939620215933096nt_nat: ( int > nat ) > $o ).
thf(sy_c_Ring__Hom_Oone__hom__1__axioms_001t__Int__Oint_001tf__b,type,
ring_o6550733107361182759_int_b: ( int > b ) > $o ).
thf(sy_c_Ring__Hom_Oone__hom__1__axioms_001t__Nat__Onat_001t__Int__Oint,type,
ring_o7180095437805828264at_int: ( nat > int ) > $o ).
thf(sy_c_Ring__Hom_Oone__hom__1__axioms_001t__Nat__Onat_001t__Nat__Onat,type,
ring_o7182585908314878540at_nat: ( nat > nat ) > $o ).
thf(sy_c_Ring__Hom_Oone__hom__1__axioms_001t__Nat__Onat_001tf__b,type,
ring_o8795878075031628675_nat_b: ( nat > b ) > $o ).
thf(sy_c_Ring__Hom_Oone__hom__1__axioms_001tf__b_001t__Int__Oint,type,
ring_o5996900432300686465_b_int: ( b > int ) > $o ).
thf(sy_c_Ring__Hom_Oone__hom__1__axioms_001tf__b_001t__Nat__Onat,type,
ring_o5999390902809736741_b_nat: ( b > nat ) > $o ).
thf(sy_c_Ring__Hom_Oone__hom__1__axioms_001tf__b_001tf__b,type,
ring_o8422147925713169386ms_b_b: ( b > b ) > $o ).
thf(sy_c_Set_OCollect_001t__Matrix__Ovec_Itf__b_J,type,
collect_vec_b: ( vec_b > $o ) > set_vec_b ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__Matrix__Ovec_Itf__b_J,type,
insert_vec_b: vec_b > set_vec_b > set_vec_b ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Set_Ois__singleton_001t__Matrix__Ovec_Itf__b_J,type,
is_singleton_vec_b: set_vec_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001tf__b,type,
is_singleton_b: set_b > $o ).
thf(sy_c_Set_Othe__elem_001tf__b,type,
the_elem_b: set_b > b ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Matrix__Ovec_Itf__b_J,type,
member_vec_b: vec_b > set_vec_b > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_j,type,
j: nat ).
thf(sy_v_k____,type,
k: b ).
thf(sy_v_matrix,type,
matrix: mat_b ).
% Relevant facts (1262)
thf(fact_0__092_060open_062k_A_061_A_I1_058_058_Hb_J_092_060close_062,axiom,
k = one_one_b ).
% \<open>k = (1::'b)\<close>
thf(fact_1_kn0,axiom,
k != zero_zero_b ).
% kn0
thf(fact_2_zero__one__matrix__axioms,axiom,
incide7367983062745021297trix_b @ matrix ).
% zero_one_matrix_axioms
thf(fact_3__092_060open_062non__empty__col_AM_Aj_092_060close_062,axiom,
incide3034858701194040400_col_b @ matrix @ j ).
% \<open>non_empty_col M j\<close>
thf(fact_4_kin,axiom,
member_b @ k @ ( vec_set_b2 @ ( col_b @ matrix @ j ) ) ).
% kin
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062k_O_A_092_060lbrakk_062k_A_092_060noteq_062_A_I0_058_058_Hb_J_059_Ak_A_092_060in_062_E_Acol_AM_Aj_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [K: b] :
( ( K != zero_zero_b )
=> ~ ( member_b @ K @ ( vec_set_b2 @ ( col_b @ matrix @ j ) ) ) ) ).
% \<open>\<And>thesis. (\<And>k. \<lbrakk>k \<noteq> (0::'b); k \<in>$ col M j\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_assms,axiom,
ord_less_nat @ j @ ( dim_col_b @ matrix ) ).
% assms
thf(fact_7__092_060open_062k_A_092_060in_062_Aelements__mat_AM_092_060close_062,axiom,
member_b @ k @ ( elements_mat_b @ matrix ) ).
% \<open>k \<in> elements_mat M\<close>
thf(fact_8_one__reorient,axiom,
! [X: b] :
( ( one_one_b = X )
= ( X = one_one_b ) ) ).
% one_reorient
thf(fact_9_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_10_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_11_non__empty__col__def,axiom,
( incide6854414339478298687ol_nat
= ( ^ [M: mat_nat,J: nat] :
? [K2: nat] :
( ( K2 != zero_zero_nat )
& ( member_nat @ K2 @ ( vec_set_nat @ ( col_nat @ M @ J ) ) ) ) ) ) ).
% non_empty_col_def
thf(fact_12_non__empty__col__def,axiom,
( incide6851923868969248411ol_int
= ( ^ [M: mat_int,J: nat] :
? [K2: int] :
( ( K2 != zero_zero_int )
& ( member_int @ K2 @ ( vec_set_int @ ( col_int @ M @ J ) ) ) ) ) ) ).
% non_empty_col_def
thf(fact_13_non__empty__col__def,axiom,
( incide8049862060206209947l_real
= ( ^ [M: mat_real,J: nat] :
? [K2: real] :
( ( K2 != zero_zero_real )
& ( member_real @ K2 @ ( vec_set_real @ ( col_real @ M @ J ) ) ) ) ) ) ).
% non_empty_col_def
thf(fact_14_non__empty__col__def,axiom,
( incide3034858701194040400_col_b
= ( ^ [M: mat_b,J: nat] :
? [K2: b] :
( ( K2 != zero_zero_b )
& ( member_b @ K2 @ ( vec_set_b2 @ ( col_b @ M @ J ) ) ) ) ) ) ).
% non_empty_col_def
thf(fact_15_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_16_one__hom__1__axioms_Ointro,axiom,
! [Hom: b > b] :
( ! [X2: b] :
( ( ( Hom @ X2 )
= one_one_b )
=> ( X2 = one_one_b ) )
=> ( ring_o8422147925713169386ms_b_b @ Hom ) ) ).
% one_hom_1_axioms.intro
thf(fact_17_one__hom__1__axioms_Ointro,axiom,
! [Hom: nat > b] :
( ! [X2: nat] :
( ( ( Hom @ X2 )
= one_one_b )
=> ( X2 = one_one_nat ) )
=> ( ring_o8795878075031628675_nat_b @ Hom ) ) ).
% one_hom_1_axioms.intro
thf(fact_18_one__hom__1__axioms_Ointro,axiom,
! [Hom: int > b] :
( ! [X2: int] :
( ( ( Hom @ X2 )
= one_one_b )
=> ( X2 = one_one_int ) )
=> ( ring_o6550733107361182759_int_b @ Hom ) ) ).
% one_hom_1_axioms.intro
thf(fact_19_one__hom__1__axioms_Ointro,axiom,
! [Hom: b > nat] :
( ! [X2: b] :
( ( ( Hom @ X2 )
= one_one_nat )
=> ( X2 = one_one_b ) )
=> ( ring_o5999390902809736741_b_nat @ Hom ) ) ).
% one_hom_1_axioms.intro
thf(fact_20_one__hom__1__axioms_Ointro,axiom,
! [Hom: nat > nat] :
( ! [X2: nat] :
( ( ( Hom @ X2 )
= one_one_nat )
=> ( X2 = one_one_nat ) )
=> ( ring_o7182585908314878540at_nat @ Hom ) ) ).
% one_hom_1_axioms.intro
thf(fact_21_one__hom__1__axioms_Ointro,axiom,
! [Hom: int > nat] :
( ! [X2: int] :
( ( ( Hom @ X2 )
= one_one_nat )
=> ( X2 = one_one_int ) )
=> ( ring_o8181939620215933096nt_nat @ Hom ) ) ).
% one_hom_1_axioms.intro
thf(fact_22_one__hom__1__axioms_Ointro,axiom,
! [Hom: b > int] :
( ! [X2: b] :
( ( ( Hom @ X2 )
= one_one_int )
=> ( X2 = one_one_b ) )
=> ( ring_o5996900432300686465_b_int @ Hom ) ) ).
% one_hom_1_axioms.intro
thf(fact_23_one__hom__1__axioms_Ointro,axiom,
! [Hom: nat > int] :
( ! [X2: nat] :
( ( ( Hom @ X2 )
= one_one_int )
=> ( X2 = one_one_nat ) )
=> ( ring_o7180095437805828264at_int @ Hom ) ) ).
% one_hom_1_axioms.intro
thf(fact_24_one__hom__1__axioms_Ointro,axiom,
! [Hom: int > int] :
( ! [X2: int] :
( ( ( Hom @ X2 )
= one_one_int )
=> ( X2 = one_one_int ) )
=> ( ring_o8179449149706882820nt_int @ Hom ) ) ).
% one_hom_1_axioms.intro
thf(fact_25_one__hom__1__axioms__def,axiom,
( ring_o8422147925713169386ms_b_b
= ( ^ [Hom2: b > b] :
! [X3: b] :
( ( ( Hom2 @ X3 )
= one_one_b )
=> ( X3 = one_one_b ) ) ) ) ).
% one_hom_1_axioms_def
thf(fact_26_one__hom__1__axioms__def,axiom,
( ring_o8795878075031628675_nat_b
= ( ^ [Hom2: nat > b] :
! [X3: nat] :
( ( ( Hom2 @ X3 )
= one_one_b )
=> ( X3 = one_one_nat ) ) ) ) ).
% one_hom_1_axioms_def
thf(fact_27_one__hom__1__axioms__def,axiom,
( ring_o6550733107361182759_int_b
= ( ^ [Hom2: int > b] :
! [X3: int] :
( ( ( Hom2 @ X3 )
= one_one_b )
=> ( X3 = one_one_int ) ) ) ) ).
% one_hom_1_axioms_def
thf(fact_28_one__hom__1__axioms__def,axiom,
( ring_o5999390902809736741_b_nat
= ( ^ [Hom2: b > nat] :
! [X3: b] :
( ( ( Hom2 @ X3 )
= one_one_nat )
=> ( X3 = one_one_b ) ) ) ) ).
% one_hom_1_axioms_def
thf(fact_29_one__hom__1__axioms__def,axiom,
( ring_o7182585908314878540at_nat
= ( ^ [Hom2: nat > nat] :
! [X3: nat] :
( ( ( Hom2 @ X3 )
= one_one_nat )
=> ( X3 = one_one_nat ) ) ) ) ).
% one_hom_1_axioms_def
thf(fact_30_one__hom__1__axioms__def,axiom,
( ring_o8181939620215933096nt_nat
= ( ^ [Hom2: int > nat] :
! [X3: int] :
( ( ( Hom2 @ X3 )
= one_one_nat )
=> ( X3 = one_one_int ) ) ) ) ).
% one_hom_1_axioms_def
thf(fact_31_one__hom__1__axioms__def,axiom,
( ring_o5996900432300686465_b_int
= ( ^ [Hom2: b > int] :
! [X3: b] :
( ( ( Hom2 @ X3 )
= one_one_int )
=> ( X3 = one_one_b ) ) ) ) ).
% one_hom_1_axioms_def
thf(fact_32_one__hom__1__axioms__def,axiom,
( ring_o7180095437805828264at_int
= ( ^ [Hom2: nat > int] :
! [X3: nat] :
( ( ( Hom2 @ X3 )
= one_one_int )
=> ( X3 = one_one_nat ) ) ) ) ).
% one_hom_1_axioms_def
thf(fact_33_one__hom__1__axioms__def,axiom,
( ring_o8179449149706882820nt_int
= ( ^ [Hom2: int > int] :
! [X3: int] :
( ( ( Hom2 @ X3 )
= one_one_int )
=> ( X3 = one_one_int ) ) ) ) ).
% one_hom_1_axioms_def
thf(fact_34_col__elems__ss01,axiom,
! [J2: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_b @ matrix ) )
=> ( ord_less_eq_set_b @ ( vec_set_b2 @ ( col_b @ matrix @ J2 ) ) @ ( insert_b @ zero_zero_b @ ( insert_b @ one_one_b @ bot_bot_set_b ) ) ) ) ).
% col_elems_ss01
thf(fact_35_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_36_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_37_elems01,axiom,
ord_less_eq_set_b @ ( elements_mat_b @ matrix ) @ ( insert_b @ zero_zero_b @ ( insert_b @ one_one_b @ bot_bot_set_b ) ) ).
% elems01
thf(fact_38_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_39_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_40_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_41_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_42_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_43_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_44_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_45_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_46_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_47_zero__one__matrix_Ointro,axiom,
! [Matrix: mat_nat] :
( ( ord_less_eq_set_nat @ ( elements_mat_nat @ Matrix ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( incide4966654671090901726ix_nat @ Matrix ) ) ).
% zero_one_matrix.intro
thf(fact_48_zero__one__matrix_Ointro,axiom,
! [Matrix: mat_int] :
( ( ord_less_eq_set_int @ ( elements_mat_int @ Matrix ) @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) )
=> ( incide4964164200581851450ix_int @ Matrix ) ) ).
% zero_one_matrix.intro
thf(fact_49_zero__one__matrix_Ointro,axiom,
! [Matrix: mat_real] :
( ( ord_less_eq_set_real @ ( elements_mat_real @ Matrix ) @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) )
=> ( incide4475037519619858106x_real @ Matrix ) ) ).
% zero_one_matrix.intro
thf(fact_50_zero__one__matrix_Ointro,axiom,
! [Matrix: mat_b] :
( ( ord_less_eq_set_b @ ( elements_mat_b @ Matrix ) @ ( insert_b @ zero_zero_b @ ( insert_b @ one_one_b @ bot_bot_set_b ) ) )
=> ( incide7367983062745021297trix_b @ Matrix ) ) ).
% zero_one_matrix.intro
thf(fact_51_zero__one__matrix_Oelems01,axiom,
! [Matrix: mat_nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ord_less_eq_set_nat @ ( elements_mat_nat @ Matrix ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ) ).
% zero_one_matrix.elems01
thf(fact_52_zero__one__matrix_Oelems01,axiom,
! [Matrix: mat_int] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ord_less_eq_set_int @ ( elements_mat_int @ Matrix ) @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) ) ) ).
% zero_one_matrix.elems01
thf(fact_53_zero__one__matrix_Oelems01,axiom,
! [Matrix: mat_real] :
( ( incide4475037519619858106x_real @ Matrix )
=> ( ord_less_eq_set_real @ ( elements_mat_real @ Matrix ) @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) ) ) ).
% zero_one_matrix.elems01
thf(fact_54_zero__one__matrix_Oelems01,axiom,
! [Matrix: mat_b] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ord_less_eq_set_b @ ( elements_mat_b @ Matrix ) @ ( insert_b @ zero_zero_b @ ( insert_b @ one_one_b @ bot_bot_set_b ) ) ) ) ).
% zero_one_matrix.elems01
thf(fact_55_zero__one__matrix_Ocol__elems__ss01,axiom,
! [Matrix: mat_nat,J2: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ Matrix ) )
=> ( ord_less_eq_set_nat @ ( vec_set_nat @ ( col_nat @ Matrix @ J2 ) ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ) ) ).
% zero_one_matrix.col_elems_ss01
thf(fact_56_zero__one__matrix_Ocol__elems__ss01,axiom,
! [Matrix: mat_int,J2: nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_int @ Matrix ) )
=> ( ord_less_eq_set_int @ ( vec_set_int @ ( col_int @ Matrix @ J2 ) ) @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) ) ) ) ).
% zero_one_matrix.col_elems_ss01
thf(fact_57_zero__one__matrix_Ocol__elems__ss01,axiom,
! [Matrix: mat_real,J2: nat] :
( ( incide4475037519619858106x_real @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_real @ Matrix ) )
=> ( ord_less_eq_set_real @ ( vec_set_real @ ( col_real @ Matrix @ J2 ) ) @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) ) ) ) ).
% zero_one_matrix.col_elems_ss01
thf(fact_58_zero__one__matrix_Ocol__elems__ss01,axiom,
! [Matrix: mat_b,J2: nat] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_b @ Matrix ) )
=> ( ord_less_eq_set_b @ ( vec_set_b2 @ ( col_b @ Matrix @ J2 ) ) @ ( insert_b @ zero_zero_b @ ( insert_b @ one_one_b @ bot_bot_set_b ) ) ) ) ) ).
% zero_one_matrix.col_elems_ss01
thf(fact_59_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_60_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_61_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_62_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_63_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_64_zero__reorient,axiom,
! [X: b] :
( ( zero_zero_b = X )
= ( X = zero_zero_b ) ) ).
% zero_reorient
thf(fact_65_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_66_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_67_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_68_zero__one__matrix__def,axiom,
( incide4966654671090901726ix_nat
= ( ^ [Matrix2: mat_nat] : ( ord_less_eq_set_nat @ ( elements_mat_nat @ Matrix2 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ) ) ).
% zero_one_matrix_def
thf(fact_69_zero__one__matrix__def,axiom,
( incide4964164200581851450ix_int
= ( ^ [Matrix2: mat_int] : ( ord_less_eq_set_int @ ( elements_mat_int @ Matrix2 ) @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) ) ) ) ).
% zero_one_matrix_def
thf(fact_70_zero__one__matrix__def,axiom,
( incide4475037519619858106x_real
= ( ^ [Matrix2: mat_real] : ( ord_less_eq_set_real @ ( elements_mat_real @ Matrix2 ) @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) ) ) ) ).
% zero_one_matrix_def
thf(fact_71_zero__one__matrix__def,axiom,
( incide7367983062745021297trix_b
= ( ^ [Matrix2: mat_b] : ( ord_less_eq_set_b @ ( elements_mat_b @ Matrix2 ) @ ( insert_b @ zero_zero_b @ ( insert_b @ one_one_b @ bot_bot_set_b ) ) ) ) ) ).
% zero_one_matrix_def
thf(fact_72_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_73_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_74_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_75_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_76_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_77_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_78_transpose__entries,axiom,
ord_less_eq_set_b @ ( elements_mat_b @ ( transpose_mat_b @ matrix ) ) @ ( insert_b @ zero_zero_b @ ( insert_b @ one_one_b @ bot_bot_set_b ) ) ).
% transpose_entries
thf(fact_79_singleton__insert__inj__eq,axiom,
! [B: b,A: b,A2: set_b] :
( ( ( insert_b @ B @ bot_bot_set_b )
= ( insert_b @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_80_singleton__insert__inj__eq_H,axiom,
! [A: b,A2: set_b,B: b] :
( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ bot_bot_set_b ) )
= ( ( A = B )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_81_col__elems__subset__mat,axiom,
! [I: nat,N2: mat_b] :
( ( ord_less_nat @ I @ ( dim_col_b @ N2 ) )
=> ( ord_less_eq_set_b @ ( vec_set_b2 @ ( col_b @ N2 @ I ) ) @ ( elements_mat_b @ N2 ) ) ) ).
% col_elems_subset_mat
thf(fact_82_vec__contains__col__elements__mat,axiom,
! [J2: nat,M3: mat_nat,A: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_nat @ M3 ) )
=> ( ( member_nat @ A @ ( vec_set_nat @ ( col_nat @ M3 @ J2 ) ) )
=> ( member_nat @ A @ ( elements_mat_nat @ M3 ) ) ) ) ).
% vec_contains_col_elements_mat
thf(fact_83_vec__contains__col__elements__mat,axiom,
! [J2: nat,M3: mat_vec_b,A: vec_b] :
( ( ord_less_nat @ J2 @ ( dim_col_vec_b @ M3 ) )
=> ( ( member_vec_b @ A @ ( vec_set_vec_b @ ( col_vec_b @ M3 @ J2 ) ) )
=> ( member_vec_b @ A @ ( elements_mat_vec_b @ M3 ) ) ) ) ).
% vec_contains_col_elements_mat
thf(fact_84_vec__contains__col__elements__mat,axiom,
! [J2: nat,M3: mat_b,A: b] :
( ( ord_less_nat @ J2 @ ( dim_col_b @ M3 ) )
=> ( ( member_b @ A @ ( vec_set_b2 @ ( col_b @ M3 @ J2 ) ) )
=> ( member_b @ A @ ( elements_mat_b @ M3 ) ) ) ) ).
% vec_contains_col_elements_mat
thf(fact_85_insert__subset,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_86_insert__subset,axiom,
! [X: vec_b,A2: set_vec_b,B2: set_vec_b] :
( ( ord_le4862985661309304830_vec_b @ ( insert_vec_b @ X @ A2 ) @ B2 )
= ( ( member_vec_b @ X @ B2 )
& ( ord_le4862985661309304830_vec_b @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_87_insert__subset,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( ( member_b @ X @ B2 )
& ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_88_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_89_singletonI,axiom,
! [A: vec_b] : ( member_vec_b @ A @ ( insert_vec_b @ A @ bot_bot_set_vec_b ) ) ).
% singletonI
thf(fact_90_singletonI,axiom,
! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_91_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_92_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
! [A: vec_b,P: vec_b > $o] :
( ( member_vec_b @ A @ ( collect_vec_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_94_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_95_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_96_Collect__mem__eq,axiom,
! [A2: set_vec_b] :
( ( collect_vec_b
@ ^ [X3: vec_b] : ( member_vec_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_97_subset__empty,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_98_empty__subsetI,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% empty_subsetI
thf(fact_99_subset__singletonD,axiom,
! [A2: set_b,X: b] :
( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) )
=> ( ( A2 = bot_bot_set_b )
| ( A2
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ).
% subset_singletonD
thf(fact_100_subset__singleton__iff,axiom,
! [X4: set_b,A: b] :
( ( ord_less_eq_set_b @ X4 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( ( X4 = bot_bot_set_b )
| ( X4
= ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_101_zero__less__one__class_Ozero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_less_one
thf(fact_102_zero__less__one__class_Ozero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_less_one
thf(fact_103_zero__less__one__class_Ozero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_less_one
thf(fact_104_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X3: b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_105_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X3: b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_106_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_107_all__not__in__conv,axiom,
! [A2: set_vec_b] :
( ( ! [X3: vec_b] :
~ ( member_vec_b @ X3 @ A2 ) )
= ( A2 = bot_bot_set_vec_b ) ) ).
% all_not_in_conv
thf(fact_108_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X3: b] :
~ ( member_b @ X3 @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_109_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_110_empty__iff,axiom,
! [C: vec_b] :
~ ( member_vec_b @ C @ bot_bot_set_vec_b ) ).
% empty_iff
thf(fact_111_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_112_psubsetI,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_b @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_113_subset__antisym,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_114_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_115_subsetI,axiom,
! [A2: set_vec_b,B2: set_vec_b] :
( ! [X2: vec_b] :
( ( member_vec_b @ X2 @ A2 )
=> ( member_vec_b @ X2 @ B2 ) )
=> ( ord_le4862985661309304830_vec_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_116_subsetI,axiom,
! [A2: set_b,B2: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ X2 @ B2 ) )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_117_insert__absorb2,axiom,
! [X: b,A2: set_b] :
( ( insert_b @ X @ ( insert_b @ X @ A2 ) )
= ( insert_b @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_118_insert__iff,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
= ( ( A = B )
| ( member_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_119_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_120_insert__iff,axiom,
! [A: vec_b,B: vec_b,A2: set_vec_b] :
( ( member_vec_b @ A @ ( insert_vec_b @ B @ A2 ) )
= ( ( A = B )
| ( member_vec_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_121_insertCI,axiom,
! [A: b,B2: set_b,B: b] :
( ( ~ ( member_b @ A @ B2 )
=> ( A = B ) )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_122_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_123_insertCI,axiom,
! [A: vec_b,B2: set_vec_b,B: vec_b] :
( ( ~ ( member_vec_b @ A @ B2 )
=> ( A = B ) )
=> ( member_vec_b @ A @ ( insert_vec_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_124_not__psubset__empty,axiom,
! [A2: set_b] :
~ ( ord_less_set_b @ A2 @ bot_bot_set_b ) ).
% not_psubset_empty
thf(fact_125_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_set_b @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_126_subset__psubset__trans,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_set_b @ B2 @ C2 )
=> ( ord_less_set_b @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_127_subset__not__subset__eq,axiom,
( ord_less_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
& ~ ( ord_less_eq_set_b @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_128_psubset__subset__trans,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ord_less_set_b @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_129_psubset__imp__subset,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_130_psubset__eq,axiom,
( ord_less_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_131_psubsetE,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_132_transpose__mat__elems,axiom,
( elements_mat_b
= ( ^ [A3: mat_b] : ( elements_mat_b @ ( transpose_mat_b @ A3 ) ) ) ) ).
% transpose_mat_elems
thf(fact_133_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_134_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_135_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_136_ex__in__conv,axiom,
! [A2: set_vec_b] :
( ( ? [X3: vec_b] : ( member_vec_b @ X3 @ A2 ) )
= ( A2 != bot_bot_set_vec_b ) ) ).
% ex_in_conv
thf(fact_137_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X3: b] : ( member_b @ X3 @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_138_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_139_equals0I,axiom,
! [A2: set_vec_b] :
( ! [Y2: vec_b] :
~ ( member_vec_b @ Y2 @ A2 )
=> ( A2 = bot_bot_set_vec_b ) ) ).
% equals0I
thf(fact_140_equals0I,axiom,
! [A2: set_b] :
( ! [Y2: b] :
~ ( member_b @ Y2 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_141_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_142_equals0D,axiom,
! [A2: set_vec_b,A: vec_b] :
( ( A2 = bot_bot_set_vec_b )
=> ~ ( member_vec_b @ A @ A2 ) ) ).
% equals0D
thf(fact_143_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_144_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_145_emptyE,axiom,
! [A: vec_b] :
~ ( member_vec_b @ A @ bot_bot_set_vec_b ) ).
% emptyE
thf(fact_146_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_147_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X3: b] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_148_set__eq__subset,axiom,
( ( ^ [Y3: set_b,Z: set_b] : ( Y3 = Z ) )
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
& ( ord_less_eq_set_b @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_149_subset__trans,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ord_less_eq_set_b @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_150_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X2: b] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_151_subset__refl,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ A2 @ A2 ) ).
% subset_refl
thf(fact_152_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_153_subset__iff,axiom,
( ord_le4862985661309304830_vec_b
= ( ^ [A3: set_vec_b,B3: set_vec_b] :
! [T: vec_b] :
( ( member_vec_b @ T @ A3 )
=> ( member_vec_b @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_154_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
! [T: b] :
( ( member_b @ T @ A3 )
=> ( member_b @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_155_equalityD2,axiom,
! [A2: set_b,B2: set_b] :
( ( A2 = B2 )
=> ( ord_less_eq_set_b @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_156_equalityD1,axiom,
! [A2: set_b,B2: set_b] :
( ( A2 = B2 )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_157_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_158_subset__eq,axiom,
( ord_le4862985661309304830_vec_b
= ( ^ [A3: set_vec_b,B3: set_vec_b] :
! [X3: vec_b] :
( ( member_vec_b @ X3 @ A3 )
=> ( member_vec_b @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_159_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
! [X3: b] :
( ( member_b @ X3 @ A3 )
=> ( member_b @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_160_equalityE,axiom,
! [A2: set_b,B2: set_b] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ~ ( ord_less_eq_set_b @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_161_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_162_subsetD,axiom,
! [A2: set_vec_b,B2: set_vec_b,C: vec_b] :
( ( ord_le4862985661309304830_vec_b @ A2 @ B2 )
=> ( ( member_vec_b @ C @ A2 )
=> ( member_vec_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_163_subsetD,axiom,
! [A2: set_b,B2: set_b,C: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_164_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_165_in__mono,axiom,
! [A2: set_vec_b,B2: set_vec_b,X: vec_b] :
( ( ord_le4862985661309304830_vec_b @ A2 @ B2 )
=> ( ( member_vec_b @ X @ A2 )
=> ( member_vec_b @ X @ B2 ) ) ) ).
% in_mono
thf(fact_166_in__mono,axiom,
! [A2: set_b,B2: set_b,X: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ X @ A2 )
=> ( member_b @ X @ B2 ) ) ) ).
% in_mono
thf(fact_167_mk__disjoint__insert,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ? [B4: set_b] :
( ( A2
= ( insert_b @ A @ B4 ) )
& ~ ( member_b @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_168_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B4: set_nat] :
( ( A2
= ( insert_nat @ A @ B4 ) )
& ~ ( member_nat @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_169_mk__disjoint__insert,axiom,
! [A: vec_b,A2: set_vec_b] :
( ( member_vec_b @ A @ A2 )
=> ? [B4: set_vec_b] :
( ( A2
= ( insert_vec_b @ A @ B4 ) )
& ~ ( member_vec_b @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_170_insert__commute,axiom,
! [X: b,Y: b,A2: set_b] :
( ( insert_b @ X @ ( insert_b @ Y @ A2 ) )
= ( insert_b @ Y @ ( insert_b @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_171_insert__eq__iff,axiom,
! [A: b,A2: set_b,B: b,B2: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ~ ( member_b @ B @ B2 )
=> ( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_b] :
( ( A2
= ( insert_b @ B @ C3 ) )
& ~ ( member_b @ B @ C3 )
& ( B2
= ( insert_b @ A @ C3 ) )
& ~ ( member_b @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_172_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B2
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_173_insert__eq__iff,axiom,
! [A: vec_b,A2: set_vec_b,B: vec_b,B2: set_vec_b] :
( ~ ( member_vec_b @ A @ A2 )
=> ( ~ ( member_vec_b @ B @ B2 )
=> ( ( ( insert_vec_b @ A @ A2 )
= ( insert_vec_b @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_vec_b] :
( ( A2
= ( insert_vec_b @ B @ C3 ) )
& ~ ( member_vec_b @ B @ C3 )
& ( B2
= ( insert_vec_b @ A @ C3 ) )
& ~ ( member_vec_b @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_174_insert__absorb,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_175_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_176_insert__absorb,axiom,
! [A: vec_b,A2: set_vec_b] :
( ( member_vec_b @ A @ A2 )
=> ( ( insert_vec_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_177_insert__ident,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ~ ( member_b @ X @ B2 )
=> ( ( ( insert_b @ X @ A2 )
= ( insert_b @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_178_insert__ident,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_179_insert__ident,axiom,
! [X: vec_b,A2: set_vec_b,B2: set_vec_b] :
( ~ ( member_vec_b @ X @ A2 )
=> ( ~ ( member_vec_b @ X @ B2 )
=> ( ( ( insert_vec_b @ X @ A2 )
= ( insert_vec_b @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_180_Set_Oset__insert,axiom,
! [X: b,A2: set_b] :
( ( member_b @ X @ A2 )
=> ~ ! [B4: set_b] :
( ( A2
= ( insert_b @ X @ B4 ) )
=> ( member_b @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_181_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B4: set_nat] :
( ( A2
= ( insert_nat @ X @ B4 ) )
=> ( member_nat @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_182_Set_Oset__insert,axiom,
! [X: vec_b,A2: set_vec_b] :
( ( member_vec_b @ X @ A2 )
=> ~ ! [B4: set_vec_b] :
( ( A2
= ( insert_vec_b @ X @ B4 ) )
=> ( member_vec_b @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_183_insertI2,axiom,
! [A: b,B2: set_b,B: b] :
( ( member_b @ A @ B2 )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_184_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_185_insertI2,axiom,
! [A: vec_b,B2: set_vec_b,B: vec_b] :
( ( member_vec_b @ A @ B2 )
=> ( member_vec_b @ A @ ( insert_vec_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_186_insertI1,axiom,
! [A: b,B2: set_b] : ( member_b @ A @ ( insert_b @ A @ B2 ) ) ).
% insertI1
thf(fact_187_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_188_insertI1,axiom,
! [A: vec_b,B2: set_vec_b] : ( member_vec_b @ A @ ( insert_vec_b @ A @ B2 ) ) ).
% insertI1
thf(fact_189_insertE,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
=> ( ( A != B )
=> ( member_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_190_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_191_insertE,axiom,
! [A: vec_b,B: vec_b,A2: set_vec_b] :
( ( member_vec_b @ A @ ( insert_vec_b @ B @ A2 ) )
=> ( ( A != B )
=> ( member_vec_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_192_zero__neq__one,axiom,
zero_zero_b != one_one_b ).
% zero_neq_one
thf(fact_193_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_194_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_195_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_196_singleton__inject,axiom,
! [A: b,B: b] :
( ( ( insert_b @ A @ bot_bot_set_b )
= ( insert_b @ B @ bot_bot_set_b ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_197_insert__not__empty,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ A2 )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_198_doubleton__eq__iff,axiom,
! [A: b,B: b,C: b,D: b] :
( ( ( insert_b @ A @ ( insert_b @ B @ bot_bot_set_b ) )
= ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_199_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_200_singleton__iff,axiom,
! [B: vec_b,A: vec_b] :
( ( member_vec_b @ B @ ( insert_vec_b @ A @ bot_bot_set_vec_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_201_singleton__iff,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_202_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_203_singletonD,axiom,
! [B: vec_b,A: vec_b] :
( ( member_vec_b @ B @ ( insert_vec_b @ A @ bot_bot_set_vec_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_204_singletonD,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_205_subset__insertI2,axiom,
! [A2: set_b,B2: set_b,B: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_206_subset__insertI,axiom,
! [B2: set_b,A: b] : ( ord_less_eq_set_b @ B2 @ ( insert_b @ A @ B2 ) ) ).
% subset_insertI
thf(fact_207_subset__insert,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_208_subset__insert,axiom,
! [X: vec_b,A2: set_vec_b,B2: set_vec_b] :
( ~ ( member_vec_b @ X @ A2 )
=> ( ( ord_le4862985661309304830_vec_b @ A2 @ ( insert_vec_b @ X @ B2 ) )
= ( ord_le4862985661309304830_vec_b @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_209_subset__insert,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B2 ) )
= ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_210_insert__mono,axiom,
! [C2: set_b,D2: set_b,A: b] :
( ( ord_less_eq_set_b @ C2 @ D2 )
=> ( ord_less_eq_set_b @ ( insert_b @ A @ C2 ) @ ( insert_b @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_211_zero__one__matrix_Otranspose__entries,axiom,
! [Matrix: mat_nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ord_less_eq_set_nat @ ( elements_mat_nat @ ( transpose_mat_nat @ Matrix ) ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ) ).
% zero_one_matrix.transpose_entries
thf(fact_212_zero__one__matrix_Otranspose__entries,axiom,
! [Matrix: mat_int] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ord_less_eq_set_int @ ( elements_mat_int @ ( transpose_mat_int @ Matrix ) ) @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) ) ) ).
% zero_one_matrix.transpose_entries
thf(fact_213_zero__one__matrix_Otranspose__entries,axiom,
! [Matrix: mat_real] :
( ( incide4475037519619858106x_real @ Matrix )
=> ( ord_less_eq_set_real @ ( elements_mat_real @ ( transpose_mat_real @ Matrix ) ) @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) ) ) ).
% zero_one_matrix.transpose_entries
thf(fact_214_zero__one__matrix_Otranspose__entries,axiom,
! [Matrix: mat_b] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ord_less_eq_set_b @ ( elements_mat_b @ ( transpose_mat_b @ Matrix ) ) @ ( insert_b @ zero_zero_b @ ( insert_b @ one_one_b @ bot_bot_set_b ) ) ) ) ).
% zero_one_matrix.transpose_entries
thf(fact_215_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_216_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_217_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_218_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_219_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_220_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_221_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_222_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_223_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_224_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_225_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_226_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_227_Matrix_Otranspose__transpose,axiom,
! [A2: mat_b] :
( ( transpose_mat_b @ ( transpose_mat_b @ A2 ) )
= A2 ) ).
% Matrix.transpose_transpose
thf(fact_228_transpose__mat__eq,axiom,
! [A2: mat_b,B2: mat_b] :
( ( ( transpose_mat_b @ A2 )
= ( transpose_mat_b @ B2 ) )
= ( A2 = B2 ) ) ).
% transpose_mat_eq
thf(fact_229_the__elem__eq,axiom,
! [X: b] :
( ( the_elem_b @ ( insert_b @ X @ bot_bot_set_b ) )
= X ) ).
% the_elem_eq
thf(fact_230_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_231_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_232_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_233_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_234_dual__order_Orefl,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ A @ A ) ).
% dual_order.refl
thf(fact_235_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_236_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_237_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_238_order__refl,axiom,
! [X: set_b] : ( ord_less_eq_set_b @ X @ X ) ).
% order_refl
thf(fact_239_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_240_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_241_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_242_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_243_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_244_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_245_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_246_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_247_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_248_psubsetD,axiom,
! [A2: set_b,B2: set_b,C: b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_249_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_250_psubsetD,axiom,
! [A2: set_vec_b,B2: set_vec_b,C: vec_b] :
( ( ord_less_set_vec_b @ A2 @ B2 )
=> ( ( member_vec_b @ C @ A2 )
=> ( member_vec_b @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_251_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_252_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_253_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_254_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_255_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_256_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_257_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_258_le__cases3,axiom,
! [X: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_259_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_b,Z: set_b] : ( Y3 = Z ) )
= ( ^ [X3: set_b,Y4: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y4 )
& ( ord_less_eq_set_b @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_260_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_261_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_262_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_263_ord__eq__le__trans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( A = B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_264_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_265_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_266_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_267_ord__le__eq__trans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_b @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_268_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_269_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_270_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_271_order__antisym,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_272_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_273_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_274_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_275_order_Otrans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ A @ C ) ) ) ).
% order.trans
thf(fact_276_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_277_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_278_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_279_order__trans,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ Y @ Z2 )
=> ( ord_less_eq_set_b @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_280_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_281_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_282_order__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_283_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: nat,B5: nat] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_284_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: int,B5: int] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_285_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B5: real] :
( ( ord_less_eq_real @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: real,B5: real] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_286_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_b,Z: set_b] : ( Y3 = Z ) )
= ( ^ [A5: set_b,B6: set_b] :
( ( ord_less_eq_set_b @ B6 @ A5 )
& ( ord_less_eq_set_b @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_287_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ( ord_less_eq_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_288_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A5: int,B6: int] :
( ( ord_less_eq_int @ B6 @ A5 )
& ( ord_less_eq_int @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_289_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A5: real,B6: real] :
( ( ord_less_eq_real @ B6 @ A5 )
& ( ord_less_eq_real @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_290_dual__order_Oantisym,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( ord_less_eq_set_b @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_291_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_292_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_293_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_294_dual__order_Otrans,axiom,
! [B: set_b,A: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( ord_less_eq_set_b @ C @ B )
=> ( ord_less_eq_set_b @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_295_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_296_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_297_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_298_antisym,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_299_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_300_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_301_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_302_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_b,Z: set_b] : ( Y3 = Z ) )
= ( ^ [A5: set_b,B6: set_b] :
( ( ord_less_eq_set_b @ A5 @ B6 )
& ( ord_less_eq_set_b @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_303_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ( ord_less_eq_nat @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_304_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A5: int,B6: int] :
( ( ord_less_eq_int @ A5 @ B6 )
& ( ord_less_eq_int @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_305_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A5: real,B6: real] :
( ( ord_less_eq_real @ A5 @ B6 )
& ( ord_less_eq_real @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_306_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_307_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_308_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_309_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_310_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_311_order__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_312_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_313_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_314_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_315_order__subst1,axiom,
! [A: set_b,F: nat > set_b,B: nat,C: nat] :
( ( ord_less_eq_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_316_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_317_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_318_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_319_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_320_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_321_order__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_322_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_323_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_324_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_325_order__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > nat,C: nat] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_326_order__eq__refl,axiom,
! [X: set_b,Y: set_b] :
( ( X = Y )
=> ( ord_less_eq_set_b @ X @ Y ) ) ).
% order_eq_refl
thf(fact_327_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_328_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_329_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_330_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_331_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_332_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_333_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_334_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_335_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_336_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_337_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_338_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_339_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_340_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_341_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_342_ord__eq__le__subst,axiom,
! [A: nat,F: set_b > nat,B: set_b,C: set_b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X2: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_343_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_344_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_345_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_346_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_347_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_348_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_349_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_350_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_351_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_352_ord__le__eq__subst,axiom,
! [A: set_b,B: set_b,F: set_b > nat,C: nat] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_353_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_354_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_355_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_356_order__antisym__conv,axiom,
! [Y: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ( ( ord_less_eq_set_b @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_357_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_358_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_359_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_360_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_361_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_362_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_363_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_364_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_365_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_366_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_367_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_368_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_369_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_370_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_371_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_372_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_373_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_374_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_375_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_376_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_377_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_378_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_379_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_380_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_381_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_382_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_383_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_384_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_385_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_386_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_387_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_388_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_389_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_390_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_391_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_392_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_nat @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B5: nat] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_393_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B5: int] :
( ( ord_less_int @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B5: int] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_394_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B5: real] :
( ( ord_less_real @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B5: real] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_395_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_396_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_397_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_398_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_399_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_400_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_401_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_402_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_403_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_404_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_405_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_406_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_407_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_408_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_409_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_410_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_411_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_412_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_413_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_414_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_415_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_416_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_417_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_418_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_419_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_420_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_421_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_422_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_423_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_424_order__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_425_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_426_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_427_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_428_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_429_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_430_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_431_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_432_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_433_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_434_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_435_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_436_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_437_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_438_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_439_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_440_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_441_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_442_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_443_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_444_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_445_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_446_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_447_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_448_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_449_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_450_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_451_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_452_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_453_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_454_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_455_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_456_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_457_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_458_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_459_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_460_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_461_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_462_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_463_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_464_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_465_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_466_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_467_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_468_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_469_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_470_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_471_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_472_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_473_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_474_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_475_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_476_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_477_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_478_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_479_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_480_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_481_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_482_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_483_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_484_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_485_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_486_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_487_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_488_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_489_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_490_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_491_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_492_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_493_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_494_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_495_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_496_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_497_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_498_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ~ ( P @ I2 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_499_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_500_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_501_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_502_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_503_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_504_leD,axiom,
! [Y: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ~ ( ord_less_set_b @ X @ Y ) ) ).
% leD
thf(fact_505_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_506_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_507_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_508_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_509_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_510_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_511_nless__le,axiom,
! [A: set_b,B: set_b] :
( ( ~ ( ord_less_set_b @ A @ B ) )
= ( ~ ( ord_less_eq_set_b @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_512_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_513_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_514_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_515_antisym__conv1,axiom,
! [X: set_b,Y: set_b] :
( ~ ( ord_less_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_516_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_517_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_518_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_519_antisym__conv2,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ~ ( ord_less_set_b @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_520_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_521_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_522_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_523_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_524_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_525_less__le__not__le,axiom,
( ord_less_set_b
= ( ^ [X3: set_b,Y4: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y4 )
& ~ ( ord_less_eq_set_b @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_526_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_527_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_528_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_529_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_530_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_531_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_532_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_b
= ( ^ [A5: set_b,B6: set_b] :
( ( ord_less_set_b @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_533_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_nat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_534_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B6: int] :
( ( ord_less_int @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_535_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B6: real] :
( ( ord_less_real @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_536_order_Ostrict__iff__order,axiom,
( ord_less_set_b
= ( ^ [A5: set_b,B6: set_b] :
( ( ord_less_eq_set_b @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_537_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_538_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B6: int] :
( ( ord_less_eq_int @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_539_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B6: real] :
( ( ord_less_eq_real @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_540_order_Ostrict__trans1,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_set_b @ B @ C )
=> ( ord_less_set_b @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_541_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_542_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_543_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_544_order_Ostrict__trans2,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_set_b @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_545_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_546_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_547_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_548_order_Ostrict__iff__not,axiom,
( ord_less_set_b
= ( ^ [A5: set_b,B6: set_b] :
( ( ord_less_eq_set_b @ A5 @ B6 )
& ~ ( ord_less_eq_set_b @ B6 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_549_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ~ ( ord_less_eq_nat @ B6 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_550_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B6: int] :
( ( ord_less_eq_int @ A5 @ B6 )
& ~ ( ord_less_eq_int @ B6 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_551_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B6: real] :
( ( ord_less_eq_real @ A5 @ B6 )
& ~ ( ord_less_eq_real @ B6 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_552_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_553_dense__le__bounded,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_554_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_b
= ( ^ [B6: set_b,A5: set_b] :
( ( ord_less_set_b @ B6 @ A5 )
| ( A5 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_555_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_nat @ B6 @ A5 )
| ( A5 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_556_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B6: int,A5: int] :
( ( ord_less_int @ B6 @ A5 )
| ( A5 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_557_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B6: real,A5: real] :
( ( ord_less_real @ B6 @ A5 )
| ( A5 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_558_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_b
= ( ^ [B6: set_b,A5: set_b] :
( ( ord_less_eq_set_b @ B6 @ A5 )
& ( A5 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_559_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ( A5 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_560_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B6: int,A5: int] :
( ( ord_less_eq_int @ B6 @ A5 )
& ( A5 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_561_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B6: real,A5: real] :
( ( ord_less_eq_real @ B6 @ A5 )
& ( A5 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_562_dual__order_Ostrict__trans1,axiom,
! [B: set_b,A: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( ord_less_set_b @ C @ B )
=> ( ord_less_set_b @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_563_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_564_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_565_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_566_dual__order_Ostrict__trans2,axiom,
! [B: set_b,A: set_b,C: set_b] :
( ( ord_less_set_b @ B @ A )
=> ( ( ord_less_eq_set_b @ C @ B )
=> ( ord_less_set_b @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_567_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_568_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_569_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_570_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_b
= ( ^ [B6: set_b,A5: set_b] :
( ( ord_less_eq_set_b @ B6 @ A5 )
& ~ ( ord_less_eq_set_b @ A5 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_571_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_572_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B6: int,A5: int] :
( ( ord_less_eq_int @ B6 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_573_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B6: real,A5: real] :
( ( ord_less_eq_real @ B6 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_574_order_Ostrict__implies__order,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_set_b @ A @ B )
=> ( ord_less_eq_set_b @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_575_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_576_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_577_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_578_dual__order_Ostrict__implies__order,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_set_b @ B @ A )
=> ( ord_less_eq_set_b @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_579_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_580_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_581_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_582_order__le__less,axiom,
( ord_less_eq_set_b
= ( ^ [X3: set_b,Y4: set_b] :
( ( ord_less_set_b @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_583_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_584_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_585_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_586_order__less__le,axiom,
( ord_less_set_b
= ( ^ [X3: set_b,Y4: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_587_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_588_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_589_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_590_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_591_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_592_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_593_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_594_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_595_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_596_order__less__imp__le,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_set_b @ X @ Y )
=> ( ord_less_eq_set_b @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_597_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_598_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_599_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_600_order__le__neq__trans,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_b @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_601_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_602_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_603_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_604_order__neq__le__trans,axiom,
! [A: set_b,B: set_b] :
( ( A != B )
=> ( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_set_b @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_605_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_606_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_607_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_608_order__le__less__trans,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_set_b @ Y @ Z2 )
=> ( ord_less_set_b @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_609_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_610_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_611_order__le__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_612_order__less__le__trans,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( ord_less_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ Y @ Z2 )
=> ( ord_less_set_b @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_613_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_614_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_615_order__less__le__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_616_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_617_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_618_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_619_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_620_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_621_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_622_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_623_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_624_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_625_order__le__less__subst1,axiom,
! [A: set_b,F: nat > set_b,B: nat,C: nat] :
( ( ord_less_eq_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_b @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_626_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_627_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_628_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_629_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_630_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_631_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_632_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_633_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_634_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_635_order__le__less__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > nat,C: nat] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_636_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_637_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_638_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_639_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_640_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_641_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_642_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_643_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_644_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_645_order__less__le__subst1,axiom,
! [A: nat,F: set_b > nat,B: set_b,C: set_b] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X2: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_646_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_647_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_648_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_649_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_650_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_651_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_652_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_653_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_654_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_655_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_b,C: set_b] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_b @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_656_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_657_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_658_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_659_order__le__imp__less__or__eq,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_set_b @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_660_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_661_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_662_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_663_bot_Oextremum,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% bot.extremum
thf(fact_664_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_665_bot_Oextremum__unique,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% bot.extremum_unique
thf(fact_666_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_667_bot_Oextremum__uniqueI,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
=> ( A = bot_bot_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_668_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_669_bot_Oextremum__strict,axiom,
! [A: set_b] :
~ ( ord_less_set_b @ A @ bot_bot_set_b ) ).
% bot.extremum_strict
thf(fact_670_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_671_bot_Onot__eq__extremum,axiom,
! [A: set_b] :
( ( A != bot_bot_set_b )
= ( ord_less_set_b @ bot_bot_set_b @ A ) ) ).
% bot.not_eq_extremum
thf(fact_672_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_673_is__singleton__the__elem,axiom,
( is_singleton_b
= ( ^ [A3: set_b] :
( A3
= ( insert_b @ ( the_elem_b @ A3 ) @ bot_bot_set_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_674_row__elems__ss01,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( dim_row_b @ matrix ) )
=> ( ord_less_eq_set_b @ ( vec_set_b2 @ ( row_b @ matrix @ I ) ) @ ( insert_b @ zero_zero_b @ ( insert_b @ one_one_b @ bot_bot_set_b ) ) ) ) ).
% row_elems_ss01
thf(fact_675_is__singletonI,axiom,
! [X: b] : ( is_singleton_b @ ( insert_b @ X @ bot_bot_set_b ) ) ).
% is_singletonI
thf(fact_676_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: nat > b,X: nat] :
( ( matrix8711691017198569721_nat_b @ Hom )
=> ( ( member_nat @ X @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( Hom @ X )
= zero_zero_b )
= ( X = zero_zero_nat ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_677_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: nat > nat,X: nat] :
( ( matrix3195545623030104918at_nat @ Hom )
=> ( ( member_nat @ X @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( Hom @ X )
= zero_zero_nat )
= ( X = zero_zero_nat ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_678_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: nat > int,X: nat] :
( ( matrix3193055152521054642at_int @ Hom )
=> ( ( member_nat @ X @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( Hom @ X )
= zero_zero_int )
= ( X = zero_zero_nat ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_679_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: nat > real,X: nat] :
( ( matrix6800728457931064114t_real @ Hom )
=> ( ( member_nat @ X @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( Hom @ X )
= zero_zero_real )
= ( X = zero_zero_nat ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_680_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: int > b,X: int] :
( ( matrix6466546049528123805_int_b @ Hom )
=> ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) )
=> ( ( ( Hom @ X )
= zero_zero_b )
= ( X = zero_zero_int ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_681_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: int > nat,X: int] :
( ( matrix4194899334931159474nt_nat @ Hom )
=> ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) )
=> ( ( ( Hom @ X )
= zero_zero_nat )
= ( X = zero_zero_int ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_682_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: int > int,X: int] :
( ( matrix4192408864422109198nt_int @ Hom )
=> ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) )
=> ( ( ( Hom @ X )
= zero_zero_int )
= ( X = zero_zero_int ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_683_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: int > real,X: int] :
( ( matrix8987650032624085902t_real @ Hom )
=> ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) )
=> ( ( ( Hom @ X )
= zero_zero_real )
= ( X = zero_zero_int ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_684_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: real > b,X: real] :
( ( matrix2454726049922035613real_b @ Hom )
=> ( ( member_real @ X @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) )
=> ( ( ( Hom @ X )
= zero_zero_b )
= ( X = zero_zero_real ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_685_inj__on__01__hom_Oinj__0__iff,axiom,
! [Hom: real > nat,X: real] :
( ( matrix2144664994205186994al_nat @ Hom )
=> ( ( member_real @ X @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) )
=> ( ( ( Hom @ X )
= zero_zero_nat )
= ( X = zero_zero_real ) ) ) ) ).
% inj_on_01_hom.inj_0_iff
thf(fact_686_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: nat > b,X: nat] :
( ( matrix8711691017198569721_nat_b @ Hom )
=> ( ( member_nat @ X @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( Hom @ X )
= one_one_b )
= ( X = one_one_nat ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_687_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: nat > nat,X: nat] :
( ( matrix3195545623030104918at_nat @ Hom )
=> ( ( member_nat @ X @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( Hom @ X )
= one_one_nat )
= ( X = one_one_nat ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_688_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: nat > int,X: nat] :
( ( matrix3193055152521054642at_int @ Hom )
=> ( ( member_nat @ X @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
=> ( ( ( Hom @ X )
= one_one_int )
= ( X = one_one_nat ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_689_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: int > b,X: int] :
( ( matrix6466546049528123805_int_b @ Hom )
=> ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) )
=> ( ( ( Hom @ X )
= one_one_b )
= ( X = one_one_int ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_690_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: int > nat,X: int] :
( ( matrix4194899334931159474nt_nat @ Hom )
=> ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) )
=> ( ( ( Hom @ X )
= one_one_nat )
= ( X = one_one_int ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_691_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: int > int,X: int] :
( ( matrix4192408864422109198nt_int @ Hom )
=> ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) )
=> ( ( ( Hom @ X )
= one_one_int )
= ( X = one_one_int ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_692_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: real > b,X: real] :
( ( matrix2454726049922035613real_b @ Hom )
=> ( ( member_real @ X @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) )
=> ( ( ( Hom @ X )
= one_one_b )
= ( X = one_one_real ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_693_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: real > nat,X: real] :
( ( matrix2144664994205186994al_nat @ Hom )
=> ( ( member_real @ X @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) )
=> ( ( ( Hom @ X )
= one_one_nat )
= ( X = one_one_real ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_694_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: real > int,X: real] :
( ( matrix2142174523696136718al_int @ Hom )
=> ( ( member_real @ X @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) )
=> ( ( ( Hom @ X )
= one_one_int )
= ( X = one_one_real ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_695_inj__on__01__hom_Oinj__1__iff,axiom,
! [Hom: b > b,X: b] :
( ( matrix6790486239265177588om_b_b @ Hom )
=> ( ( member_b @ X @ ( insert_b @ zero_zero_b @ ( insert_b @ one_one_b @ bot_bot_set_b ) ) )
=> ( ( ( Hom @ X )
= one_one_b )
= ( X = one_one_b ) ) ) ) ).
% inj_on_01_hom.inj_1_iff
thf(fact_696_insert__subsetI,axiom,
! [X: nat,A2: set_nat,X4: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_697_insert__subsetI,axiom,
! [X: vec_b,A2: set_vec_b,X4: set_vec_b] :
( ( member_vec_b @ X @ A2 )
=> ( ( ord_le4862985661309304830_vec_b @ X4 @ A2 )
=> ( ord_le4862985661309304830_vec_b @ ( insert_vec_b @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_698_insert__subsetI,axiom,
! [X: b,A2: set_b,X4: set_b] :
( ( member_b @ X @ A2 )
=> ( ( ord_less_eq_set_b @ X4 @ A2 )
=> ( ord_less_eq_set_b @ ( insert_b @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_699_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_700_subset__emptyI,axiom,
! [A2: set_vec_b] :
( ! [X2: vec_b] :
~ ( member_vec_b @ X2 @ A2 )
=> ( ord_le4862985661309304830_vec_b @ A2 @ bot_bot_set_vec_b ) ) ).
% subset_emptyI
thf(fact_701_subset__emptyI,axiom,
! [A2: set_b] :
( ! [X2: b] :
~ ( member_b @ X2 @ A2 )
=> ( ord_less_eq_set_b @ A2 @ bot_bot_set_b ) ) ).
% subset_emptyI
thf(fact_702_minf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T2 @ X6 ) ) ).
% minf(8)
thf(fact_703_minf_I8_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_eq_int @ T2 @ X6 ) ) ).
% minf(8)
thf(fact_704_minf_I8_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ~ ( ord_less_eq_real @ T2 @ X6 ) ) ).
% minf(8)
thf(fact_705_index__transpose__mat_I3_J,axiom,
! [A2: mat_b] :
( ( dim_col_b @ ( transpose_mat_b @ A2 ) )
= ( dim_row_b @ A2 ) ) ).
% index_transpose_mat(3)
thf(fact_706_index__transpose__mat_I2_J,axiom,
! [A2: mat_b] :
( ( dim_row_b @ ( transpose_mat_b @ A2 ) )
= ( dim_col_b @ A2 ) ) ).
% index_transpose_mat(2)
thf(fact_707_col__transpose,axiom,
! [I: nat,A2: mat_b] :
( ( ord_less_nat @ I @ ( dim_row_b @ A2 ) )
=> ( ( col_b @ ( transpose_mat_b @ A2 ) @ I )
= ( row_b @ A2 @ I ) ) ) ).
% col_transpose
thf(fact_708_row__transpose,axiom,
! [J2: nat,A2: mat_b] :
( ( ord_less_nat @ J2 @ ( dim_col_b @ A2 ) )
=> ( ( row_b @ ( transpose_mat_b @ A2 ) @ J2 )
= ( col_b @ A2 @ J2 ) ) ) ).
% row_transpose
thf(fact_709_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_710_le__trans,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K3 )
=> ( ord_less_eq_nat @ I @ K3 ) ) ) ).
% le_trans
thf(fact_711_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_712_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_713_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_714_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K3: nat,B: nat] :
( ( P @ K3 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_715_eq__rowI,axiom,
! [B2: mat_b,A2: mat_b] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_b @ B2 ) )
=> ( ( row_b @ A2 @ I3 )
= ( row_b @ B2 @ I3 ) ) )
=> ( ( ( dim_row_b @ A2 )
= ( dim_row_b @ B2 ) )
=> ( ( ( dim_col_b @ A2 )
= ( dim_col_b @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% eq_rowI
thf(fact_716_vec__contains__row__elements__mat,axiom,
! [I: nat,M3: mat_nat,A: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ M3 ) )
=> ( ( member_nat @ A @ ( vec_set_nat @ ( row_nat @ M3 @ I ) ) )
=> ( member_nat @ A @ ( elements_mat_nat @ M3 ) ) ) ) ).
% vec_contains_row_elements_mat
thf(fact_717_vec__contains__row__elements__mat,axiom,
! [I: nat,M3: mat_vec_b,A: vec_b] :
( ( ord_less_nat @ I @ ( dim_row_vec_b @ M3 ) )
=> ( ( member_vec_b @ A @ ( vec_set_vec_b @ ( row_vec_b @ M3 @ I ) ) )
=> ( member_vec_b @ A @ ( elements_mat_vec_b @ M3 ) ) ) ) ).
% vec_contains_row_elements_mat
thf(fact_718_vec__contains__row__elements__mat,axiom,
! [I: nat,M3: mat_b,A: b] :
( ( ord_less_nat @ I @ ( dim_row_b @ M3 ) )
=> ( ( member_b @ A @ ( vec_set_b2 @ ( row_b @ M3 @ I ) ) )
=> ( member_b @ A @ ( elements_mat_b @ M3 ) ) ) ) ).
% vec_contains_row_elements_mat
thf(fact_719_row__elems__subset__mat,axiom,
! [I: nat,N2: mat_b] :
( ( ord_less_nat @ I @ ( dim_row_b @ N2 ) )
=> ( ord_less_eq_set_b @ ( vec_set_b2 @ ( row_b @ N2 @ I ) ) @ ( elements_mat_b @ N2 ) ) ) ).
% row_elems_subset_mat
thf(fact_720_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_721_is__singletonI_H,axiom,
! [A2: set_vec_b] :
( ( A2 != bot_bot_set_vec_b )
=> ( ! [X2: vec_b,Y2: vec_b] :
( ( member_vec_b @ X2 @ A2 )
=> ( ( member_vec_b @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_vec_b @ A2 ) ) ) ).
% is_singletonI'
thf(fact_722_is__singletonI_H,axiom,
! [A2: set_b] :
( ( A2 != bot_bot_set_b )
=> ( ! [X2: b,Y2: b] :
( ( member_b @ X2 @ A2 )
=> ( ( member_b @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_b @ A2 ) ) ) ).
% is_singletonI'
thf(fact_723_mat__col__eqI,axiom,
! [B2: mat_b,A2: mat_b] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_col_b @ B2 ) )
=> ( ( col_b @ A2 @ I3 )
= ( col_b @ B2 @ I3 ) ) )
=> ( ( ( dim_row_b @ A2 )
= ( dim_row_b @ B2 ) )
=> ( ( ( dim_col_b @ A2 )
= ( dim_col_b @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% mat_col_eqI
thf(fact_724_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_725_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_726_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_727_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_728_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_729_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_730_pinf_I3_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T2 ) ) ).
% pinf(3)
thf(fact_731_pinf_I3_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( X6 != T2 ) ) ).
% pinf(3)
thf(fact_732_pinf_I3_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( X6 != T2 ) ) ).
% pinf(3)
thf(fact_733_pinf_I4_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T2 ) ) ).
% pinf(4)
thf(fact_734_pinf_I4_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( X6 != T2 ) ) ).
% pinf(4)
thf(fact_735_pinf_I4_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( X6 != T2 ) ) ).
% pinf(4)
thf(fact_736_pinf_I5_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T2 ) ) ).
% pinf(5)
thf(fact_737_pinf_I5_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_int @ X6 @ T2 ) ) ).
% pinf(5)
thf(fact_738_pinf_I5_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ~ ( ord_less_real @ X6 @ T2 ) ) ).
% pinf(5)
thf(fact_739_pinf_I7_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_nat @ T2 @ X6 ) ) ).
% pinf(7)
thf(fact_740_pinf_I7_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_int @ T2 @ X6 ) ) ).
% pinf(7)
thf(fact_741_pinf_I7_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ord_less_real @ T2 @ X6 ) ) ).
% pinf(7)
thf(fact_742_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_743_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_744_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_745_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_746_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_747_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_748_minf_I3_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T2 ) ) ).
% minf(3)
thf(fact_749_minf_I3_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( X6 != T2 ) ) ).
% minf(3)
thf(fact_750_minf_I3_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( X6 != T2 ) ) ).
% minf(3)
thf(fact_751_minf_I4_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T2 ) ) ).
% minf(4)
thf(fact_752_minf_I4_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( X6 != T2 ) ) ).
% minf(4)
thf(fact_753_minf_I4_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( X6 != T2 ) ) ).
% minf(4)
thf(fact_754_minf_I5_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_nat @ X6 @ T2 ) ) ).
% minf(5)
thf(fact_755_minf_I5_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_int @ X6 @ T2 ) ) ).
% minf(5)
thf(fact_756_minf_I5_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ord_less_real @ X6 @ T2 ) ) ).
% minf(5)
thf(fact_757_minf_I7_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_nat @ T2 @ X6 ) ) ).
% minf(7)
thf(fact_758_minf_I7_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_int @ T2 @ X6 ) ) ).
% minf(7)
thf(fact_759_minf_I7_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ~ ( ord_less_real @ T2 @ X6 ) ) ).
% minf(7)
thf(fact_760_zero__one__matrix_Orow__elems__ss01,axiom,
! [Matrix: mat_nat,I: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_row_nat @ Matrix ) )
=> ( ord_less_eq_set_nat @ ( vec_set_nat @ ( row_nat @ Matrix @ I ) ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ) ) ).
% zero_one_matrix.row_elems_ss01
thf(fact_761_zero__one__matrix_Orow__elems__ss01,axiom,
! [Matrix: mat_int,I: nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ Matrix ) )
=> ( ord_less_eq_set_int @ ( vec_set_int @ ( row_int @ Matrix @ I ) ) @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) ) ) ) ).
% zero_one_matrix.row_elems_ss01
thf(fact_762_zero__one__matrix_Orow__elems__ss01,axiom,
! [Matrix: mat_real,I: nat] :
( ( incide4475037519619858106x_real @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_row_real @ Matrix ) )
=> ( ord_less_eq_set_real @ ( vec_set_real @ ( row_real @ Matrix @ I ) ) @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) ) ) ) ).
% zero_one_matrix.row_elems_ss01
thf(fact_763_zero__one__matrix_Orow__elems__ss01,axiom,
! [Matrix: mat_b,I: nat] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_row_b @ Matrix ) )
=> ( ord_less_eq_set_b @ ( vec_set_b2 @ ( row_b @ Matrix @ I ) ) @ ( insert_b @ zero_zero_b @ ( insert_b @ one_one_b @ bot_bot_set_b ) ) ) ) ) ).
% zero_one_matrix.row_elems_ss01
thf(fact_764_is__singleton__def,axiom,
( is_singleton_b
= ( ^ [A3: set_b] :
? [X3: b] :
( A3
= ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ).
% is_singleton_def
thf(fact_765_is__singletonE,axiom,
! [A2: set_b] :
( ( is_singleton_b @ A2 )
=> ~ ! [X2: b] :
( A2
!= ( insert_b @ X2 @ bot_bot_set_b ) ) ) ).
% is_singletonE
thf(fact_766_pinf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T2 ) ) ).
% pinf(6)
thf(fact_767_pinf_I6_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T2 ) ) ).
% pinf(6)
thf(fact_768_pinf_I6_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ~ ( ord_less_eq_real @ X6 @ T2 ) ) ).
% pinf(6)
thf(fact_769_pinf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T2 @ X6 ) ) ).
% pinf(8)
thf(fact_770_pinf_I8_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_eq_int @ T2 @ X6 ) ) ).
% pinf(8)
thf(fact_771_pinf_I8_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ord_less_eq_real @ T2 @ X6 ) ) ).
% pinf(8)
thf(fact_772_minf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T2 ) ) ).
% minf(6)
thf(fact_773_minf_I6_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_eq_int @ X6 @ T2 ) ) ).
% minf(6)
thf(fact_774_minf_I6_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ord_less_eq_real @ X6 @ T2 ) ) ).
% minf(6)
thf(fact_775_in__map__col__valid__index__M,axiom,
! [J2: nat,I: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_b @ matrix ) )
=> ( ( member_nat @ I @ ( incide5355957740755015149lock_b @ ( col_b @ matrix @ J2 ) ) )
=> ( ord_less_nat @ I @ ( dim_row_b @ matrix ) ) ) ) ).
% in_map_col_valid_index_M
thf(fact_776_row__nth__0__or__1__iff,axiom,
! [J2: nat,I: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_b @ matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_b @ matrix ) )
=> ( ( ( vec_index_b @ ( row_b @ matrix @ I ) @ J2 )
= zero_zero_b )
= ( ( vec_index_b @ ( row_b @ matrix @ I ) @ J2 )
!= one_one_b ) ) ) ) ).
% row_nth_0_or_1_iff
thf(fact_777_col__nth__0__or__1__iff,axiom,
! [J2: nat,I: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_b @ matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_b @ matrix ) )
=> ( ( ( vec_index_b @ ( col_b @ matrix @ J2 ) @ I )
= zero_zero_b )
= ( ( vec_index_b @ ( col_b @ matrix @ J2 ) @ I )
!= one_one_b ) ) ) ) ).
% col_nth_0_or_1_iff
thf(fact_778_proper__inc__mat__def,axiom,
( incide2997380824311827482_mat_b
= ( ^ [M: mat_b] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_row_b @ M ) )
& ( ord_less_nat @ zero_zero_nat @ ( dim_col_b @ M ) ) ) ) ) ).
% proper_inc_mat_def
thf(fact_779_zero__one__matrix_Oin__map__col__valid__index__M,axiom,
! [Matrix: mat_b,J2: nat,I: nat] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_b @ Matrix ) )
=> ( ( member_nat @ I @ ( incide5355957740755015149lock_b @ ( col_b @ Matrix @ J2 ) ) )
=> ( ord_less_nat @ I @ ( dim_row_b @ Matrix ) ) ) ) ) ).
% zero_one_matrix.in_map_col_valid_index_M
thf(fact_780_Collect__empty__eq__bot,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( P = bot_bot_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_781_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_782_bot__empty__eq,axiom,
( bot_bot_vec_b_o
= ( ^ [X3: vec_b] : ( member_vec_b @ X3 @ bot_bot_set_vec_b ) ) ) ).
% bot_empty_eq
thf(fact_783_bot__empty__eq,axiom,
( bot_bot_b_o
= ( ^ [X3: b] : ( member_b @ X3 @ bot_bot_set_b ) ) ) ).
% bot_empty_eq
thf(fact_784_non__empty__col__obtains,axiom,
! [M3: mat_nat,J2: nat] :
( ( incide6854414339478298687ol_nat @ M3 @ J2 )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ M3 ) )
=> ( ( vec_index_nat @ ( col_nat @ M3 @ J2 ) @ I3 )
= zero_zero_nat ) ) ) ).
% non_empty_col_obtains
thf(fact_785_non__empty__col__obtains,axiom,
! [M3: mat_int,J2: nat] :
( ( incide6851923868969248411ol_int @ M3 @ J2 )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_int @ M3 ) )
=> ( ( vec_index_int @ ( col_int @ M3 @ J2 ) @ I3 )
= zero_zero_int ) ) ) ).
% non_empty_col_obtains
thf(fact_786_non__empty__col__obtains,axiom,
! [M3: mat_real,J2: nat] :
( ( incide8049862060206209947l_real @ M3 @ J2 )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ M3 ) )
=> ( ( vec_index_real @ ( col_real @ M3 @ J2 ) @ I3 )
= zero_zero_real ) ) ) ).
% non_empty_col_obtains
thf(fact_787_non__empty__col__obtains,axiom,
! [M3: mat_b,J2: nat] :
( ( incide3034858701194040400_col_b @ M3 @ J2 )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_b @ M3 ) )
=> ( ( vec_index_b @ ( col_b @ M3 @ J2 ) @ I3 )
= zero_zero_b ) ) ) ).
% non_empty_col_obtains
thf(fact_788_zero__one__matrix_Orow__nth__0__or__1__iff,axiom,
! [Matrix: mat_nat,J2: nat,I: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ Matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_nat @ Matrix ) )
=> ( ( ( vec_index_nat @ ( row_nat @ Matrix @ I ) @ J2 )
= zero_zero_nat )
= ( ( vec_index_nat @ ( row_nat @ Matrix @ I ) @ J2 )
!= one_one_nat ) ) ) ) ) ).
% zero_one_matrix.row_nth_0_or_1_iff
thf(fact_789_zero__one__matrix_Orow__nth__0__or__1__iff,axiom,
! [Matrix: mat_int,J2: nat,I: nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_int @ Matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ Matrix ) )
=> ( ( ( vec_index_int @ ( row_int @ Matrix @ I ) @ J2 )
= zero_zero_int )
= ( ( vec_index_int @ ( row_int @ Matrix @ I ) @ J2 )
!= one_one_int ) ) ) ) ) ).
% zero_one_matrix.row_nth_0_or_1_iff
thf(fact_790_zero__one__matrix_Orow__nth__0__or__1__iff,axiom,
! [Matrix: mat_real,J2: nat,I: nat] :
( ( incide4475037519619858106x_real @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_real @ Matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_real @ Matrix ) )
=> ( ( ( vec_index_real @ ( row_real @ Matrix @ I ) @ J2 )
= zero_zero_real )
= ( ( vec_index_real @ ( row_real @ Matrix @ I ) @ J2 )
!= one_one_real ) ) ) ) ) ).
% zero_one_matrix.row_nth_0_or_1_iff
thf(fact_791_zero__one__matrix_Orow__nth__0__or__1__iff,axiom,
! [Matrix: mat_b,J2: nat,I: nat] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_b @ Matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_b @ Matrix ) )
=> ( ( ( vec_index_b @ ( row_b @ Matrix @ I ) @ J2 )
= zero_zero_b )
= ( ( vec_index_b @ ( row_b @ Matrix @ I ) @ J2 )
!= one_one_b ) ) ) ) ) ).
% zero_one_matrix.row_nth_0_or_1_iff
thf(fact_792_zero__one__matrix_Ocol__nth__0__or__1__iff,axiom,
! [Matrix: mat_nat,J2: nat,I: nat] :
( ( incide4966654671090901726ix_nat @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ Matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_nat @ Matrix ) )
=> ( ( ( vec_index_nat @ ( col_nat @ Matrix @ J2 ) @ I )
= zero_zero_nat )
= ( ( vec_index_nat @ ( col_nat @ Matrix @ J2 ) @ I )
!= one_one_nat ) ) ) ) ) ).
% zero_one_matrix.col_nth_0_or_1_iff
thf(fact_793_zero__one__matrix_Ocol__nth__0__or__1__iff,axiom,
! [Matrix: mat_int,J2: nat,I: nat] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_int @ Matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ Matrix ) )
=> ( ( ( vec_index_int @ ( col_int @ Matrix @ J2 ) @ I )
= zero_zero_int )
= ( ( vec_index_int @ ( col_int @ Matrix @ J2 ) @ I )
!= one_one_int ) ) ) ) ) ).
% zero_one_matrix.col_nth_0_or_1_iff
thf(fact_794_zero__one__matrix_Ocol__nth__0__or__1__iff,axiom,
! [Matrix: mat_real,J2: nat,I: nat] :
( ( incide4475037519619858106x_real @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_real @ Matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_real @ Matrix ) )
=> ( ( ( vec_index_real @ ( col_real @ Matrix @ J2 ) @ I )
= zero_zero_real )
= ( ( vec_index_real @ ( col_real @ Matrix @ J2 ) @ I )
!= one_one_real ) ) ) ) ) ).
% zero_one_matrix.col_nth_0_or_1_iff
thf(fact_795_zero__one__matrix_Ocol__nth__0__or__1__iff,axiom,
! [Matrix: mat_b,J2: nat,I: nat] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_b @ Matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_b @ Matrix ) )
=> ( ( ( vec_index_b @ ( col_b @ Matrix @ J2 ) @ I )
= zero_zero_b )
= ( ( vec_index_b @ ( col_b @ Matrix @ J2 ) @ I )
!= one_one_b ) ) ) ) ) ).
% zero_one_matrix.col_nth_0_or_1_iff
thf(fact_796_all__ones__index,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_b @ ( matrix8789069900454870053_vec_b @ N ) @ I )
= one_one_b ) ) ).
% all_ones_index
thf(fact_797_all__ones__index,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_nat @ ( matrix2751262895470517546ec_nat @ N ) @ I )
= one_one_nat ) ) ).
% all_ones_index
thf(fact_798_all__ones__index,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_int @ ( matrix2748772424961467270ec_int @ N ) @ I )
= one_one_int ) ) ).
% all_ones_index
thf(fact_799_map__col__to__block__elem,axiom,
! [I: nat,C: vec_nat] :
( ( ord_less_nat @ I @ ( dim_vec_nat @ C ) )
=> ( ( member_nat @ I @ ( incide3975725477190312290ck_nat @ C ) )
= ( ( vec_index_nat @ C @ I )
= one_one_nat ) ) ) ).
% map_col_to_block_elem
thf(fact_800_map__col__to__block__elem,axiom,
! [I: nat,C: vec_int] :
( ( ord_less_nat @ I @ ( dim_vec_int @ C ) )
=> ( ( member_nat @ I @ ( incide3973235006681262014ck_int @ C ) )
= ( ( vec_index_int @ C @ I )
= one_one_int ) ) ) ).
% map_col_to_block_elem
thf(fact_801_map__col__to__block__elem,axiom,
! [I: nat,C: vec_b] :
( ( ord_less_nat @ I @ ( dim_vec_b @ C ) )
=> ( ( member_nat @ I @ ( incide5355957740755015149lock_b @ C ) )
= ( ( vec_index_b @ C @ I )
= one_one_b ) ) ) ).
% map_col_to_block_elem
thf(fact_802_map__col__to__block__size,axiom,
! [J2: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_b @ matrix ) )
=> ( ( finite_card_nat @ ( incide5355957740755015149lock_b @ ( col_b @ matrix @ J2 ) ) )
= ( incide1360831186763368190size_b @ matrix @ J2 ) ) ) ).
% map_col_to_block_size
thf(fact_803_diag__mat__transpose,axiom,
! [A2: mat_b] :
( ( ( dim_row_b @ A2 )
= ( dim_col_b @ A2 ) )
=> ( ( diag_mat_b @ ( transpose_mat_b @ A2 ) )
= ( diag_mat_b @ A2 ) ) ) ).
% diag_mat_transpose
thf(fact_804_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N @ K )
=> ( P @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K @ I2 )
=> ( P @ I2 ) )
=> ( P @ K ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_805_in__map__col__valid__index,axiom,
! [I: nat,C: vec_int] :
( ( member_nat @ I @ ( incide3973235006681262014ck_int @ C ) )
=> ( ord_less_nat @ I @ ( dim_vec_int @ C ) ) ) ).
% in_map_col_valid_index
thf(fact_806_in__map__col__valid__index,axiom,
! [I: nat,C: vec_b] :
( ( member_nat @ I @ ( incide5355957740755015149lock_b @ C ) )
=> ( ord_less_nat @ I @ ( dim_vec_b @ C ) ) ) ).
% in_map_col_valid_index
thf(fact_807_dim__vec__all__ones,axiom,
! [N: nat] :
( ( dim_vec_int @ ( matrix2748772424961467270ec_int @ N ) )
= N ) ).
% dim_vec_all_ones
thf(fact_808_dim__vec__all__ones,axiom,
! [N: nat] :
( ( dim_vec_b @ ( matrix8789069900454870053_vec_b @ N ) )
= N ) ).
% dim_vec_all_ones
thf(fact_809_eq__vecI,axiom,
! [W2: vec_b,V: vec_b] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_b @ W2 ) )
=> ( ( vec_index_b @ V @ I3 )
= ( vec_index_b @ W2 @ I3 ) ) )
=> ( ( ( dim_vec_b @ V )
= ( dim_vec_b @ W2 ) )
=> ( V = W2 ) ) ) ).
% eq_vecI
thf(fact_810_eq__vecI,axiom,
! [W2: vec_int,V: vec_int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_int @ W2 ) )
=> ( ( vec_index_int @ V @ I3 )
= ( vec_index_int @ W2 @ I3 ) ) )
=> ( ( ( dim_vec_int @ V )
= ( dim_vec_int @ W2 ) )
=> ( V = W2 ) ) ) ).
% eq_vecI
thf(fact_811_dim__col,axiom,
! [A2: mat_int,I: nat] :
( ( dim_vec_int @ ( col_int @ A2 @ I ) )
= ( dim_row_int @ A2 ) ) ).
% dim_col
thf(fact_812_dim__col,axiom,
! [A2: mat_b,I: nat] :
( ( dim_vec_b @ ( col_b @ A2 @ I ) )
= ( dim_row_b @ A2 ) ) ).
% dim_col
thf(fact_813_index__row_I2_J,axiom,
! [A2: mat_int,I: nat] :
( ( dim_vec_int @ ( row_int @ A2 @ I ) )
= ( dim_col_int @ A2 ) ) ).
% index_row(2)
thf(fact_814_index__row_I2_J,axiom,
! [A2: mat_b,I: nat] :
( ( dim_vec_b @ ( row_b @ A2 @ I ) )
= ( dim_col_b @ A2 ) ) ).
% index_row(2)
thf(fact_815_vec__eq__iff,axiom,
( ( ^ [Y3: vec_b,Z: vec_b] : ( Y3 = Z ) )
= ( ^ [X3: vec_b,Y4: vec_b] :
( ( ( dim_vec_b @ X3 )
= ( dim_vec_b @ Y4 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_b @ Y4 ) )
=> ( ( vec_index_b @ X3 @ I4 )
= ( vec_index_b @ Y4 @ I4 ) ) ) ) ) ) ).
% vec_eq_iff
thf(fact_816_vec__eq__iff,axiom,
( ( ^ [Y3: vec_int,Z: vec_int] : ( Y3 = Z ) )
= ( ^ [X3: vec_int,Y4: vec_int] :
( ( ( dim_vec_int @ X3 )
= ( dim_vec_int @ Y4 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_int @ Y4 ) )
=> ( ( vec_index_int @ X3 @ I4 )
= ( vec_index_int @ Y4 @ I4 ) ) ) ) ) ) ).
% vec_eq_iff
thf(fact_817_less__eq__vec__def,axiom,
( ord_le7003571194052948520_set_b
= ( ^ [V2: vec_set_b,W3: vec_set_b] :
( ( ( dim_vec_set_b @ V2 )
= ( dim_vec_set_b @ W3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_set_b @ W3 ) )
=> ( ord_less_eq_set_b @ ( vec_index_set_b @ V2 @ I4 ) @ ( vec_index_set_b @ W3 @ I4 ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_818_less__eq__vec__def,axiom,
( ord_less_eq_vec_nat
= ( ^ [V2: vec_nat,W3: vec_nat] :
( ( ( dim_vec_nat @ V2 )
= ( dim_vec_nat @ W3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_nat @ W3 ) )
=> ( ord_less_eq_nat @ ( vec_index_nat @ V2 @ I4 ) @ ( vec_index_nat @ W3 @ I4 ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_819_less__eq__vec__def,axiom,
( ord_less_eq_vec_int
= ( ^ [V2: vec_int,W3: vec_int] :
( ( ( dim_vec_int @ V2 )
= ( dim_vec_int @ W3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_int @ W3 ) )
=> ( ord_less_eq_int @ ( vec_index_int @ V2 @ I4 ) @ ( vec_index_int @ W3 @ I4 ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_820_less__eq__vec__def,axiom,
( ord_less_eq_vec_real
= ( ^ [V2: vec_real,W3: vec_real] :
( ( ( dim_vec_real @ V2 )
= ( dim_vec_real @ W3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_vec_real @ W3 ) )
=> ( ord_less_eq_real @ ( vec_index_real @ V2 @ I4 ) @ ( vec_index_real @ W3 @ I4 ) ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_821_vec__contains__obtains__index,axiom,
! [A: nat,V: vec_nat] :
( ( member_nat @ A @ ( vec_set_nat @ V ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_nat @ V ) )
=> ( ( vec_index_nat @ V @ I3 )
!= A ) ) ) ).
% vec_contains_obtains_index
thf(fact_822_vec__contains__obtains__index,axiom,
! [A: vec_b,V: vec_vec_b] :
( ( member_vec_b @ A @ ( vec_set_vec_b @ V ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_vec_b @ V ) )
=> ( ( vec_index_vec_b @ V @ I3 )
!= A ) ) ) ).
% vec_contains_obtains_index
thf(fact_823_vec__contains__obtains__index,axiom,
! [A: b,V: vec_b] :
( ( member_b @ A @ ( vec_set_b2 @ V ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_b @ V ) )
=> ( ( vec_index_b @ V @ I3 )
!= A ) ) ) ).
% vec_contains_obtains_index
thf(fact_824_vec__contains__obtains__index,axiom,
! [A: int,V: vec_int] :
( ( member_int @ A @ ( vec_set_int @ V ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_int @ V ) )
=> ( ( vec_index_int @ V @ I3 )
!= A ) ) ) ).
% vec_contains_obtains_index
thf(fact_825_vec__setI,axiom,
! [V: vec_nat,I: nat,A: nat] :
( ( ( vec_index_nat @ V @ I )
= A )
=> ( ( ord_less_nat @ I @ ( dim_vec_nat @ V ) )
=> ( member_nat @ A @ ( vec_set_nat @ V ) ) ) ) ).
% vec_setI
thf(fact_826_vec__setI,axiom,
! [V: vec_vec_b,I: nat,A: vec_b] :
( ( ( vec_index_vec_b @ V @ I )
= A )
=> ( ( ord_less_nat @ I @ ( dim_vec_vec_b @ V ) )
=> ( member_vec_b @ A @ ( vec_set_vec_b @ V ) ) ) ) ).
% vec_setI
thf(fact_827_vec__setI,axiom,
! [V: vec_b,I: nat,A: b] :
( ( ( vec_index_b @ V @ I )
= A )
=> ( ( ord_less_nat @ I @ ( dim_vec_b @ V ) )
=> ( member_b @ A @ ( vec_set_b2 @ V ) ) ) ) ).
% vec_setI
thf(fact_828_vec__setI,axiom,
! [V: vec_int,I: nat,A: int] :
( ( ( vec_index_int @ V @ I )
= A )
=> ( ( ord_less_nat @ I @ ( dim_vec_int @ V ) )
=> ( member_int @ A @ ( vec_set_int @ V ) ) ) ) ).
% vec_setI
thf(fact_829_vec__setE,axiom,
! [A: nat,V: vec_nat] :
( ( member_nat @ A @ ( vec_set_nat @ V ) )
=> ~ ! [I3: nat] :
( ( ( vec_index_nat @ V @ I3 )
= A )
=> ~ ( ord_less_nat @ I3 @ ( dim_vec_nat @ V ) ) ) ) ).
% vec_setE
thf(fact_830_vec__setE,axiom,
! [A: vec_b,V: vec_vec_b] :
( ( member_vec_b @ A @ ( vec_set_vec_b @ V ) )
=> ~ ! [I3: nat] :
( ( ( vec_index_vec_b @ V @ I3 )
= A )
=> ~ ( ord_less_nat @ I3 @ ( dim_vec_vec_b @ V ) ) ) ) ).
% vec_setE
thf(fact_831_vec__setE,axiom,
! [A: b,V: vec_b] :
( ( member_b @ A @ ( vec_set_b2 @ V ) )
=> ~ ! [I3: nat] :
( ( ( vec_index_b @ V @ I3 )
= A )
=> ~ ( ord_less_nat @ I3 @ ( dim_vec_b @ V ) ) ) ) ).
% vec_setE
thf(fact_832_vec__setE,axiom,
! [A: int,V: vec_int] :
( ( member_int @ A @ ( vec_set_int @ V ) )
=> ~ ! [I3: nat] :
( ( ( vec_index_int @ V @ I3 )
= A )
=> ~ ( ord_less_nat @ I3 @ ( dim_vec_int @ V ) ) ) ) ).
% vec_setE
thf(fact_833_zero__one__matrix_Oin__map__col__valid__index,axiom,
! [Matrix: mat_b,I: nat,C: vec_int] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( member_nat @ I @ ( incide3973235006681262014ck_int @ C ) )
=> ( ord_less_nat @ I @ ( dim_vec_int @ C ) ) ) ) ).
% zero_one_matrix.in_map_col_valid_index
thf(fact_834_zero__one__matrix_Oin__map__col__valid__index,axiom,
! [Matrix: mat_b,I: nat,C: vec_b] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( member_nat @ I @ ( incide5355957740755015149lock_b @ C ) )
=> ( ord_less_nat @ I @ ( dim_vec_b @ C ) ) ) ) ).
% zero_one_matrix.in_map_col_valid_index
thf(fact_835_zero__one__matrix_Omap__col__to__block__size,axiom,
! [Matrix: mat_b,J2: nat] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( ord_less_nat @ J2 @ ( dim_col_b @ Matrix ) )
=> ( ( finite_card_nat @ ( incide5355957740755015149lock_b @ ( col_b @ Matrix @ J2 ) ) )
= ( incide1360831186763368190size_b @ Matrix @ J2 ) ) ) ) ).
% zero_one_matrix.map_col_to_block_size
thf(fact_836_zero__one__matrix_Omap__col__to__block__elem,axiom,
! [Matrix: mat_b,I: nat,C: vec_nat] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_vec_nat @ C ) )
=> ( ( member_nat @ I @ ( incide3975725477190312290ck_nat @ C ) )
= ( ( vec_index_nat @ C @ I )
= one_one_nat ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem
thf(fact_837_zero__one__matrix_Omap__col__to__block__elem,axiom,
! [Matrix: mat_b,I: nat,C: vec_int] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_vec_int @ C ) )
=> ( ( member_nat @ I @ ( incide3973235006681262014ck_int @ C ) )
= ( ( vec_index_int @ C @ I )
= one_one_int ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem
thf(fact_838_zero__one__matrix_Omap__col__to__block__elem,axiom,
! [Matrix: mat_b,I: nat,C: vec_b] :
( ( incide7367983062745021297trix_b @ Matrix )
=> ( ( ord_less_nat @ I @ ( dim_vec_b @ C ) )
=> ( ( member_nat @ I @ ( incide5355957740755015149lock_b @ C ) )
= ( ( vec_index_b @ C @ I )
= one_one_b ) ) ) ) ).
% zero_one_matrix.map_col_to_block_elem
thf(fact_839_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_840_card_Oempty,axiom,
( ( finite_card_b @ bot_bot_set_b )
= zero_zero_nat ) ).
% card.empty
thf(fact_841_index__update__vec1,axiom,
! [I: nat,V: vec_b,A: b] :
( ( ord_less_nat @ I @ ( dim_vec_b @ V ) )
=> ( ( vec_index_b @ ( update_vec_b @ V @ I @ A ) @ I )
= A ) ) ).
% index_update_vec1
thf(fact_842_index__update__vec1,axiom,
! [I: nat,V: vec_int,A: int] :
( ( ord_less_nat @ I @ ( dim_vec_int @ V ) )
=> ( ( vec_index_int @ ( update_vec_int @ V @ I @ A ) @ I )
= A ) ) ).
% index_update_vec1
thf(fact_843_card__1__singletonE,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= one_one_nat )
=> ~ ! [X2: nat] :
( A2
!= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_844_card__1__singletonE,axiom,
! [A2: set_b] :
( ( ( finite_card_b @ A2 )
= one_one_nat )
=> ~ ! [X2: b] :
( A2
!= ( insert_b @ X2 @ bot_bot_set_b ) ) ) ).
% card_1_singletonE
thf(fact_845_trans__mat__rep__block__size__sym_I1_J,axiom,
! [J2: nat,M3: mat_b] :
( ( ord_less_nat @ J2 @ ( dim_col_b @ M3 ) )
=> ( ( incide1360831186763368190size_b @ M3 @ J2 )
= ( incide1817117124879203335_num_b @ ( transpose_mat_b @ M3 ) @ J2 ) ) ) ).
% trans_mat_rep_block_size_sym(1)
thf(fact_846_trans__mat__rep__block__size__sym_I2_J,axiom,
! [I: nat,M3: mat_b] :
( ( ord_less_nat @ I @ ( dim_row_b @ M3 ) )
=> ( ( incide1817117124879203335_num_b @ M3 @ I )
= ( incide1360831186763368190size_b @ ( transpose_mat_b @ M3 ) @ I ) ) ) ).
% trans_mat_rep_block_size_sym(2)
thf(fact_847_is__singleton__altdef,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( ( finite_card_nat @ A3 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_848_dim__update__vec,axiom,
! [V: vec_int,I: nat,A: int] :
( ( dim_vec_int @ ( update_vec_int @ V @ I @ A ) )
= ( dim_vec_int @ V ) ) ).
% dim_update_vec
thf(fact_849_dim__update__vec,axiom,
! [V: vec_b,I: nat,A: b] :
( ( dim_vec_b @ ( update_vec_b @ V @ I @ A ) )
= ( dim_vec_b @ V ) ) ).
% dim_update_vec
thf(fact_850_index__update__vec2,axiom,
! [I5: nat,I: nat,V: vec_b,A: b] :
( ( I5 != I )
=> ( ( vec_index_b @ ( update_vec_b @ V @ I @ A ) @ I5 )
= ( vec_index_b @ V @ I5 ) ) ) ).
% index_update_vec2
thf(fact_851_index__update__vec2,axiom,
! [I5: nat,I: nat,V: vec_int,A: int] :
( ( I5 != I )
=> ( ( vec_index_int @ ( update_vec_int @ V @ I @ A ) @ I5 )
= ( vec_index_int @ V @ I5 ) ) ) ).
% index_update_vec2
thf(fact_852_card__insert__le,axiom,
! [A2: set_b,X: b] : ( ord_less_eq_nat @ ( finite_card_b @ A2 ) @ ( finite_card_b @ ( insert_b @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_853_card__insert__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_854_elem__exists__non__empty__set,axiom,
! [A2: set_b] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_b @ A2 ) )
=> ~ ! [X2: b] :
~ ( member_b @ X2 @ A2 ) ) ).
% elem_exists_non_empty_set
thf(fact_855_elem__exists__non__empty__set,axiom,
! [A2: set_vec_b] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_vec_b @ A2 ) )
=> ~ ! [X2: vec_b] :
~ ( member_vec_b @ X2 @ A2 ) ) ).
% elem_exists_non_empty_set
thf(fact_856_elem__exists__non__empty__set,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ~ ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) ) ).
% elem_exists_non_empty_set
thf(fact_857_row__mat__of__row__fun,axiom,
! [I: nat,Nr: nat,F: nat > vec_int,Nc: nat] :
( ( ord_less_nat @ I @ Nr )
=> ( ( ( dim_vec_int @ ( F @ I ) )
= Nc )
=> ( ( row_int @ ( mat_of_row_fun_int @ Nr @ Nc @ F ) @ I )
= ( F @ I ) ) ) ) ).
% row_mat_of_row_fun
thf(fact_858_row__mat__of__row__fun,axiom,
! [I: nat,Nr: nat,F: nat > vec_b,Nc: nat] :
( ( ord_less_nat @ I @ Nr )
=> ( ( ( dim_vec_b @ ( F @ I ) )
= Nc )
=> ( ( row_b @ ( mat_of_row_fun_b @ Nr @ Nc @ F ) @ I )
= ( F @ I ) ) ) ) ).
% row_mat_of_row_fun
thf(fact_859_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K: nat] :
? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( F @ K @ I2 ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ! [K4: nat] :
? [K5: nat] :
( ( ord_less_eq_nat @ K4 @ K5 )
& ( F @ K5 @ I3 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_860_mat__rep__num__sum__alt,axiom,
! [M3: mat_real,I: nat] :
( ( ord_less_eq_set_real @ ( elements_mat_real @ M3 ) @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) )
=> ( ( ord_less_nat @ I @ ( dim_row_real @ M3 ) )
=> ( ( semiri5074537144036343181t_real @ ( incide5781393841671188388m_real @ M3 @ I ) )
= ( matrix1363837090280519610c_real @ ( row_real @ M3 @ I ) ) ) ) ) ).
% mat_rep_num_sum_alt
thf(fact_861_mat__rep__num__sum__alt,axiom,
! [M3: mat_int,I: nat] :
( ( ord_less_eq_set_int @ ( elements_mat_int @ M3 ) @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ M3 ) )
=> ( ( semiri1314217659103216013at_int @ ( incide7000514267430604580um_int @ M3 @ I ) )
= ( matrix3634415343793898042ec_int @ ( row_int @ M3 @ I ) ) ) ) ) ).
% mat_rep_num_sum_alt
thf(fact_862_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_863_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A6 ) )
= ( ord_less_int @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_864_verit__comp__simplify1_I3_J,axiom,
! [B7: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B7 @ A6 ) )
= ( ord_less_real @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_865_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_866_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_867_dim__row__mat_I2_J,axiom,
! [Nr: nat,Nc: nat,G: nat > vec_b] :
( ( dim_row_b @ ( mat_of_row_fun_b @ Nr @ Nc @ G ) )
= Nr ) ).
% dim_row_mat(2)
thf(fact_868_dim__col__mat_I2_J,axiom,
! [Nr: nat,Nc: nat,G: nat > vec_b] :
( ( dim_col_b @ ( mat_of_row_fun_b @ Nr @ Nc @ G ) )
= Nc ) ).
% dim_col_mat(2)
thf(fact_869_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_870_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_871_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_872_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_873_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_874_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_875_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_876_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_877_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_878_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_879_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_880_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_881_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_882_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_883_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_884_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_885_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_886_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_887_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_888_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_889_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_890_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_891_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_892_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_893_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_894_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_895_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_896_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_897_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_898_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_899_sum__vec__one__zero,axiom,
! [V: vec_int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_int @ V ) )
=> ( ord_less_eq_int @ ( vec_index_int @ V @ I3 ) @ one_one_int ) )
=> ( ord_less_eq_int @ ( matrix3634415343793898042ec_int @ V ) @ ( semiri1314217659103216013at_int @ ( dim_vec_int @ V ) ) ) ) ).
% sum_vec_one_zero
thf(fact_900_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B6: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_901_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B6: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_902_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_903_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_904_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_905_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_906_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_907_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_908_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_909_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_910_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_911_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_912_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_913_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_914_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_915_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_916_of__nat__mono,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).
% of_nat_mono
thf(fact_917_of__nat__mono,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).
% of_nat_mono
thf(fact_918_of__nat__mono,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J2 ) ) ) ).
% of_nat_mono
thf(fact_919_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_920_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_921_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_922_verit__comp__simplify1_I2_J,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_923_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_924_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_925_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_926_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_927_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_928_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_929_mat__block__size__sum__alt,axiom,
! [M3: mat_real,J2: nat] :
( ( ord_less_eq_set_real @ ( elements_mat_real @ M3 ) @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) )
=> ( ( ord_less_nat @ J2 @ ( dim_col_real @ M3 ) )
=> ( ( semiri5074537144036343181t_real @ ( incide8656787765868073069e_real @ M3 @ J2 ) )
= ( matrix1363837090280519610c_real @ ( col_real @ M3 @ J2 ) ) ) ) ) ).
% mat_block_size_sum_alt
thf(fact_930_mat__block__size__sum__alt,axiom,
! [M3: mat_int,J2: nat] :
( ( ord_less_eq_set_int @ ( elements_mat_int @ M3 ) @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) )
=> ( ( ord_less_nat @ J2 @ ( dim_col_int @ M3 ) )
=> ( ( semiri1314217659103216013at_int @ ( incide7086334917823153261ze_int @ M3 @ J2 ) )
= ( matrix3634415343793898042ec_int @ ( col_int @ M3 @ J2 ) ) ) ) ) ).
% mat_block_size_sum_alt
thf(fact_931_pos__int__cases,axiom,
! [K3: int] :
( ( ord_less_int @ zero_zero_int @ K3 )
=> ~ ! [N4: nat] :
( ( K3
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% pos_int_cases
thf(fact_932_zero__less__imp__eq__int,axiom,
! [K3: int] :
( ( ord_less_int @ zero_zero_int @ K3 )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( K3
= ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_933_reals__Archimedean2,axiom,
! [X: real] :
? [N4: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% reals_Archimedean2
thf(fact_934_real__arch__simple,axiom,
! [X: real] :
? [N4: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% real_arch_simple
thf(fact_935_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A @ C4 )
& ( ord_less_eq_nat @ C4 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C4 ) )
=> ( P @ X6 ) )
& ! [D3: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_936_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C4: int] :
( ( ord_less_eq_int @ A @ C4 )
& ( ord_less_eq_int @ C4 @ B )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A @ X6 )
& ( ord_less_int @ X6 @ C4 ) )
=> ( P @ X6 ) )
& ! [D3: int] :
( ! [X2: int] :
( ( ( ord_less_eq_int @ A @ X2 )
& ( ord_less_int @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_int @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_937_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C4: real] :
( ( ord_less_eq_real @ A @ C4 )
& ( ord_less_eq_real @ C4 @ B )
& ! [X6: real] :
( ( ( ord_less_eq_real @ A @ X6 )
& ( ord_less_real @ X6 @ C4 ) )
=> ( P @ X6 ) )
& ! [D3: real] :
( ! [X2: real] :
( ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_real @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_real @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_938_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_939_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_940_ex__gt__or__lt,axiom,
! [A: real] :
? [B5: real] :
( ( ord_less_real @ A @ B5 )
| ( ord_less_real @ B5 @ A ) ) ).
% ex_gt_or_lt
thf(fact_941_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_942_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_943_zero__min,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_min
thf(fact_944_lift__01__vec__simp_I1_J,axiom,
! [V: vec_int] :
( ( dim_vec_int @ ( matrix8301520909418075407nt_int @ V ) )
= ( dim_vec_int @ V ) ) ).
% lift_01_vec_simp(1)
thf(fact_945_lift__01__vec__simp_I1_J,axiom,
! [V: vec_b] :
( ( dim_vec_int @ ( matrix1311240063772730166_b_int @ V ) )
= ( dim_vec_b @ V ) ) ).
% lift_01_vec_simp(1)
thf(fact_946_lift__01__vec__simp_I1_J,axiom,
! [V: vec_int] :
( ( dim_vec_b @ ( matrix1865072738833226460_int_b @ V ) )
= ( dim_vec_int @ V ) ) ).
% lift_01_vec_simp(1)
thf(fact_947_lift__01__vec__simp_I1_J,axiom,
! [V: vec_b] :
( ( dim_vec_b @ ( matrix7059812428859951221ec_b_b @ V ) )
= ( dim_vec_b @ V ) ) ).
% lift_01_vec_simp(1)
thf(fact_948_dim__vec__first,axiom,
! [V: vec_int,N: nat] :
( ( dim_vec_int @ ( vec_first_int @ V @ N ) )
= N ) ).
% dim_vec_first
thf(fact_949_dim__vec__first,axiom,
! [V: vec_b,N: nat] :
( ( dim_vec_b @ ( vec_first_b @ V @ N ) )
= N ) ).
% dim_vec_first
thf(fact_950_dim__vec__last,axiom,
! [V: vec_int,N: nat] :
( ( dim_vec_int @ ( vec_last_int @ V @ N ) )
= N ) ).
% dim_vec_last
thf(fact_951_dim__vec__last,axiom,
! [V: vec_b,N: nat] :
( ( dim_vec_b @ ( vec_last_b @ V @ N ) )
= N ) ).
% dim_vec_last
thf(fact_952_lift__01__vec__simp_I2_J,axiom,
! [I: nat,V: vec_b] :
( ( ord_less_nat @ I @ ( dim_vec_b @ V ) )
=> ( ( vec_index_b @ ( matrix7059812428859951221ec_b_b @ V ) @ I )
= ( matrix4781043112069605324ne_b_b @ ( vec_index_b @ V @ I ) ) ) ) ).
% lift_01_vec_simp(2)
thf(fact_953_lift__01__vec__simp_I2_J,axiom,
! [I: nat,V: vec_int] :
( ( ord_less_nat @ I @ ( dim_vec_int @ V ) )
=> ( ( vec_index_b @ ( matrix1865072738833226460_int_b @ V ) @ I )
= ( matrix6038540757728371653_int_b @ ( vec_index_int @ V @ I ) ) ) ) ).
% lift_01_vec_simp(2)
thf(fact_954_lift__01__vec__simp_I2_J,axiom,
! [I: nat,V: vec_b] :
( ( ord_less_nat @ I @ ( dim_vec_b @ V ) )
=> ( ( vec_index_int @ ( matrix1311240063772730166_b_int @ V ) @ I )
= ( matrix5484708082667875359_b_int @ ( vec_index_b @ V @ I ) ) ) ) ).
% lift_01_vec_simp(2)
thf(fact_955_lift__01__vec__simp_I2_J,axiom,
! [I: nat,V: vec_int] :
( ( ord_less_nat @ I @ ( dim_vec_int @ V ) )
=> ( ( vec_index_int @ ( matrix8301520909418075407nt_int @ V ) @ I )
= ( matrix1697308990001484774nt_int @ ( vec_index_int @ V @ I ) ) ) ) ).
% lift_01_vec_simp(2)
thf(fact_956_index__unit__vec_I1_J,axiom,
! [I: nat,N: nat,J2: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ( ( J2 = I )
=> ( ( vec_index_b @ ( unit_vec_b @ N @ I ) @ J2 )
= one_one_b ) )
& ( ( J2 != I )
=> ( ( vec_index_b @ ( unit_vec_b @ N @ I ) @ J2 )
= zero_zero_b ) ) ) ) ) ).
% index_unit_vec(1)
thf(fact_957_index__unit__vec_I1_J,axiom,
! [I: nat,N: nat,J2: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ( ( J2 = I )
=> ( ( vec_index_nat @ ( unit_vec_nat @ N @ I ) @ J2 )
= one_one_nat ) )
& ( ( J2 != I )
=> ( ( vec_index_nat @ ( unit_vec_nat @ N @ I ) @ J2 )
= zero_zero_nat ) ) ) ) ) ).
% index_unit_vec(1)
thf(fact_958_index__unit__vec_I1_J,axiom,
! [I: nat,N: nat,J2: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ( ( J2 = I )
=> ( ( vec_index_int @ ( unit_vec_int @ N @ I ) @ J2 )
= one_one_int ) )
& ( ( J2 != I )
=> ( ( vec_index_int @ ( unit_vec_int @ N @ I ) @ J2 )
= zero_zero_int ) ) ) ) ) ).
% index_unit_vec(1)
thf(fact_959_index__unit__vec_I1_J,axiom,
! [I: nat,N: nat,J2: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ( ( J2 = I )
=> ( ( vec_index_real @ ( unit_vec_real @ N @ I ) @ J2 )
= one_one_real ) )
& ( ( J2 != I )
=> ( ( vec_index_real @ ( unit_vec_real @ N @ I ) @ J2 )
= zero_zero_real ) ) ) ) ) ).
% index_unit_vec(1)
thf(fact_960_card__insert__le__m1,axiom,
! [N: nat,Y: set_b,X: b] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_b @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_b @ ( insert_b @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_961_card__insert__le__m1,axiom,
! [N: nat,Y: set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_962_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_963_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_964_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_965_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_966_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_967_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_968_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_969_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_970_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_971_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_972_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_973_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_974_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_975_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_976_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_977_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_978_of__zero__hom_Ohom__zero,axiom,
( ( matrix4781043112069605324ne_b_b @ zero_zero_b )
= zero_zero_b ) ).
% of_zero_hom.hom_zero
thf(fact_979_of__zero__hom_Ohom__zero,axiom,
( ( matrix5487198553176925635_b_nat @ zero_zero_b )
= zero_zero_nat ) ).
% of_zero_hom.hom_zero
thf(fact_980_of__zero__hom_Ohom__zero,axiom,
( ( matrix5484708082667875359_b_int @ zero_zero_b )
= zero_zero_int ) ).
% of_zero_hom.hom_zero
thf(fact_981_of__zero__hom_Ohom__zero,axiom,
( ( matrix2280091663418064671b_real @ zero_zero_b )
= zero_zero_real ) ).
% of_zero_hom.hom_zero
thf(fact_982_of__zero__hom_Ohom__zero,axiom,
( ( matrix8283685725398817569_nat_b @ zero_zero_nat )
= zero_zero_b ) ).
% of_zero_hom.hom_zero
thf(fact_983_of__zero__hom_Ohom__zero,axiom,
( ( matrix700445748609480494at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_zero_hom.hom_zero
thf(fact_984_of__zero__hom_Ohom__zero,axiom,
( ( matrix697955278100430218at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_zero_hom.hom_zero
thf(fact_985_of__zero__hom_Ohom__zero,axiom,
( ( matrix8742843541027031818t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_zero_hom.hom_zero
thf(fact_986_of__zero__hom_Ohom__zero,axiom,
( ( matrix6038540757728371653_int_b @ zero_zero_int )
= zero_zero_b ) ).
% of_zero_hom.hom_zero
thf(fact_987_of__zero__hom_Ohom__zero,axiom,
( ( matrix1699799460510535050nt_nat @ zero_zero_int )
= zero_zero_nat ) ).
% of_zero_hom.hom_zero
thf(fact_988_of__zero__hom_Ohom__0__iff,axiom,
! [X: b] :
( ( ( matrix4781043112069605324ne_b_b @ X )
= zero_zero_b )
= ( X = zero_zero_b ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_989_of__zero__hom_Ohom__0__iff,axiom,
! [X: nat] :
( ( ( matrix8283685725398817569_nat_b @ X )
= zero_zero_b )
= ( X = zero_zero_nat ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_990_of__zero__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( matrix6038540757728371653_int_b @ X )
= zero_zero_b )
= ( X = zero_zero_int ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_991_of__zero__hom_Ohom__0__iff,axiom,
! [X: real] :
( ( ( matrix6537263852557659589real_b @ X )
= zero_zero_b )
= ( X = zero_zero_real ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_992_of__zero__hom_Ohom__0__iff,axiom,
! [X: b] :
( ( ( matrix5487198553176925635_b_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_b ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_993_of__zero__hom_Ohom__0__iff,axiom,
! [X: nat] :
( ( ( matrix700445748609480494at_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_nat ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_994_of__zero__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( matrix1699799460510535050nt_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_int ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_995_of__zero__hom_Ohom__0__iff,axiom,
! [X: real] :
( ( ( matrix4086780077301154698al_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_real ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_996_of__zero__hom_Ohom__0__iff,axiom,
! [X: b] :
( ( ( matrix5484708082667875359_b_int @ X )
= zero_zero_int )
= ( X = zero_zero_b ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_997_of__zero__hom_Ohom__0__iff,axiom,
! [X: nat] :
( ( ( matrix697955278100430218at_int @ X )
= zero_zero_int )
= ( X = zero_zero_nat ) ) ).
% of_zero_hom.hom_0_iff
thf(fact_998_of__zero__neq__one__0,axiom,
( ( matrix4781043112069605324ne_b_b @ zero_zero_b )
= zero_zero_b ) ).
% of_zero_neq_one_0
thf(fact_999_of__zero__neq__one__0,axiom,
( ( matrix5487198553176925635_b_nat @ zero_zero_b )
= zero_zero_nat ) ).
% of_zero_neq_one_0
thf(fact_1000_of__zero__neq__one__0,axiom,
( ( matrix5484708082667875359_b_int @ zero_zero_b )
= zero_zero_int ) ).
% of_zero_neq_one_0
thf(fact_1001_of__zero__neq__one__0,axiom,
( ( matrix2280091663418064671b_real @ zero_zero_b )
= zero_zero_real ) ).
% of_zero_neq_one_0
thf(fact_1002_of__zero__neq__one__0,axiom,
( ( matrix8283685725398817569_nat_b @ zero_zero_nat )
= zero_zero_b ) ).
% of_zero_neq_one_0
thf(fact_1003_of__zero__neq__one__0,axiom,
( ( matrix700445748609480494at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_zero_neq_one_0
thf(fact_1004_of__zero__neq__one__0,axiom,
( ( matrix697955278100430218at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_zero_neq_one_0
thf(fact_1005_of__zero__neq__one__0,axiom,
( ( matrix8742843541027031818t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_zero_neq_one_0
thf(fact_1006_of__zero__neq__one__0,axiom,
( ( matrix6038540757728371653_int_b @ zero_zero_int )
= zero_zero_b ) ).
% of_zero_neq_one_0
thf(fact_1007_of__zero__neq__one__0,axiom,
( ( matrix1699799460510535050nt_nat @ zero_zero_int )
= zero_zero_nat ) ).
% of_zero_neq_one_0
thf(fact_1008_of__zero__neq__one__0__iff,axiom,
! [X: b] :
( ( ( matrix4781043112069605324ne_b_b @ X )
= zero_zero_b )
= ( X = zero_zero_b ) ) ).
% of_zero_neq_one_0_iff
thf(fact_1009_of__zero__neq__one__0__iff,axiom,
! [X: nat] :
( ( ( matrix8283685725398817569_nat_b @ X )
= zero_zero_b )
= ( X = zero_zero_nat ) ) ).
% of_zero_neq_one_0_iff
thf(fact_1010_of__zero__neq__one__0__iff,axiom,
! [X: int] :
( ( ( matrix6038540757728371653_int_b @ X )
= zero_zero_b )
= ( X = zero_zero_int ) ) ).
% of_zero_neq_one_0_iff
thf(fact_1011_of__zero__neq__one__0__iff,axiom,
! [X: real] :
( ( ( matrix6537263852557659589real_b @ X )
= zero_zero_b )
= ( X = zero_zero_real ) ) ).
% of_zero_neq_one_0_iff
thf(fact_1012_of__zero__neq__one__0__iff,axiom,
! [X: b] :
( ( ( matrix5487198553176925635_b_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_b ) ) ).
% of_zero_neq_one_0_iff
thf(fact_1013_of__zero__neq__one__0__iff,axiom,
! [X: nat] :
( ( ( matrix700445748609480494at_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_nat ) ) ).
% of_zero_neq_one_0_iff
thf(fact_1014_of__zero__neq__one__0__iff,axiom,
! [X: int] :
( ( ( matrix1699799460510535050nt_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_int ) ) ).
% of_zero_neq_one_0_iff
thf(fact_1015_of__zero__neq__one__0__iff,axiom,
! [X: real] :
( ( ( matrix4086780077301154698al_nat @ X )
= zero_zero_nat )
= ( X = zero_zero_real ) ) ).
% of_zero_neq_one_0_iff
thf(fact_1016_of__zero__neq__one__0__iff,axiom,
! [X: b] :
( ( ( matrix5484708082667875359_b_int @ X )
= zero_zero_int )
= ( X = zero_zero_b ) ) ).
% of_zero_neq_one_0_iff
thf(fact_1017_of__zero__neq__one__0__iff,axiom,
! [X: nat] :
( ( ( matrix697955278100430218at_int @ X )
= zero_zero_int )
= ( X = zero_zero_nat ) ) ).
% of_zero_neq_one_0_iff
thf(fact_1018_of__zero__neq__one__1,axiom,
( ( matrix4781043112069605324ne_b_b @ one_one_b )
= one_one_b ) ).
% of_zero_neq_one_1
thf(fact_1019_of__zero__neq__one__1,axiom,
( ( matrix5487198553176925635_b_nat @ one_one_b )
= one_one_nat ) ).
% of_zero_neq_one_1
thf(fact_1020_of__zero__neq__one__1,axiom,
( ( matrix5484708082667875359_b_int @ one_one_b )
= one_one_int ) ).
% of_zero_neq_one_1
thf(fact_1021_of__zero__neq__one__1,axiom,
( ( matrix8283685725398817569_nat_b @ one_one_nat )
= one_one_b ) ).
% of_zero_neq_one_1
thf(fact_1022_of__zero__neq__one__1,axiom,
( ( matrix700445748609480494at_nat @ one_one_nat )
= one_one_nat ) ).
% of_zero_neq_one_1
thf(fact_1023_of__zero__neq__one__1,axiom,
( ( matrix697955278100430218at_int @ one_one_nat )
= one_one_int ) ).
% of_zero_neq_one_1
thf(fact_1024_of__zero__neq__one__1,axiom,
( ( matrix6038540757728371653_int_b @ one_one_int )
= one_one_b ) ).
% of_zero_neq_one_1
thf(fact_1025_of__zero__neq__one__1,axiom,
( ( matrix1699799460510535050nt_nat @ one_one_int )
= one_one_nat ) ).
% of_zero_neq_one_1
thf(fact_1026_of__zero__neq__one__1,axiom,
( ( matrix1697308990001484774nt_int @ one_one_int )
= one_one_int ) ).
% of_zero_neq_one_1
thf(fact_1027_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix4781043112069605324ne_b_b @ one_one_b )
= one_one_b ) ).
% of_inj_on_01_hom.hom_one
thf(fact_1028_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix5487198553176925635_b_nat @ one_one_b )
= one_one_nat ) ).
% of_inj_on_01_hom.hom_one
thf(fact_1029_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix5484708082667875359_b_int @ one_one_b )
= one_one_int ) ).
% of_inj_on_01_hom.hom_one
thf(fact_1030_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix8283685725398817569_nat_b @ one_one_nat )
= one_one_b ) ).
% of_inj_on_01_hom.hom_one
thf(fact_1031_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix700445748609480494at_nat @ one_one_nat )
= one_one_nat ) ).
% of_inj_on_01_hom.hom_one
thf(fact_1032_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix697955278100430218at_int @ one_one_nat )
= one_one_int ) ).
% of_inj_on_01_hom.hom_one
thf(fact_1033_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix6038540757728371653_int_b @ one_one_int )
= one_one_b ) ).
% of_inj_on_01_hom.hom_one
thf(fact_1034_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix1699799460510535050nt_nat @ one_one_int )
= one_one_nat ) ).
% of_inj_on_01_hom.hom_one
thf(fact_1035_of__inj__on__01__hom_Ohom__one,axiom,
( ( matrix1697308990001484774nt_int @ one_one_int )
= one_one_int ) ).
% of_inj_on_01_hom.hom_one
thf(fact_1036_index__unit__vec_I3_J,axiom,
! [N: nat,I: nat] :
( ( dim_vec_int @ ( unit_vec_int @ N @ I ) )
= N ) ).
% index_unit_vec(3)
thf(fact_1037_index__unit__vec_I3_J,axiom,
! [N: nat,I: nat] :
( ( dim_vec_b @ ( unit_vec_b @ N @ I ) )
= N ) ).
% index_unit_vec(3)
thf(fact_1038_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1039_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1040_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1041_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1042_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1043_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_1044_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1045_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1046_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1047_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_1048_index__minus__vec_I1_J,axiom,
! [I: nat,V_2: vec_int,V_1: vec_int] :
( ( ord_less_nat @ I @ ( dim_vec_int @ V_2 ) )
=> ( ( vec_index_int @ ( minus_minus_vec_int @ V_1 @ V_2 ) @ I )
= ( minus_minus_int @ ( vec_index_int @ V_1 @ I ) @ ( vec_index_int @ V_2 @ I ) ) ) ) ).
% index_minus_vec(1)
thf(fact_1049_index__minus__vec_I1_J,axiom,
! [I: nat,V_2: vec_real,V_1: vec_real] :
( ( ord_less_nat @ I @ ( dim_vec_real @ V_2 ) )
=> ( ( vec_index_real @ ( minus_minus_vec_real @ V_1 @ V_2 ) @ I )
= ( minus_minus_real @ ( vec_index_real @ V_1 @ I ) @ ( vec_index_real @ V_2 @ I ) ) ) ) ).
% index_minus_vec(1)
thf(fact_1050_index__unit__vec_I2_J,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_b @ ( unit_vec_b @ N @ I ) @ I )
= one_one_b ) ) ).
% index_unit_vec(2)
thf(fact_1051_index__unit__vec_I2_J,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_nat @ ( unit_vec_nat @ N @ I ) @ I )
= one_one_nat ) ) ).
% index_unit_vec(2)
thf(fact_1052_index__unit__vec_I2_J,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( vec_index_int @ ( unit_vec_int @ N @ I ) @ I )
= one_one_int ) ) ).
% index_unit_vec(2)
thf(fact_1053_less__imp__diff__less,axiom,
! [J2: nat,K3: nat,N: nat] :
( ( ord_less_nat @ J2 @ K3 )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K3 ) ) ).
% less_imp_diff_less
thf(fact_1054_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1055_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1056_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1057_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_1058_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1059_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1060_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_1061_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_1062_diff__commute,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K3 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K3 ) @ J2 ) ) ).
% diff_commute
thf(fact_1063_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1064_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1065_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1066_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1067_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1068_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1069_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1070_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1071_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1072_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1073_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1074_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1075_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1076_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1077_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_1078_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_1079_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A5: int,B6: int] :
( ( minus_minus_int @ A5 @ B6 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1080_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A5: real,B6: real] :
( ( minus_minus_real @ A5 @ B6 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1081_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1082_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1083_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1084_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1085_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_1086_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1087_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1088_Nat_Odiff__diff__eq,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M2 )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1089_le__diff__iff,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M2 )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1090_eq__diff__iff,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M2 )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ( minus_minus_nat @ M2 @ K3 )
= ( minus_minus_nat @ N @ K3 ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1091_of__zero__hom_Ohom__0,axiom,
! [X: int] :
( ( ( matrix1697308990001484774nt_int @ X )
= zero_zero_int )
=> ( X = zero_zero_int ) ) ).
% of_zero_hom.hom_0
thf(fact_1092_of__zero__hom_Ohom__0,axiom,
! [X: real] :
( ( ( matrix4084289606792104422al_int @ X )
= zero_zero_int )
=> ( X = zero_zero_real ) ) ).
% of_zero_hom.hom_0
thf(fact_1093_of__zero__hom_Ohom__0,axiom,
! [X: b] :
( ( ( matrix2280091663418064671b_real @ X )
= zero_zero_real )
=> ( X = zero_zero_b ) ) ).
% of_zero_hom.hom_0
thf(fact_1094_of__zero__hom_Ohom__0,axiom,
! [X: nat] :
( ( ( matrix8742843541027031818t_real @ X )
= zero_zero_real )
=> ( X = zero_zero_nat ) ) ).
% of_zero_hom.hom_0
thf(fact_1095_of__zero__hom_Ohom__0,axiom,
! [X: int] :
( ( ( matrix1706393078865277798t_real @ X )
= zero_zero_real )
=> ( X = zero_zero_int ) ) ).
% of_zero_hom.hom_0
thf(fact_1096_of__zero__hom_Ohom__0,axiom,
! [X: real] :
( ( ( matrix3070681271257819494l_real @ X )
= zero_zero_real )
=> ( X = zero_zero_real ) ) ).
% of_zero_hom.hom_0
thf(fact_1097_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1098_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_1099_less__diff__iff,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M2 )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1100_map__col__to__block__elem__not,axiom,
! [C: vec_b,I: nat] :
( ( member_vec_b @ C @ ( set_vec_b2 @ ( cols_b @ matrix ) ) )
=> ( ( ord_less_nat @ I @ ( dim_vec_b @ C ) )
=> ( ( ~ ( member_nat @ I @ ( incide5355957740755015149lock_b @ C ) ) )
= ( ( vec_index_b @ C @ I )
= zero_zero_b ) ) ) ) ).
% map_col_to_block_elem_not
thf(fact_1101_M__not__zero__simp,axiom,
! [J2: nat,I: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_b @ matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_b @ matrix ) )
=> ( ( ( index_mat_b @ matrix @ ( product_Pair_nat_nat @ I @ J2 ) )
!= zero_zero_b )
=> ( ( index_mat_b @ matrix @ ( product_Pair_nat_nat @ I @ J2 ) )
= one_one_b ) ) ) ) ).
% M_not_zero_simp
thf(fact_1102_M__not__one__simp,axiom,
! [J2: nat,I: nat] :
( ( ord_less_nat @ J2 @ ( dim_col_b @ matrix ) )
=> ( ( ord_less_nat @ I @ ( dim_row_b @ matrix ) )
=> ( ( ( index_mat_b @ matrix @ ( product_Pair_nat_nat @ I @ J2 ) )
!= one_one_b )
=> ( ( index_mat_b @ matrix @ ( product_Pair_nat_nat @ I @ J2 ) )
= zero_zero_b ) ) ) ) ).
% M_not_one_simp
thf(fact_1103_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1104_int__less__induct,axiom,
! [I: int,K3: int,P: int > $o] :
( ( ord_less_int @ I @ K3 )
=> ( ( P @ ( minus_minus_int @ K3 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K3 )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1105_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1106_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1107_mult__cancel2,axiom,
! [M2: nat,K3: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K3 )
= ( times_times_nat @ N @ K3 ) )
= ( ( M2 = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1108_mult__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K3 @ M2 )
= ( times_times_nat @ K3 @ N ) )
= ( ( M2 = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1109_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1110_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1111_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1112_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1113_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1114_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1115_mult__less__cancel2,axiom,
! [M2: nat,K3: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K3 ) @ ( times_times_nat @ N @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1116_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1117_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1118_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1119_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_1120_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1121_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1122_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_1123_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_1124_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K3 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K3 ) ) ).
% Suc_diff_diff
thf(fact_1125_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1126_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1127_mult__le__cancel2,axiom,
! [M2: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K3 ) @ ( times_times_nat @ N @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1128_one__le__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1129_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1130_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1131_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1132_Suc__mult__less__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K3 ) @ M2 ) @ ( times_times_nat @ ( suc @ K3 ) @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1133_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_1134_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1135_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1136_Nat_OlessE,axiom,
! [I: nat,K3: nat] :
( ( ord_less_nat @ I @ K3 )
=> ( ( K3
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K3
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1137_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_1138_Suc__lessE,axiom,
! [I: nat,K3: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K3 )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K3
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1139_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1140_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_1141_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1142_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1143_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_1144_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1145_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1146_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1147_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_1148_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_1149_less__trans__Suc,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ord_less_nat @ ( suc @ I ) @ K3 ) ) ) ).
% less_trans_Suc
thf(fact_1150_less__Suc__induct,axiom,
! [I: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K )
=> ( P @ I3 @ K ) ) ) ) )
=> ( P @ I @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_1151_strict__inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1152_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1153_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1154_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1155_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M7: nat] :
( N
= ( suc @ M7 ) ) ) ).
% not0_implies_Suc
thf(fact_1156_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_1157_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_1158_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1159_zero__induct,axiom,
! [P: nat > $o,K3: nat] :
( ( P @ K3 )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1160_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] :
( ( P @ X2 @ Y2 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_1161_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1162_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1163_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1164_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1165_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1166_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1167_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N4: nat] :
( ~ ( P @ N4 )
& ( P @ ( suc @ N4 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1168_Suc__mult__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K3 ) @ M2 )
= ( times_times_nat @ ( suc @ K3 ) @ N ) )
= ( M2 = N ) ) ).
% Suc_mult_cancel1
thf(fact_1169_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1170_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1171_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1172_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X2 @ Z3 ) ) )
=> ( ! [N4: nat] : ( R @ N4 @ ( suc @ N4 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1173_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1174_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1175_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1176_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1177_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1178_Suc__le__D,axiom,
! [N: nat,M8: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M8 )
=> ? [M7: nat] :
( M8
= ( suc @ M7 ) ) ) ).
% Suc_le_D
thf(fact_1179_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1180_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1181_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_1182_Suc__mult__le__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K3 ) @ M2 ) @ ( times_times_nat @ ( suc @ K3 ) @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1183_diff__mult__distrib2,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( times_times_nat @ K3 @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1184_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K3 )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K3 ) @ ( times_times_nat @ N @ K3 ) ) ) ).
% diff_mult_distrib
thf(fact_1185_mult__le__mono2,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1186_mult__le__mono1,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J2 @ K3 ) ) ) ).
% mult_le_mono1
thf(fact_1187_mult__le__mono,axiom,
! [I: nat,J2: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1188_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1189_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1190_zero__induct__lemma,axiom,
! [P: nat > $o,K3: nat,I: nat] :
( ( P @ K3 )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K3 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1191_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1192_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M7: nat] :
( N
= ( suc @ M7 ) ) ) ).
% gr0_implies_Suc
thf(fact_1193_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J: nat] :
( ( M2
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1194_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_1195_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_1196_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_1197_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1198_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1199_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1200_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1201_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_1202_inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ J2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J2 )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1203_dec__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ I )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J2 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_1204_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1205_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1206_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1207_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1208_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M7: nat] :
( ( ord_less_nat @ zero_zero_nat @ M7 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M7 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1209_mult__less__mono1,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J2 @ K3 ) ) ) ) ).
% mult_less_mono1
thf(fact_1210_mult__less__mono2,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1211_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1212_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1213_unit__vecs__first_Ocases,axiom,
! [X: product_prod_nat_nat] :
( ! [N4: nat] :
( X
!= ( product_Pair_nat_nat @ N4 @ zero_zero_nat ) )
=> ~ ! [N4: nat,I3: nat] :
( X
!= ( product_Pair_nat_nat @ N4 @ ( suc @ I3 ) ) ) ) ).
% unit_vecs_first.cases
thf(fact_1214_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1215_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1216_zmult__zless__mono2,axiom,
! [I: int,J2: int,K3: int] :
( ( ord_less_int @ I @ J2 )
=> ( ( ord_less_int @ zero_zero_int @ K3 )
=> ( ord_less_int @ ( times_times_int @ K3 @ I ) @ ( times_times_int @ K3 @ J2 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1217_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K )
=> ~ ( P @ I2 ) )
& ( P @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1218_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1219_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1220_pos__zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1221_plusinfinity,axiom,
! [D: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int,K: int] :
( ( P4 @ X2 )
= ( P4 @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D ) ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1222_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int,K: int] :
( ( P1 @ X2 )
= ( P1 @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D ) ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P1 @ X2 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1223_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1224_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1225_zmult__zless__mono2__lemma,axiom,
! [I: int,J2: int,K3: nat] :
( ( ord_less_int @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K3 ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K3 ) @ J2 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1226_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K3: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K3 )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1227_nat__mult__le__cancel__disj,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1228_nat__mult__less__cancel__disj,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1229_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N4: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1230_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1231_nat__mult__eq__cancel__disj,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K3 @ M2 )
= ( times_times_nat @ K3 @ N ) )
= ( ( K3 = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1232_nat__mult__less__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1233_nat__mult__eq__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ( times_times_nat @ K3 @ M2 )
= ( times_times_nat @ K3 @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1234_nat__mult__le__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1235_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1236_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A4: real,B5: real,C4: real] :
( ( P @ A4 @ B5 )
=> ( ( P @ B5 @ C4 )
=> ( ( ord_less_eq_real @ A4 @ B5 )
=> ( ( ord_less_eq_real @ B5 @ C4 )
=> ( P @ A4 @ C4 ) ) ) ) )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A4: real,B5: real] :
( ( ( ord_less_eq_real @ A4 @ X2 )
& ( ord_less_eq_real @ X2 @ B5 )
& ( ord_less_real @ ( minus_minus_real @ B5 @ A4 ) @ D3 ) )
=> ( P @ A4 @ B5 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1237_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N4: nat] :
( X
!= ( suc @ N4 ) ) ) ).
% list_decode.cases
thf(fact_1238_le__Suc__eq_H,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ ( suc @ Y ) )
= ( ( X = zero_zero_nat )
| ? [X7: nat] :
( ( X
= ( suc @ X7 ) )
& ( ord_less_eq_nat @ X7 @ Y ) ) ) ) ).
% le_Suc_eq'
thf(fact_1239_ex__leq__Suc,axiom,
! [J2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ ( suc @ J2 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% ex_leq_Suc
thf(fact_1240_ex__less__Suc,axiom,
! [J2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ J2 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ J2 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% ex_less_Suc
thf(fact_1241_inf__concat__simple_Ocases,axiom,
! [X: produc8199716216217303280at_nat] :
( ! [F2: nat > nat] :
( X
!= ( produc72220940542539688at_nat @ F2 @ zero_zero_nat ) )
=> ~ ! [F2: nat > nat,N4: nat] :
( X
!= ( produc72220940542539688at_nat @ F2 @ ( suc @ N4 ) ) ) ) ).
% inf_concat_simple.cases
thf(fact_1242_adjust__idx__rev__def,axiom,
( missin3815256168798769645dx_rev
= ( ^ [I4: nat,J: nat] : ( if_nat @ ( ord_less_nat @ J @ I4 ) @ J @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% adjust_idx_rev_def
thf(fact_1243_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K2: nat,M4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M4 @ K2 ) @ ( product_Pair_nat_nat @ M4 @ ( minus_minus_nat @ K2 @ M4 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M4 @ ( suc @ K2 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_1244_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_1245_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K3: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K3 )
=> ( ( ord_less_eq_int @ K3 @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K3 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1246_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1247_nat__intermed__int__val,axiom,
! [M2: nat,N: nat,F: nat > int,K3: int] :
( ! [I3: nat] :
( ( ( ord_less_eq_nat @ M2 @ I3 )
& ( ord_less_nat @ I3 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_int @ ( F @ M2 ) @ K3 )
=> ( ( ord_less_eq_int @ K3 @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ M2 @ I3 )
& ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K3 ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1248_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K3: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K3 )
=> ( ( ord_less_eq_int @ K3 @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K3 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1249_nat__add__left__cancel__less,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K3 @ M2 ) @ ( plus_plus_nat @ K3 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1250_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1251_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1252_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_1253_nat__add__left__cancel__le,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K3 @ M2 ) @ ( plus_plus_nat @ K3 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1254_diff__diff__left,axiom,
! [I: nat,J2: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K3 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J2 @ K3 ) ) ) ).
% diff_diff_left
thf(fact_1255_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1256_mult__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ M2 @ ( suc @ N ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc_right
thf(fact_1257_Nat_Odiff__diff__right,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J2 @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K3 ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1258_Nat_Oadd__diff__assoc2,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K3 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K3 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1259_Nat_Oadd__diff__assoc,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K3 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1260_diff__Suc__diff__eq1,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J2 @ K3 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K3 ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1261_diff__Suc__diff__eq2,axiom,
! [K3: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K3 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K3 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
% Helper facts (5)
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 ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_b @ one_one_b @ ( vec_set_b2 @ ( col_b @ matrix @ j ) ) ).
%------------------------------------------------------------------------------