TPTP Problem File: SLH0768^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Fishers_Inequality/0034_Incidence_Matrices/prob_01653_071479__28061448_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1480 ( 682 unt; 206 typ; 0 def)
% Number of atoms : 3175 (1203 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 10223 ( 193 ~; 60 |; 155 &;8770 @)
% ( 0 <=>;1045 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 30 ( 29 usr)
% Number of type conns : 295 ( 295 >; 0 *; 0 +; 0 <<)
% Number of symbols : 180 ( 177 usr; 23 con; 0-4 aty)
% Number of variables : 2718 ( 55 ^;2645 !; 18 ?;2718 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:48:38.115
%------------------------------------------------------------------------------
% Could-be-implicit typings (29)
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
set_list_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
list_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Real__Oreal_J_J,type,
set_mat_real: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Omat_It__Int__Oint_J_J,type,
list_mat_int: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Nat__Onat_J_J,type,
set_mat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Int__Oint_J_J,type,
set_mat_int: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
multiset_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
list_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Int__Oint_J,type,
multiset_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Matrix__Omat_It__Real__Oreal_J,type,
mat_real: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
multiset_a: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Int__Oint_J,type,
mat_int: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__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__a,type,
a: $tType ).
% Explicit typings (177)
thf(sy_c_Block__Designs_Oconstant__rep__design_001tf__a,type,
block_6028206285060069402sign_a: set_a > multiset_set_a > nat > $o ).
thf(sy_c_Block__Designs_Opairwise__balance_001tf__a,type,
block_5355636846524985331ance_a: set_a > multiset_set_a > nat > $o ).
thf(sy_c_Design__Basics_Odesign_001tf__a,type,
design_design_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Basics_Oincidence__system_Oblock__complement_001tf__a,type,
design6447616907850319326ment_a: set_a > set_a > set_a ).
thf(sy_c_Design__Basics_Oincidence__system_Ocomplement__blocks_001tf__a,type,
design8640656491286871389ocks_a: set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Design__Basics_Oincidence__system_Odesign__support_001tf__a,type,
design5397942185814921632port_a: multiset_set_a > set_set_a ).
thf(sy_c_Design__Basics_Oincidence__system_Oincident_001tf__a,type,
design3210447939978979927dent_a: multiset_set_a > a > set_a > $o ).
thf(sy_c_Design__Basics_Oincidence__system_Oreplication__numbers_001tf__a,type,
design8835372594653258411bers_a: set_a > multiset_set_a > set_nat ).
thf(sy_c_Design__Basics_Oincidence__system_Osys__block__sizes_001tf__a,type,
design1769254222028858111izes_a: multiset_set_a > set_nat ).
thf(sy_c_Design__Basics_On__intersect__number_001tf__a,type,
design735257067508376852mber_a: set_a > nat > set_a > nat ).
thf(sy_c_Design__Basics_Oproper__design_001tf__a,type,
design7287791228148780576sign_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Basics_Osimple__incidence__system_001tf__a,type,
design1338723777345758283stem_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Extras_Oregular__pairwise__balance_001tf__a,type,
design8863377621459125358ance_a: set_a > multiset_set_a > nat > nat > $o ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__block_001tf__a,type,
design4001997691126659652lock_a: multiset_set_a > set_a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point_001tf__a,type,
design2964366272795260673oint_a: set_a > a > set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point__to__blocks_001tf__a,type,
design2935547469388721088ocks_a: multiset_set_a > a > set_set_a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__block_001tf__a,type,
design1146539425385464078lock_a: multiset_set_a > set_a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__point_001tf__a,type,
design108908007054065099oint_a: set_a > a > set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__point__blocks_001tf__a,type,
design6411949732824333445ocks_a: multiset_set_a > a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Ostr__del__block_001tf__a,type,
design4241783006516448631lock_a: multiset_set_a > set_a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Ostr__del__point__blocks_001tf__a,type,
design5657747894866638574ocks_a: multiset_set_a > a > multiset_set_a ).
thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
finite_card_int: set_int > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_Itf__a_J,type,
finite_card_list_a: set_list_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Matrix__Omat_It__Int__Oint_J,type,
finite_card_mat_int: set_mat_int > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Int__Oint_J,type,
minus_minus_mat_int: mat_int > mat_int > mat_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
minus_706656509937749387_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Matrix__Omat_It__Int__Oint_J_J,type,
minus_7323808874771660652at_int: set_mat_int > set_mat_int > set_mat_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
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_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Int__Oint_J,type,
plus_plus_mat_int: mat_int > mat_int > mat_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Nat__Onat_J,type,
plus_plus_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Real__Oreal_J,type,
plus_plus_mat_real: mat_real > mat_real > mat_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
plus_p2331992037799027419_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Int__Oint_J,type,
times_times_mat_int: mat_int > mat_int > mat_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__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__Multiset__Omultiset_It__Int__Oint_J,type,
zero_z3170743180189231877et_int: multiset_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
zero_z7348594199698428585et_nat: multiset_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
zero_z5079479921072680283_set_a: multiset_set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
zero_zero_multiset_a: multiset_a ).
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_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Incidence__Matrices_Oinc__mat__of_001t__Set__Oset_Itf__a_J_001t__Int__Oint,type,
incide2330559700202178271_a_int: list_set_a > list_set_set_a > mat_int ).
thf(sy_c_Incidence__Matrices_Oinc__mat__of_001tf__a_001t__Int__Oint,type,
incide7016682120514301311_a_int: list_a > list_set_a > mat_int ).
thf(sy_c_Incidence__Matrices_Ois__incidence__matrix_001t__Int__Oint_001tf__a,type,
incide3487307869859529607_int_a: mat_int > set_a > multiset_set_a > $o ).
thf(sy_c_Incidence__Matrices_Omat__point__index_001t__Int__Oint,type,
incide1709061621920695677ex_int: mat_int > set_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_Onon__empty__col_001t__Int__Oint,type,
incide6851923868969248411ol_int: mat_int > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__constant__rep_001tf__a,type,
incide6922509864216205631_rep_a: list_a > list_set_a > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__design_001tf__a,type,
incide2848671379600480836sign_a: list_a > list_set_a > $o ).
thf(sy_c_Incidence__Matrices_Oordered__pairwise__balance_001tf__a,type,
incide6880889959311561818ance_a: list_a > list_set_a > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__proper__design_001tf__a,type,
incide3676903341588786676sign_a: list_a > list_set_a > $o ).
thf(sy_c_Incidence__Matrices_Oordered__regular__pairwise__balance_001tf__a,type,
incide5282158983398259552ance_a: list_a > list_set_a > nat > nat > $o ).
thf(sy_c_Incidence__Matrices_Oproper__inc__mat_001t__Int__Oint,type,
incide294466202882093137at_int: mat_int > $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__ring__1_001t__Int__Oint,type,
incide6080938071136783841_1_int: mat_int > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Omat_It__Int__Oint_J,type,
set_mat_int2: list_mat_int > set_mat_int ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_Itf__a_J,type,
set_set_a2: list_set_a > set_set_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Int__Oint,type,
carrier_mat_int: nat > nat > set_mat_int ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Nat__Onat,type,
carrier_mat_nat: nat > nat > set_mat_nat ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Real__Oreal,type,
carrier_mat_real: nat > nat > set_mat_real ).
thf(sy_c_Matrix_Odim__col_001t__Int__Oint,type,
dim_col_int: mat_int > nat ).
thf(sy_c_Matrix_Odim__row_001t__Int__Oint,type,
dim_row_int: mat_int > 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__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_Oinverts__mat_001t__Int__Oint,type,
inverts_mat_int: mat_int > mat_int > $o ).
thf(sy_c_Matrix_Oone__mat_001t__Int__Oint,type,
one_mat_int: nat > mat_int ).
thf(sy_c_Matrix_Osmult__mat_001t__Int__Oint,type,
smult_mat_int: int > mat_int > mat_int ).
thf(sy_c_Matrix_Osmult__mat_001t__Nat__Onat,type,
smult_mat_nat: nat > mat_nat > mat_nat ).
thf(sy_c_Matrix_Osmult__mat_001t__Real__Oreal,type,
smult_mat_real: real > mat_real > mat_real ).
thf(sy_c_Matrix_Otranspose__mat_001t__Int__Oint,type,
transpose_mat_int: mat_int > mat_int ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__mat_001t__Int__Oint,type,
matrix8485685120660989714at_int: nat > mat_int ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__mat_001t__Int__Oint_001t__Int__Oint,type,
matrix323868623736973467nt_int: mat_int > mat_int ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__mat_001t__Int__Oint_001t__Nat__Onat,type,
matrix326359094246023743nt_nat: mat_int > mat_nat ).
thf(sy_c_Matrix__Vector__Extras_Olift__01__mat_001t__Int__Oint_001t__Real__Oreal,type,
matrix1811533390045330843t_real: mat_int > mat_real ).
thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
mset_nat: list_nat > multiset_nat ).
thf(sy_c_Multiset_Omset_001t__Set__Oset_Itf__a_J,type,
mset_set_a: list_set_a > multiset_set_a ).
thf(sy_c_Multiset_Omset_001tf__a,type,
mset_a: list_a > multiset_a ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Set__Oset_Itf__a_J,type,
count_set_a: multiset_set_a > set_a > nat ).
thf(sy_c_Multiset_Orepeat__mset_001t__Set__Oset_Itf__a_J,type,
repeat_mset_set_a: nat > multiset_set_a > multiset_set_a ).
thf(sy_c_Multiset_Oreplicate__mset_001t__Set__Oset_Itf__a_J,type,
replicate_mset_set_a: nat > set_a > multiset_set_a ).
thf(sy_c_Multiset_Oset__mset_001t__Int__Oint,type,
set_mset_int: multiset_int > set_int ).
thf(sy_c_Multiset_Oset__mset_001t__Nat__Onat,type,
set_mset_nat: multiset_nat > set_nat ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_Itf__a_J,type,
set_mset_set_a: multiset_set_a > set_set_a ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_a ).
thf(sy_c_Multiset_Osubset__mset_001t__Set__Oset_Itf__a_J,type,
subset_mset_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Multiset__Permutations_Opermutations__of__set_001t__Nat__Onat,type,
multis1655833086286526861et_nat: set_nat > set_list_nat ).
thf(sy_c_Multiset__Permutations_Opermutations__of__set_001t__Set__Oset_Itf__a_J,type,
multis2257881577744371681_set_a: set_set_a > set_list_set_a ).
thf(sy_c_Multiset__Permutations_Opermutations__of__set_001tf__a,type,
multis2428024204330136193_set_a: set_a > set_list_a ).
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_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Omat_It__Int__Oint_J_J,type,
size_s6799427505661098007at_int: list_mat_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
size_size_list_set_a: list_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
size_s6566526139600085008_set_a: multiset_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
size_size_multiset_a: multiset_a > 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__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Matrix__Omat_It__Int__Oint_J_J,type,
bot_bot_set_mat_int: set_mat_int ).
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_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
ord_le5765082015083327056_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
ord_le7905258569527593284_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Omat_It__Int__Oint_J_J,type,
ord_le5299038897506728741at_int: set_mat_int > set_mat_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_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_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_It__Int__Oint_J,type,
collect_mat_int: ( mat_int > $o ) > set_mat_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Matrix__Omat_It__Int__Oint_J,type,
insert_mat_int: mat_int > set_mat_int > set_mat_int ).
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_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Sub__Designs_Osub__design_001tf__a,type,
sub_sub_design_a: set_a > multiset_set_a > set_a > multiset_set_a > $o ).
thf(sy_c_Sub__Designs_Osub__incidence__system_001tf__a,type,
sub_su7923802003039619913stem_a: set_a > multiset_set_a > set_a > multiset_set_a > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
member_list_set_a: list_set_a > set_list_set_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Int__Oint_J,type,
member_mat_int: mat_int > set_mat_int > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Nat__Onat_J,type,
member_mat_nat: mat_nat > set_mat_nat > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Real__Oreal_J,type,
member_mat_real: mat_real > set_mat_real > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v__092_060B_062s,type,
b_s: list_set_a ).
thf(sy_v__092_060Lambda_062,type,
lambda: nat ).
thf(sy_v__092_060V_062s,type,
v_s: list_a ).
thf(sy_v__092_060r_062,type,
r: nat ).
% Relevant facts (1270)
thf(fact_0_ordered__regular__pairwise__balance__axioms,axiom,
incide5282158983398259552ance_a @ v_s @ b_s @ lambda @ r ).
% ordered_regular_pairwise_balance_axioms
thf(fact_1_mat__is__proper,axiom,
incide294466202882093137at_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ).
% mat_is_proper
thf(fact_2_ordered__constant__rep__axioms,axiom,
incide6922509864216205631_rep_a @ v_s @ b_s @ r ).
% ordered_constant_rep_axioms
thf(fact_3_zero__one__matrix__ring__1__axioms,axiom,
incide6080938071136783841_1_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ).
% zero_one_matrix_ring_1_axioms
thf(fact_4_ordered__pairwise__balance__axioms,axiom,
incide6880889959311561818ance_a @ v_s @ b_s @ lambda ).
% ordered_pairwise_balance_axioms
thf(fact_5_ordered__design__axioms,axiom,
incide2848671379600480836sign_a @ v_s @ b_s ).
% ordered_design_axioms
thf(fact_6_ordered__proper__design__axioms,axiom,
incide3676903341588786676sign_a @ v_s @ b_s ).
% ordered_proper_design_axioms
thf(fact_7_transpose__N__mult__dim_I2_J,axiom,
( ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% transpose_N_mult_dim(2)
thf(fact_8_transpose__N__mult__dim_I1_J,axiom,
( ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% transpose_N_mult_dim(1)
thf(fact_9_transpose__one,axiom,
! [N: nat] :
( ( transpose_mat_int @ ( one_mat_int @ N ) )
= ( one_mat_int @ N ) ) ).
% transpose_one
thf(fact_10_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_11_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_12_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_13_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_14_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_15_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_16_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_17_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_18_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_19_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_20_add__diff__cancel__left,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
= ( minus_706656509937749387_set_a @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_21_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_22_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_23_add__diff__cancel__left_H,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_24_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_25_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_26_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_27_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_28_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_29_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_30_index__minus__mat_I3_J,axiom,
! [A2: mat_int,B2: mat_int] :
( ( dim_col_int @ ( minus_minus_mat_int @ A2 @ B2 ) )
= ( dim_col_int @ B2 ) ) ).
% index_minus_mat(3)
thf(fact_31_index__minus__mat_I2_J,axiom,
! [A2: mat_int,B2: mat_int] :
( ( dim_row_int @ ( minus_minus_mat_int @ A2 @ B2 ) )
= ( dim_row_int @ B2 ) ) ).
% index_minus_mat(2)
thf(fact_32_Matrix_Otranspose__transpose,axiom,
! [A2: mat_int] :
( ( transpose_mat_int @ ( transpose_mat_int @ A2 ) )
= A2 ) ).
% Matrix.transpose_transpose
thf(fact_33_transpose__mat__eq,axiom,
! [A2: mat_int,B2: mat_int] :
( ( ( transpose_mat_int @ A2 )
= ( transpose_mat_int @ B2 ) )
= ( A2 = B2 ) ) ).
% transpose_mat_eq
thf(fact_34_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_35_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_36_add__diff__cancel__right_H,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_37_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_38_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_39_add__diff__cancel__right,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
= ( minus_706656509937749387_set_a @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_40_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_41_index__mult__mat_I3_J,axiom,
! [A2: mat_int,B2: mat_int] :
( ( dim_col_int @ ( times_times_mat_int @ A2 @ B2 ) )
= ( dim_col_int @ B2 ) ) ).
% index_mult_mat(3)
thf(fact_42_index__mult__mat_I2_J,axiom,
! [A2: mat_int,B2: mat_int] :
( ( dim_row_int @ ( times_times_mat_int @ A2 @ B2 ) )
= ( dim_row_int @ A2 ) ) ).
% index_mult_mat(2)
thf(fact_43_index__one__mat_I3_J,axiom,
! [N: nat] :
( ( dim_col_int @ ( one_mat_int @ N ) )
= N ) ).
% index_one_mat(3)
thf(fact_44_index__one__mat_I2_J,axiom,
! [N: nat] :
( ( dim_row_int @ ( one_mat_int @ N ) )
= N ) ).
% index_one_mat(2)
thf(fact_45_index__add__mat_I3_J,axiom,
! [A2: mat_int,B2: mat_int] :
( ( dim_col_int @ ( plus_plus_mat_int @ A2 @ B2 ) )
= ( dim_col_int @ B2 ) ) ).
% index_add_mat(3)
thf(fact_46_index__add__mat_I2_J,axiom,
! [A2: mat_int,B2: mat_int] :
( ( dim_row_int @ ( plus_plus_mat_int @ A2 @ B2 ) )
= ( dim_row_int @ B2 ) ) ).
% index_add_mat(2)
thf(fact_47_index__smult__mat_I3_J,axiom,
! [A: int,A2: mat_int] :
( ( dim_col_int @ ( smult_mat_int @ A @ A2 ) )
= ( dim_col_int @ A2 ) ) ).
% index_smult_mat(3)
thf(fact_48_index__smult__mat_I2_J,axiom,
! [A: int,A2: mat_int] :
( ( dim_row_int @ ( smult_mat_int @ A @ A2 ) )
= ( dim_row_int @ A2 ) ) ).
% index_smult_mat(2)
thf(fact_49_index__transpose__mat_I3_J,axiom,
! [A2: mat_int] :
( ( dim_col_int @ ( transpose_mat_int @ A2 ) )
= ( dim_row_int @ A2 ) ) ).
% index_transpose_mat(3)
thf(fact_50_index__transpose__mat_I2_J,axiom,
! [A2: mat_int] :
( ( dim_row_int @ ( transpose_mat_int @ A2 ) )
= ( dim_col_int @ A2 ) ) ).
% index_transpose_mat(2)
thf(fact_51_right__mult__one__mat_H,axiom,
! [A2: mat_int,N: nat] :
( ( ( dim_col_int @ A2 )
= N )
=> ( ( times_times_mat_int @ A2 @ ( one_mat_int @ N ) )
= A2 ) ) ).
% right_mult_one_mat'
thf(fact_52_left__mult__one__mat_H,axiom,
! [A2: mat_int,N: nat] :
( ( ( dim_row_int @ A2 )
= N )
=> ( ( times_times_mat_int @ ( one_mat_int @ N ) @ A2 )
= A2 ) ) ).
% left_mult_one_mat'
thf(fact_53_dim__row__is__v,axiom,
( ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% dim_row_is_v
thf(fact_54_ordered__constant__rep_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a,R: nat] :
( ( incide6922509864216205631_rep_a @ V_s @ B_s @ R )
=> ( incide3676903341588786676sign_a @ V_s @ B_s ) ) ).
% ordered_constant_rep.axioms(1)
thf(fact_55_ordered__proper__design_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide3676903341588786676sign_a @ V_s @ B_s )
=> ( incide2848671379600480836sign_a @ V_s @ B_s ) ) ).
% ordered_proper_design.axioms(1)
thf(fact_56_ordered__regular__pairwise__balance_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a,Lambda: nat,R: nat] :
( ( incide5282158983398259552ance_a @ V_s @ B_s @ Lambda @ R )
=> ( incide6880889959311561818ance_a @ V_s @ B_s @ Lambda ) ) ).
% ordered_regular_pairwise_balance.axioms(1)
thf(fact_57_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_58_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_59_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_60_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A: mat_int,P: mat_int > $o] :
( ( member_mat_int @ A @ ( collect_mat_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( member_list_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A2: set_mat_int] :
( ( collect_mat_int
@ ^ [X: mat_int] : ( member_mat_int @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_71_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_72_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_73_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_74_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_75_ordered__proper__design_Omat__is__proper,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide3676903341588786676sign_a @ V_s @ B_s )
=> ( incide294466202882093137at_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ).
% ordered_proper_design.mat_is_proper
thf(fact_76_inc__mat__of__01__mat,axiom,
! [Vs: list_a,Bs: list_set_a] : ( incide6080938071136783841_1_int @ ( incide7016682120514301311_a_int @ Vs @ Bs ) ) ).
% inc_mat_of_01_mat
thf(fact_77_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_78_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_79_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_80_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_81_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_82_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_83_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_84_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_85_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_86_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_87_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_88_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_89_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_90_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_91_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_92_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_93_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_94_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_95_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_96_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_97_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_98_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_99_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_100_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_101_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_102_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_103_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_104_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_105_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_106_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ A @ C ) @ B )
= ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_107_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_108_diff__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_109_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_110_diff__diff__eq,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ A @ B ) @ C )
= ( minus_706656509937749387_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_111_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( ( plus_plus_real @ C @ B )
= A )
=> ( C
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_112_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_113_add__implies__diff,axiom,
! [C: multiset_set_a,B: multiset_set_a,A: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ C @ B )
= A )
=> ( C
= ( minus_706656509937749387_set_a @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_114_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_115_diff__add__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_116_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_117_add__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_118_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_119_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= C )
= ( A
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_120_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_121_mult__of__nat__commute,axiom,
! [X2: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_122_mult__of__nat__commute,axiom,
! [X2: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_123_mult__of__nat__commute,axiom,
! [X2: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_124_all__ones__mat__dim__row,axiom,
! [N: nat] :
( ( dim_row_int @ ( matrix8485685120660989714at_int @ N ) )
= N ) ).
% all_ones_mat_dim_row
thf(fact_125_all__ones__mat__dim__col,axiom,
! [N: nat] :
( ( dim_col_int @ ( matrix8485685120660989714at_int @ N ) )
= N ) ).
% all_ones_mat_dim_col
thf(fact_126_zero__one__matrix__axioms,axiom,
incide4964164200581851450ix_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ).
% zero_one_matrix_axioms
thf(fact_127_del__invalid__point,axiom,
! [P2: a] :
( ~ ( member_a @ P2 @ ( set_a2 @ v_s ) )
=> ( ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P2 )
= ( set_a2 @ v_s ) ) ) ).
% del_invalid_point
thf(fact_128_reg__index__lt__rep,axiom,
ord_less_eq_nat @ lambda @ r ).
% reg_index_lt_rep
thf(fact_129_points__indexing,axiom,
member_list_a @ v_s @ ( multis2428024204330136193_set_a @ ( set_a2 @ v_s ) ) ).
% points_indexing
thf(fact_130_inverts__mat__def,axiom,
( inverts_mat_int
= ( ^ [A4: mat_int,B4: mat_int] :
( ( times_times_mat_int @ A4 @ B4 )
= ( one_mat_int @ ( dim_row_int @ A4 ) ) ) ) ) ).
% inverts_mat_def
thf(fact_131_mult__diff__mult,axiom,
! [X2: real,Y: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ Y ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X2 @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X2 @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_132_square__diff__square__factored,axiom,
! [X2: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X2 @ Y ) @ ( minus_minus_real @ X2 @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_133_eq__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_134_eq__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_135_points__list__length,axiom,
( ( size_size_list_a @ v_s )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% points_list_length
thf(fact_136_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_137_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_138_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_139_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_140_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_141_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_142_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_143_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_144_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_145_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_146_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_147_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_148_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_149_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_150_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_151_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_152_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_153_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_154_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_155_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_156_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_157_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_158_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_159_size__neq__size__imp__neq,axiom,
! [X2: multiset_set_a,Y: multiset_set_a] :
( ( ( size_s6566526139600085008_set_a @ X2 )
!= ( size_s6566526139600085008_set_a @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_160_size__neq__size__imp__neq,axiom,
! [X2: list_set_a,Y: list_set_a] :
( ( ( size_size_list_set_a @ X2 )
!= ( size_size_list_set_a @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_161_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_162_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_163_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_164_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_165_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_166_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_167_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_168_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_169_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_170_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_171_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_172_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_173_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_174_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_175_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_176_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_177_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_178_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_179_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_180_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_181_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_182_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_183_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_184_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_185_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_186_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_187_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_188_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_189_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_190_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_191_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_192_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_193_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_194_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_195_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_196_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_197_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_198_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_199_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_200_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_201_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_202_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_203_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_204_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_205_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_206_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_207_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_208_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_209_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_210_add__le__imp__le__diff,axiom,
! [I: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_211_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_212_add__le__add__imp__diff__le,axiom,
! [I: real,K: real,N: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_213_zero__one__matrix__ring__1__def,axiom,
incide6080938071136783841_1_int = incide4964164200581851450ix_int ).
% zero_one_matrix_ring_1_def
thf(fact_214_zero__one__matrix__ring__1_Oaxioms,axiom,
! [M3: mat_int] :
( ( incide6080938071136783841_1_int @ M3 )
=> ( incide4964164200581851450ix_int @ M3 ) ) ).
% zero_one_matrix_ring_1.axioms
thf(fact_215_zero__one__matrix__ring__1_Ointro,axiom,
! [M3: mat_int] :
( ( incide4964164200581851450ix_int @ M3 )
=> ( incide6080938071136783841_1_int @ M3 ) ) ).
% zero_one_matrix_ring_1.intro
thf(fact_216_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_217_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_218_diff__le__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_219_le__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_220_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_221_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_222_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_223_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_224_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_225_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_226_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_227_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_228_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_229_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_230_inc__mat__dim__row,axiom,
! [Vs: list_set_a,Bs: list_set_set_a] :
( ( dim_row_int @ ( incide2330559700202178271_a_int @ Vs @ Bs ) )
= ( size_size_list_set_a @ Vs ) ) ).
% inc_mat_dim_row
thf(fact_231_inc__mat__dim__row,axiom,
! [Vs: list_a,Bs: list_set_a] :
( ( dim_row_int @ ( incide7016682120514301311_a_int @ Vs @ Bs ) )
= ( size_size_list_a @ Vs ) ) ).
% inc_mat_dim_row
thf(fact_232_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_233_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_234_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_235_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_236_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_237_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_238_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_239_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_240_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_241_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_242_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_243_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_244_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_245_distrib__left,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_246_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_247_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_248_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_249_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_250_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_251_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_252_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_253_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_254_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_255_right__diff__distrib_H,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_256_add__diff__add,axiom,
! [A: real,C: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_257_lift__mat__is__0__1,axiom,
incide4964164200581851450ix_int @ ( matrix323868623736973467nt_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ).
% lift_mat_is_0_1
thf(fact_258_add__delete__point__inv,axiom,
! [P2: a] :
( ~ ( member_a @ P2 @ ( set_a2 @ v_s ) )
=> ( ( design108908007054065099oint_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P2 ) @ P2 )
= ( set_a2 @ v_s ) ) ) ).
% add_delete_point_inv
thf(fact_259_del__point__order,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( set_a2 @ v_s ) )
=> ( ( finite_card_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P2 ) )
= ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) ) ).
% del_point_order
thf(fact_260_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_261_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_262_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_263_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_264_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_265_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_266_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_267_card__length,axiom,
! [Xs: list_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( set_a2 @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% card_length
thf(fact_268_card__length,axiom,
! [Xs: list_set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( set_set_a2 @ Xs ) ) @ ( size_size_list_set_a @ Xs ) ) ).
% card_length
thf(fact_269_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_270_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_271_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_272_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_273_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_274_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_275_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_276_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_277_lift__01__mat__simp_I2_J,axiom,
! [M3: mat_int] :
( ( dim_col_int @ ( matrix323868623736973467nt_int @ M3 ) )
= ( dim_col_int @ M3 ) ) ).
% lift_01_mat_simp(2)
thf(fact_278_lift__01__mat__simp_I1_J,axiom,
! [M3: mat_int] :
( ( dim_row_int @ ( matrix323868623736973467nt_int @ M3 ) )
= ( dim_row_int @ M3 ) ) ).
% lift_01_mat_simp(1)
thf(fact_279_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_280_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_281_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_282_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_283_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_284_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_285_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_286_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_287_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_288_add__existing__point,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( set_a2 @ v_s ) )
=> ( ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P2 )
= ( set_a2 @ v_s ) ) ) ).
% add_existing_point
thf(fact_289_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_290_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_291_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_292_subset__code_I1_J,axiom,
! [Xs: list_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B2 )
= ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_293_subset__code_I1_J,axiom,
! [Xs: list_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B2 )
= ( ! [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_294_subset__code_I1_J,axiom,
! [Xs: list_mat_int,B2: set_mat_int] :
( ( ord_le5299038897506728741at_int @ ( set_mat_int2 @ Xs ) @ B2 )
= ( ! [X: mat_int] :
( ( member_mat_int @ X @ ( set_mat_int2 @ Xs ) )
=> ( member_mat_int @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_295_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_296_subset__code_I1_J,axiom,
! [Xs: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B2 )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( member_a @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_297_subset__code_I1_J,axiom,
! [Xs: list_int,B2: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
= ( ! [X: int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( member_int @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_298_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_299_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_300_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_301_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_302_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_303_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_304_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_305_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_306_inc__mat__dim__col,axiom,
! [Vs: list_a,Bs: list_set_a] :
( ( dim_col_int @ ( incide7016682120514301311_a_int @ Vs @ Bs ) )
= ( size_size_list_set_a @ Bs ) ) ).
% inc_mat_dim_col
thf(fact_307_zero__one__matrix_Olift__mat__is__0__1,axiom,
! [Matrix: mat_int] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( incide4964164200581851450ix_int @ ( matrix323868623736973467nt_int @ Matrix ) ) ) ).
% zero_one_matrix.lift_mat_is_0_1
thf(fact_308_square__diff__one__factored,axiom,
! [X2: int] :
( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X2 @ one_one_int ) @ ( minus_minus_int @ X2 @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_309_square__diff__one__factored,axiom,
! [X2: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X2 @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_310_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_311_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_set_a] :
( ( size_size_list_set_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_312_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_313_neq__if__length__neq,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
!= ( size_size_list_set_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_314_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_315_ideal_Oscale__one,axiom,
! [X2: int] :
( ( times_times_int @ one_one_int @ X2 )
= X2 ) ).
% ideal.scale_one
thf(fact_316_ideal_Oscale__one,axiom,
! [X2: real] :
( ( times_times_real @ one_one_real @ X2 )
= X2 ) ).
% ideal.scale_one
thf(fact_317_block__complement__size,axiom,
! [B: set_a] :
( ( ord_less_eq_set_a @ B @ ( set_a2 @ v_s ) )
=> ( ( finite_card_a @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ ( finite_card_a @ B ) ) ) ) ).
% block_complement_size
thf(fact_318_r__lt__eq__b,axiom,
ord_less_eq_nat @ r @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).
% r_lt_eq_b
thf(fact_319_b__gt__index,axiom,
ord_less_eq_nat @ lambda @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).
% b_gt_index
thf(fact_320_length__finite__permutations__of__set,axiom,
! [Xs: list_nat,A2: set_nat] :
( ( member_list_nat @ Xs @ ( multis1655833086286526861et_nat @ A2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( finite_card_nat @ A2 ) ) ) ).
% length_finite_permutations_of_set
thf(fact_321_length__finite__permutations__of__set,axiom,
! [Xs: list_set_a,A2: set_set_a] :
( ( member_list_set_a @ Xs @ ( multis2257881577744371681_set_a @ A2 ) )
=> ( ( size_size_list_set_a @ Xs )
= ( finite_card_set_a @ A2 ) ) ) ).
% length_finite_permutations_of_set
thf(fact_322_length__finite__permutations__of__set,axiom,
! [Xs: list_a,A2: set_a] :
( ( member_list_a @ Xs @ ( multis2428024204330136193_set_a @ A2 ) )
=> ( ( size_size_list_a @ Xs )
= ( finite_card_a @ A2 ) ) ) ).
% length_finite_permutations_of_set
thf(fact_323_add__point__def,axiom,
! [P2: a] :
( ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P2 )
= ( insert_a @ P2 @ ( set_a2 @ v_s ) ) ) ).
% add_point_def
thf(fact_324_le__imp__inv,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( B
= ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% le_imp_inv
thf(fact_325_le__imp__inv,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( B
= ( plus_plus_real @ A @ ( minus_minus_real @ B @ A ) ) ) ) ).
% le_imp_inv
thf(fact_326_size__union,axiom,
! [M3: multiset_set_a,N4: multiset_set_a] :
( ( size_s6566526139600085008_set_a @ ( plus_p2331992037799027419_set_a @ M3 @ N4 ) )
= ( plus_plus_nat @ ( size_s6566526139600085008_set_a @ M3 ) @ ( size_s6566526139600085008_set_a @ N4 ) ) ) ).
% size_union
thf(fact_327_blocks__list__length,axiom,
( ( size_size_list_set_a @ b_s )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% blocks_list_length
thf(fact_328_block__complement__def,axiom,
! [B: set_a] :
( ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ B )
= ( minus_minus_set_a @ ( set_a2 @ v_s ) @ B ) ) ).
% block_complement_def
thf(fact_329_diff__diff__add__mset,axiom,
! [M3: multiset_set_a,N4: multiset_set_a,P: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M3 @ N4 ) @ P )
= ( minus_706656509937749387_set_a @ M3 @ ( plus_p2331992037799027419_set_a @ N4 @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_330_block__comp__elem__alt__left,axiom,
! [X2: a,Bl: set_a,Ps: set_a] :
( ( member_a @ X2 @ Bl )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl ) )
=> ~ ( member_a @ X2 @ Ps ) ) ) ).
% block_comp_elem_alt_left
thf(fact_331_block__comp__elem__alt__right,axiom,
! [Ps: set_a,Bl: set_a] :
( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ Ps )
=> ~ ( member_a @ X3 @ Bl ) )
=> ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl ) ) ) ) ).
% block_comp_elem_alt_right
thf(fact_332_block__complement__elem__iff,axiom,
! [Ps: set_a,Bl: set_a] :
( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl ) )
= ( ! [X: a] :
( ( member_a @ X @ Ps )
=> ~ ( member_a @ X @ Bl ) ) ) ) ) ).
% block_complement_elem_iff
thf(fact_333_block__complement__subset__points,axiom,
! [Ps: set_a,Bl: set_a] :
( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl ) )
=> ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) ) ) ).
% block_complement_subset_points
thf(fact_334_size__mset,axiom,
! [Xs: list_a] :
( ( size_size_multiset_a @ ( mset_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% size_mset
thf(fact_335_size__mset,axiom,
! [Xs: list_set_a] :
( ( size_s6566526139600085008_set_a @ ( mset_set_a @ Xs ) )
= ( size_size_list_set_a @ Xs ) ) ).
% size_mset
thf(fact_336_is__incidence__mat__true,axiom,
incide3487307869859529607_int_a @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).
% is_incidence_mat_true
thf(fact_337_dim__col__is__b,axiom,
( ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% dim_col_is_b
thf(fact_338_regular__pairwise__balance__axioms,axiom,
design8863377621459125358ance_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ lambda @ r ).
% regular_pairwise_balance_axioms
thf(fact_339_mset__eq__setD,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( mset_a @ Xs )
= ( mset_a @ Ys ) )
=> ( ( set_a2 @ Xs )
= ( set_a2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_340_mset__eq__setD,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( mset_set_a @ Xs )
= ( mset_set_a @ Ys ) )
=> ( ( set_set_a2 @ Xs )
= ( set_set_a2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_341_mset__eq__length,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( mset_a @ Xs )
= ( mset_a @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_342_mset__eq__length,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( mset_set_a @ Xs )
= ( mset_set_a @ Ys ) )
=> ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_343_ex__mset,axiom,
! [X4: multiset_set_a] :
? [Xs2: list_set_a] :
( ( mset_set_a @ Xs2 )
= X4 ) ).
% ex_mset
thf(fact_344_diff__union__cancelR,axiom,
! [M3: multiset_set_a,N4: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ M3 @ N4 ) @ N4 )
= M3 ) ).
% diff_union_cancelR
thf(fact_345_diff__union__cancelL,axiom,
! [N4: multiset_set_a,M3: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ N4 @ M3 ) @ N4 )
= M3 ) ).
% diff_union_cancelL
thf(fact_346_Multiset_Odiff__add,axiom,
! [M3: multiset_set_a,N4: multiset_set_a,Q: multiset_set_a] :
( ( minus_706656509937749387_set_a @ M3 @ ( plus_p2331992037799027419_set_a @ N4 @ Q ) )
= ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M3 @ N4 ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_347_ideal_Oscale__scale,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X2 ) ) ).
% ideal.scale_scale
thf(fact_348_ideal_Oscale__left__commute,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X2 ) ) ) ).
% ideal.scale_left_commute
thf(fact_349_leq__add__left,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( plus_plus_nat @ Y @ X2 ) ) ).
% leq_add_left
thf(fact_350_leq__add__right,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( plus_plus_nat @ X2 @ Y ) ) ).
% leq_add_right
thf(fact_351_le__imp__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ? [C2: nat] :
( B
= ( plus_plus_nat @ A @ C2 ) ) ) ).
% le_imp_add
thf(fact_352_le__imp__add,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ? [C2: real] :
( B
= ( plus_plus_real @ A @ C2 ) ) ) ).
% le_imp_add
thf(fact_353_ideal_Oscale__left__distrib,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ X2 )
= ( plus_plus_real @ ( times_times_real @ A @ X2 ) @ ( times_times_real @ B @ X2 ) ) ) ).
% ideal.scale_left_distrib
thf(fact_354_ideal_Oscale__right__distrib,axiom,
! [A: real,X2: real,Y: real] :
( ( times_times_real @ A @ ( plus_plus_real @ X2 @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A @ X2 ) @ ( times_times_real @ A @ Y ) ) ) ).
% ideal.scale_right_distrib
thf(fact_355_ideal_Oscale__left__diff__distrib,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ X2 )
= ( minus_minus_real @ ( times_times_real @ A @ X2 ) @ ( times_times_real @ B @ X2 ) ) ) ).
% ideal.scale_left_diff_distrib
thf(fact_356_ideal_Oscale__right__diff__distrib,axiom,
! [A: real,X2: real,Y: real] :
( ( times_times_real @ A @ ( minus_minus_real @ X2 @ Y ) )
= ( minus_minus_real @ ( times_times_real @ A @ X2 ) @ ( times_times_real @ A @ Y ) ) ) ).
% ideal.scale_right_diff_distrib
thf(fact_357_group__eq__aux,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ A ) )
= B ) ).
% group_eq_aux
thf(fact_358_diff__size__le__size__Diff,axiom,
! [M3: multiset_set_a,M4: multiset_set_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M3 ) @ ( size_s6566526139600085008_set_a @ M4 ) ) @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M3 @ M4 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_359_permutations__of__setD_I1_J,axiom,
! [Xs: list_a,A2: set_a] :
( ( member_list_a @ Xs @ ( multis2428024204330136193_set_a @ A2 ) )
=> ( ( set_a2 @ Xs )
= A2 ) ) ).
% permutations_of_setD(1)
thf(fact_360_add__point__existing__blocks,axiom,
! [Bs2: set_set_a,P2: a] :
( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ Bs2 )
=> ( member_a @ P2 @ Bl2 ) )
=> ( ( design2935547469388721088ocks_a @ ( mset_set_a @ b_s ) @ P2 @ Bs2 )
= ( mset_set_a @ b_s ) ) ) ).
% add_point_existing_blocks
thf(fact_361_add__point__sub__des,axiom,
! [P2: a] : ( sub_sub_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P2 ) @ ( mset_set_a @ b_s ) ) ).
% add_point_sub_des
thf(fact_362_card__Diff__insert,axiom,
! [A: int,A2: set_int,B2: set_int] :
( ( member_int @ A @ A2 )
=> ( ~ ( member_int @ A @ B2 )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_363_card__Diff__insert,axiom,
! [A: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ~ ( member_list_a @ A @ B2 )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_364_card__Diff__insert,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ A @ B2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_365_card__Diff__insert,axiom,
! [A: mat_int,A2: set_mat_int,B2: set_mat_int] :
( ( member_mat_int @ A @ A2 )
=> ( ~ ( member_mat_int @ A @ B2 )
=> ( ( finite_card_mat_int @ ( minus_7323808874771660652at_int @ A2 @ ( insert_mat_int @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_mat_int @ ( minus_7323808874771660652at_int @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_366_card__Diff__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ A @ B2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_367_card__Diff__insert,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ~ ( member_a @ A @ B2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_368_add__point__sub__sys,axiom,
! [P2: a] : ( sub_su7923802003039619913stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P2 ) @ ( mset_set_a @ b_s ) ) ).
% add_point_sub_sys
thf(fact_369_pairwise__balance__axioms,axiom,
block_5355636846524985331ance_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ lambda ).
% pairwise_balance_axioms
thf(fact_370_del__invalid__point__blocks,axiom,
! [P2: a] :
( ~ ( member_a @ P2 @ ( set_a2 @ v_s ) )
=> ( ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P2 )
= ( mset_set_a @ b_s ) ) ) ).
% del_invalid_point_blocks
thf(fact_371_constant__rep__design__axioms,axiom,
block_6028206285060069402sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ r ).
% constant_rep_design_axioms
thf(fact_372_del__point__block__count,axiom,
! [P2: a] :
( ( size_s6566526139600085008_set_a @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P2 ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% del_point_block_count
thf(fact_373_Multiset_Odiff__right__commute,axiom,
! [M3: multiset_set_a,N4: multiset_set_a,Q: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M3 @ N4 ) @ Q )
= ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M3 @ Q ) @ N4 ) ) ).
% Multiset.diff_right_commute
thf(fact_374_ordered__constant__rep_Oaxioms_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a,R: nat] :
( ( incide6922509864216205631_rep_a @ V_s @ B_s @ R )
=> ( block_6028206285060069402sign_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ R ) ) ).
% ordered_constant_rep.axioms(2)
thf(fact_375_ordered__pairwise__balance_Oaxioms_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a,Lambda: nat] :
( ( incide6880889959311561818ance_a @ V_s @ B_s @ Lambda )
=> ( block_5355636846524985331ance_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ Lambda ) ) ).
% ordered_pairwise_balance.axioms(2)
thf(fact_376_ordered__constant__rep_Ointro,axiom,
! [V_s: list_a,B_s: list_set_a,R: nat] :
( ( incide3676903341588786676sign_a @ V_s @ B_s )
=> ( ( block_6028206285060069402sign_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ R )
=> ( incide6922509864216205631_rep_a @ V_s @ B_s @ R ) ) ) ).
% ordered_constant_rep.intro
thf(fact_377_ordered__constant__rep__def,axiom,
( incide6922509864216205631_rep_a
= ( ^ [V_s2: list_a,B_s2: list_set_a,R2: nat] :
( ( incide3676903341588786676sign_a @ V_s2 @ B_s2 )
& ( block_6028206285060069402sign_a @ ( set_a2 @ V_s2 ) @ ( mset_set_a @ B_s2 ) @ R2 ) ) ) ) ).
% ordered_constant_rep_def
thf(fact_378_ordered__regular__pairwise__balance_Oaxioms_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a,Lambda: nat,R: nat] :
( ( incide5282158983398259552ance_a @ V_s @ B_s @ Lambda @ R )
=> ( design8863377621459125358ance_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ Lambda @ R ) ) ).
% ordered_regular_pairwise_balance.axioms(2)
thf(fact_379_card__insert__le,axiom,
! [A2: set_int,X2: int] : ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ ( insert_int @ X2 @ A2 ) ) ) ).
% card_insert_le
thf(fact_380_card__insert__le,axiom,
! [A2: set_a,X2: a] : ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( insert_a @ X2 @ A2 ) ) ) ).
% card_insert_le
thf(fact_381_card__insert__le,axiom,
! [A2: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X2 @ A2 ) ) ) ).
% card_insert_le
thf(fact_382_ordered__regular__pairwise__balance_Ointro,axiom,
! [V_s: list_a,B_s: list_set_a,Lambda: nat,R: nat] :
( ( incide6880889959311561818ance_a @ V_s @ B_s @ Lambda )
=> ( ( design8863377621459125358ance_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ Lambda @ R )
=> ( incide5282158983398259552ance_a @ V_s @ B_s @ Lambda @ R ) ) ) ).
% ordered_regular_pairwise_balance.intro
thf(fact_383_ordered__regular__pairwise__balance__def,axiom,
( incide5282158983398259552ance_a
= ( ^ [V_s2: list_a,B_s2: list_set_a,Lambda2: nat,R2: nat] :
( ( incide6880889959311561818ance_a @ V_s2 @ B_s2 @ Lambda2 )
& ( design8863377621459125358ance_a @ ( set_a2 @ V_s2 ) @ ( mset_set_a @ B_s2 ) @ Lambda2 @ R2 ) ) ) ) ).
% ordered_regular_pairwise_balance_def
thf(fact_384_strong__del__point__sub__des,axiom,
! [P2: a] : ( sub_sub_design_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P2 ) @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P2 ) @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% strong_del_point_sub_des
thf(fact_385_strong__del__point__sub__sys,axiom,
! [P2: a] : ( sub_su7923802003039619913stem_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P2 ) @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P2 ) @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% strong_del_point_sub_sys
thf(fact_386_v__eq0__imp__b__eq__0,axiom,
( ( ( finite_card_a @ ( set_a2 @ v_s ) )
= zero_zero_nat )
=> ( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
= zero_zero_nat ) ) ).
% v_eq0_imp_b_eq_0
thf(fact_387_delete__block__sub__des,axiom,
! [B: set_a] : ( sub_sub_design_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B ) @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% delete_block_sub_des
thf(fact_388_delete__block__sub__sys,axiom,
! [B: set_a] : ( sub_su7923802003039619913stem_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B ) @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% delete_block_sub_sys
thf(fact_389_delete__invalid__pt__strong__eq,axiom,
! [P2: a] :
( ~ ( member_a @ P2 @ ( set_a2 @ v_s ) )
=> ( ( mset_set_a @ b_s )
= ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P2 ) ) ) ).
% delete_invalid_pt_strong_eq
thf(fact_390_N__carrier__mat__01__lift,axiom,
member_mat_int @ ( matrix323868623736973467nt_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) @ ( carrier_mat_int @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% N_carrier_mat_01_lift
thf(fact_391_rep__not__zero,axiom,
r != zero_zero_nat ).
% rep_not_zero
thf(fact_392_b__non__zero,axiom,
( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
!= zero_zero_nat ) ).
% b_non_zero
thf(fact_393_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_394_ideal_Oscale__zero__left,axiom,
! [X2: int] :
( ( times_times_int @ zero_zero_int @ X2 )
= zero_zero_int ) ).
% ideal.scale_zero_left
thf(fact_395_ideal_Oscale__zero__left,axiom,
! [X2: real] :
( ( times_times_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% ideal.scale_zero_left
thf(fact_396_ideal_Oscale__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% ideal.scale_zero_right
thf(fact_397_ideal_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% ideal.scale_zero_right
thf(fact_398_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_399_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_400_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_401_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_402_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_403_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_404_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_405_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_406_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_407_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_408_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_409_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_410_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_411_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_412_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_413_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_414_add_Oright__neutral,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ A @ zero_z5079479921072680283_set_a )
= A ) ).
% add.right_neutral
thf(fact_415_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_416_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_417_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_418_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_419_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_420_add__cancel__left__left,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ B @ A )
= A )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_left_left
thf(fact_421_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_422_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_423_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_424_add__cancel__left__right,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ A @ B )
= A )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_left_right
thf(fact_425_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_426_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_427_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_428_add__cancel__right__left,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( A
= ( plus_p2331992037799027419_set_a @ B @ A ) )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_right_left
thf(fact_429_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_430_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_431_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_432_add__cancel__right__right,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( A
= ( plus_p2331992037799027419_set_a @ A @ B ) )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_right_right
thf(fact_433_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_434_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_435_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_436_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_437_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_438_add__0,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ A )
= A ) ).
% add_0
thf(fact_439_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_440_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_441_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_442_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_443_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_444_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_445_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_446_zero__diff,axiom,
! [A: multiset_set_a] :
( ( minus_706656509937749387_set_a @ zero_z5079479921072680283_set_a @ A )
= zero_z5079479921072680283_set_a ) ).
% zero_diff
thf(fact_447_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_448_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_449_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_450_diff__zero,axiom,
! [A: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A @ zero_z5079479921072680283_set_a )
= A ) ).
% diff_zero
thf(fact_451_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_452_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_453_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_454_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A @ A )
= zero_z5079479921072680283_set_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_455_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_456_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_457_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_458_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_459_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_460_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_461_N__carrier__mat,axiom,
member_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( carrier_mat_int @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% N_carrier_mat
thf(fact_462_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_463_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_464_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_465_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_466_carrier__matD_I2_J,axiom,
! [A2: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( dim_col_int @ A2 )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_467_carrier__matD_I1_J,axiom,
! [A2: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( dim_row_int @ A2 )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_468_assoc__mult__mat,axiom,
! [A2: mat_int,N_1: nat,N_2: nat,B2: mat_int,N_3: nat,C4: mat_int,N_4: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ N_1 @ N_2 ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ N_2 @ N_3 ) )
=> ( ( member_mat_int @ C4 @ ( carrier_mat_int @ N_3 @ N_4 ) )
=> ( ( times_times_mat_int @ ( times_times_mat_int @ A2 @ B2 ) @ C4 )
= ( times_times_mat_int @ A2 @ ( times_times_mat_int @ B2 @ C4 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_469_transpose__carrier__mat,axiom,
! [A2: mat_int,Nc: nat,Nr: nat] :
( ( member_mat_int @ ( transpose_mat_int @ A2 ) @ ( carrier_mat_int @ Nc @ Nr ) )
= ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_470_assoc__add__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,B2: mat_int,C4: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ C4 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( plus_plus_mat_int @ ( plus_plus_mat_int @ A2 @ B2 ) @ C4 )
= ( plus_plus_mat_int @ A2 @ ( plus_plus_mat_int @ B2 @ C4 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_471_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_472_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_473_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_474_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_475_le__add__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ A ) )
= ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% le_add_same_cancel2
thf(fact_476_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_477_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_478_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_479_le__add__same__cancel1,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ ( plus_p2331992037799027419_set_a @ A @ B ) )
= ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% le_add_same_cancel1
thf(fact_480_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_481_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_482_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_483_add__le__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ B )
= ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% add_le_same_cancel2
thf(fact_484_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_485_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_486_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_487_add__le__same__cancel1,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ B @ A ) @ B )
= ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% add_le_same_cancel1
thf(fact_488_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_489_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_490_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_491_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_492_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_493_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_494_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_495_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_496_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_497_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_498_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_499_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_500_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_501_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_502_diff__add__zero,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A @ ( plus_p2331992037799027419_set_a @ A @ B ) )
= zero_z5079479921072680283_set_a ) ).
% diff_add_zero
thf(fact_503_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_504_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_505_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_506_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_507_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_508_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_509_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_510_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_511_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_512_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_513_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_514_carrier__matI,axiom,
! [A2: mat_int,Nr: nat,Nc: nat] :
( ( ( dim_row_int @ A2 )
= Nr )
=> ( ( ( dim_col_int @ A2 )
= Nc )
=> ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_515_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_516_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_517_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_518_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_519_zero__reorient,axiom,
! [X2: multiset_set_a] :
( ( zero_z5079479921072680283_set_a = X2 )
= ( X2 = zero_z5079479921072680283_set_a ) ) ).
% zero_reorient
thf(fact_520_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_521_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_522_mult__carrier__mat,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ N @ Nc ) )
=> ( member_mat_int @ ( times_times_mat_int @ A2 @ B2 ) @ ( carrier_mat_int @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_523_zero__min,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_min
thf(fact_524_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_525_one__carrier__mat,axiom,
! [N: nat] : ( member_mat_int @ ( one_mat_int @ N ) @ ( carrier_mat_int @ N @ N ) ) ).
% one_carrier_mat
thf(fact_526_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_527_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_528_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_529_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_530_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_531_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_532_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_533_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_534_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_535_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_536_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_537_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_538_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_539_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_540_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_541_plus__eq__zero__2,axiom,
! [S: nat,T: nat] :
( ( ( plus_plus_nat @ S @ T )
= zero_zero_nat )
=> ( T = zero_zero_nat ) ) ).
% plus_eq_zero_2
thf(fact_542_plus__eq__zero,axiom,
! [S: nat,T: nat] :
( ( ( plus_plus_nat @ S @ T )
= zero_zero_nat )
=> ( S = zero_zero_nat ) ) ).
% plus_eq_zero
thf(fact_543_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_544_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_545_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_546_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_547_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_548_add_Ocomm__neutral,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ A @ zero_z5079479921072680283_set_a )
= A ) ).
% add.comm_neutral
thf(fact_549_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_550_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_551_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_552_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_553_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_554_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_555_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_556_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A3: real,B3: real] :
( ( minus_minus_real @ A3 @ B3 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_557_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_558_comm__add__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,B2: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( plus_plus_mat_int @ A2 @ B2 )
= ( plus_plus_mat_int @ B2 @ A2 ) ) ) ) ).
% comm_add_mat
thf(fact_559_add__carrier__mat,axiom,
! [B2: mat_int,Nr: nat,Nc: nat,A2: mat_int] :
( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( member_mat_int @ ( plus_plus_mat_int @ A2 @ B2 ) @ ( carrier_mat_int @ Nr @ Nc ) ) ) ).
% add_carrier_mat
thf(fact_560_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_561_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_562_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_563_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_564_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_565_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_566_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_567_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_568_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_569_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_570_smult__carrier__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,K: int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( member_mat_int @ ( smult_mat_int @ K @ A2 ) @ ( carrier_mat_int @ Nr @ Nc ) ) ) ).
% smult_carrier_mat
thf(fact_571_minus__carrier__mat,axiom,
! [B2: mat_int,Nr: nat,Nc: nat,A2: mat_int] :
( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( member_mat_int @ ( minus_minus_mat_int @ A2 @ B2 ) @ ( carrier_mat_int @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_572_carrier__mat__triv,axiom,
! [M: mat_int] : ( member_mat_int @ M @ ( carrier_mat_int @ ( dim_row_int @ M ) @ ( dim_col_int @ M ) ) ) ).
% carrier_mat_triv
thf(fact_573_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_574_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_575_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_576_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_577_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_578_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_579_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_580_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_581_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_582_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_583_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_584_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_585_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_586_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_587_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_588_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_589_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_590_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_591_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_592_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_593_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_594_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_595_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_596_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_597_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_598_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_599_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_600_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_601_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_602_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_603_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_604_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_605_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_606_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_607_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_608_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_609_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_610_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_611_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_612_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_613_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_614_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_615_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_616_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_617_add__nonpos__eq__0__iff,axiom,
! [X2: multiset_set_a,Y: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ X2 @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ Y @ zero_z5079479921072680283_set_a )
=> ( ( ( plus_p2331992037799027419_set_a @ X2 @ Y )
= zero_z5079479921072680283_set_a )
= ( ( X2 = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_618_add__nonpos__eq__0__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X2 @ Y )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_619_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_620_add__nonpos__eq__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X2 @ Y )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_621_add__nonneg__eq__0__iff,axiom,
! [X2: multiset_set_a,Y: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ X2 )
=> ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ Y )
=> ( ( ( plus_p2331992037799027419_set_a @ X2 @ Y )
= zero_z5079479921072680283_set_a )
= ( ( X2 = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_622_add__nonneg__eq__0__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X2 @ Y )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_623_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_624_add__nonneg__eq__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X2 @ Y )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_625_add__nonpos__nonpos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% add_nonpos_nonpos
thf(fact_626_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_627_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_628_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_629_add__nonneg__nonneg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_630_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_631_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_632_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_633_add__increasing2,axiom,
! [C: multiset_set_a,B: multiset_set_a,A: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ C )
=> ( ( ord_le7905258569527593284_set_a @ B @ A )
=> ( ord_le7905258569527593284_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_634_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_635_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_636_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_637_add__decreasing2,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ C @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ A @ B )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_638_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_639_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_640_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_641_add__increasing,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le7905258569527593284_set_a @ B @ C )
=> ( ord_le7905258569527593284_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_642_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_643_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_644_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_645_add__decreasing,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ C @ B )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_646_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_647_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_648_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_649_left__mult__one__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( times_times_mat_int @ ( one_mat_int @ Nr ) @ A2 )
= A2 ) ) ).
% left_mult_one_mat
thf(fact_650_right__mult__one__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( times_times_mat_int @ A2 @ ( one_mat_int @ Nc ) )
= A2 ) ) ).
% right_mult_one_mat
thf(fact_651_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_652_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_653_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_654_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_655_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_656_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_657_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_658_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_659_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_660_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_661_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_662_transpose__mult,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( transpose_mat_int @ ( times_times_mat_int @ A2 @ B2 ) )
= ( times_times_mat_int @ ( transpose_mat_int @ B2 ) @ ( transpose_mat_int @ A2 ) ) ) ) ) ).
% transpose_mult
thf(fact_663_add__mult__distrib__mat,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,C4: mat_int,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ C4 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( times_times_mat_int @ ( plus_plus_mat_int @ A2 @ B2 ) @ C4 )
= ( plus_plus_mat_int @ ( times_times_mat_int @ A2 @ C4 ) @ ( times_times_mat_int @ B2 @ C4 ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_664_mult__add__distrib__mat,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,Nc: nat,C4: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( member_mat_int @ C4 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( times_times_mat_int @ A2 @ ( plus_plus_mat_int @ B2 @ C4 ) )
= ( plus_plus_mat_int @ ( times_times_mat_int @ A2 @ B2 ) @ ( times_times_mat_int @ A2 @ C4 ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_665_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_666_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_667_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_668_mult__smult__distrib,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,Nc: nat,K: int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( times_times_mat_int @ A2 @ ( smult_mat_int @ K @ B2 ) )
= ( smult_mat_int @ K @ ( times_times_mat_int @ A2 @ B2 ) ) ) ) ) ).
% mult_smult_distrib
thf(fact_669_mult__smult__assoc__mat,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,Nc: nat,K: int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( times_times_mat_int @ ( smult_mat_int @ K @ A2 ) @ B2 )
= ( smult_mat_int @ K @ ( times_times_mat_int @ A2 @ B2 ) ) ) ) ) ).
% mult_smult_assoc_mat
thf(fact_670_transpose__add,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,B2: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( transpose_mat_int @ ( plus_plus_mat_int @ A2 @ B2 ) )
= ( plus_plus_mat_int @ ( transpose_mat_int @ A2 ) @ ( transpose_mat_int @ B2 ) ) ) ) ) ).
% transpose_add
thf(fact_671_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_672_mult__minus__distrib__mat,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,Nc: nat,C4: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( member_mat_int @ C4 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( times_times_mat_int @ A2 @ ( minus_minus_mat_int @ B2 @ C4 ) )
= ( minus_minus_mat_int @ ( times_times_mat_int @ A2 @ B2 ) @ ( times_times_mat_int @ A2 @ C4 ) ) ) ) ) ) ).
% mult_minus_distrib_mat
thf(fact_673_minus__mult__distrib__mat,axiom,
! [A2: mat_int,Nr: nat,N: nat,B2: mat_int,C4: mat_int,Nc: nat] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ C4 @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( times_times_mat_int @ ( minus_minus_mat_int @ A2 @ B2 ) @ C4 )
= ( minus_minus_mat_int @ ( times_times_mat_int @ A2 @ C4 ) @ ( times_times_mat_int @ B2 @ C4 ) ) ) ) ) ) ).
% minus_mult_distrib_mat
thf(fact_674_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_675_add__smult__distrib__left__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,B2: mat_int,K: int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( smult_mat_int @ K @ ( plus_plus_mat_int @ A2 @ B2 ) )
= ( plus_plus_mat_int @ ( smult_mat_int @ K @ A2 ) @ ( smult_mat_int @ K @ B2 ) ) ) ) ) ).
% add_smult_distrib_left_mat
thf(fact_676_transpose__minus,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,B2: mat_int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( transpose_mat_int @ ( minus_minus_mat_int @ A2 @ B2 ) )
= ( minus_minus_mat_int @ ( transpose_mat_int @ A2 ) @ ( transpose_mat_int @ B2 ) ) ) ) ) ).
% transpose_minus
thf(fact_677_minus__add__minus__mat,axiom,
! [U: mat_int,Nr: nat,Nc: nat,V: mat_int,W: mat_int] :
( ( member_mat_int @ U @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ V @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ W @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( minus_minus_mat_int @ U @ ( plus_plus_mat_int @ V @ W ) )
= ( minus_minus_mat_int @ ( minus_minus_mat_int @ U @ V ) @ W ) ) ) ) ) ).
% minus_add_minus_mat
thf(fact_678_sum__squares__ge__zero,axiom,
! [X2: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_679_sum__squares__ge__zero,axiom,
! [X2: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_680_mult__left__le__one__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_681_mult__left__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_682_mult__right__le__one__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_683_mult__right__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_684_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_685_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_686_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_687_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_688_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_689_mult__left__le,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_690_lift__01__mat__carrier,axiom,
! [M3: mat_int] : ( member_mat_int @ ( matrix323868623736973467nt_int @ M3 ) @ ( carrier_mat_int @ ( dim_row_int @ M3 ) @ ( dim_col_int @ M3 ) ) ) ).
% lift_01_mat_carrier
thf(fact_691_add__smult__distrib__right__mat,axiom,
! [A2: mat_int,Nr: nat,Nc: nat,K: int,L: int] :
( ( member_mat_int @ A2 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( smult_mat_int @ ( plus_plus_int @ K @ L ) @ A2 )
= ( plus_plus_mat_int @ ( smult_mat_int @ K @ A2 ) @ ( smult_mat_int @ L @ A2 ) ) ) ) ).
% add_smult_distrib_right_mat
thf(fact_692_add__smult__distrib__right__mat,axiom,
! [A2: mat_nat,Nr: nat,Nc: nat,K: nat,L: nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( smult_mat_nat @ ( plus_plus_nat @ K @ L ) @ A2 )
= ( plus_plus_mat_nat @ ( smult_mat_nat @ K @ A2 ) @ ( smult_mat_nat @ L @ A2 ) ) ) ) ).
% add_smult_distrib_right_mat
thf(fact_693_add__smult__distrib__right__mat,axiom,
! [A2: mat_real,Nr: nat,Nc: nat,K: real,L: real] :
( ( member_mat_real @ A2 @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ( smult_mat_real @ ( plus_plus_real @ K @ L ) @ A2 )
= ( plus_plus_mat_real @ ( smult_mat_real @ K @ A2 ) @ ( smult_mat_real @ L @ A2 ) ) ) ) ).
% add_smult_distrib_right_mat
thf(fact_694_convex__bound__le,axiom,
! [X2: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X2 @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_695_convex__bound__le,axiom,
! [X2: real,A: real,Y: real,U: real,V: real] :
( ( ord_less_eq_real @ X2 @ A )
=> ( ( ord_less_eq_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_696_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_697_add__del__block__inv,axiom,
! [Bl: set_a] :
( ( ord_less_eq_set_a @ Bl @ ( set_a2 @ v_s ) )
=> ( ( design1146539425385464078lock_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ Bl ) @ Bl )
= ( mset_set_a @ b_s ) ) ) ).
% add_del_block_inv
thf(fact_698_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_699_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_700_sum__squares__eq__zero__iff,axiom,
! [X2: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_701_sum__squares__eq__zero__iff,axiom,
! [X2: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_702_b__non__zero__imp__v__non__zero,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ).
% b_non_zero_imp_v_non_zero
thf(fact_703_del__point__def,axiom,
! [P2: a] :
( ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P2 )
= ( minus_minus_set_a @ ( set_a2 @ v_s ) @ ( insert_a @ P2 @ bot_bot_set_a ) ) ) ).
% del_point_def
thf(fact_704_design__blocks__nempty,axiom,
( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a ) ).
% design_blocks_nempty
thf(fact_705_design__points__nempty,axiom,
( ( set_a2 @ v_s )
!= bot_bot_set_a ) ).
% design_points_nempty
thf(fact_706_r__gzero,axiom,
ord_less_nat @ zero_zero_nat @ r ).
% r_gzero
thf(fact_707_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X2: multiset_set_a,Y: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ X2 @ Y )
= zero_z5079479921072680283_set_a )
= ( ( X2 = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_708_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X2: multiset_set_a,Y: multiset_set_a] :
( ( zero_z5079479921072680283_set_a
= ( plus_p2331992037799027419_set_a @ X2 @ Y ) )
= ( ( X2 = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_709_empty__eq__union,axiom,
! [M3: multiset_set_a,N4: multiset_set_a] :
( ( zero_z5079479921072680283_set_a
= ( plus_p2331992037799027419_set_a @ M3 @ N4 ) )
= ( ( M3 = zero_z5079479921072680283_set_a )
& ( N4 = zero_z5079479921072680283_set_a ) ) ) ).
% empty_eq_union
thf(fact_710_union__eq__empty,axiom,
! [M3: multiset_set_a,N4: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ M3 @ N4 )
= zero_z5079479921072680283_set_a )
= ( ( M3 = zero_z5079479921072680283_set_a )
& ( N4 = zero_z5079479921072680283_set_a ) ) ) ).
% union_eq_empty
thf(fact_711_v__non__zero,axiom,
ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) ).
% v_non_zero
thf(fact_712_b__positive,axiom,
ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).
% b_positive
thf(fact_713_block__set__nempty__imp__points,axiom,
( ( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a )
=> ( ( set_a2 @ v_s )
!= bot_bot_set_a ) ) ).
% block_set_nempty_imp_points
thf(fact_714_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_715_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_716_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_717_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_718_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_719_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_720_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_721_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_722_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_723_size__empty,axiom,
( ( size_s6566526139600085008_set_a @ zero_z5079479921072680283_set_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_724_size__eq__0__iff__empty,axiom,
! [M3: multiset_set_a] :
( ( ( size_s6566526139600085008_set_a @ M3 )
= zero_zero_nat )
= ( M3 = zero_z5079479921072680283_set_a ) ) ).
% size_eq_0_iff_empty
thf(fact_725_add__less__same__cancel1,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ B @ A ) @ B )
= ( ord_le5765082015083327056_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% add_less_same_cancel1
thf(fact_726_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_727_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_728_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_729_add__less__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ B )
= ( ord_le5765082015083327056_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% add_less_same_cancel2
thf(fact_730_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_731_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_732_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_733_less__add__same__cancel1,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ ( plus_p2331992037799027419_set_a @ A @ B ) )
= ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% less_add_same_cancel1
thf(fact_734_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_735_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_736_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_737_less__add__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ A ) )
= ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% less_add_same_cancel2
thf(fact_738_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_739_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_740_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_741_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_742_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_743_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_744_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_745_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_746_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_747_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_748_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_749_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_750_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_751_card_Oempty,axiom,
( ( finite_card_int @ bot_bot_set_int )
= zero_zero_nat ) ).
% card.empty
thf(fact_752_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_753_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_754_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_755_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_756_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_757_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_758_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_759_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_760_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_761_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_762_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_763_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_764_real__archimedian__rdiv__eq__0,axiom,
! [X2: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X2 ) @ C ) )
=> ( X2 = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_765_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N2: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_766_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_767_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_768_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_769_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
=> ( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_770_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_771_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_772_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_773_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_774_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_775_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_776_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_777_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
& ~ ( P @ M6 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_778_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( P @ M6 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_779_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_780_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_781_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_782_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_783_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_784_nonempty__has__size,axiom,
! [S2: multiset_set_a] :
( ( S2 != zero_z5079479921072680283_set_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ S2 ) ) ) ).
% nonempty_has_size
thf(fact_785_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_786_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_787_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_788_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_789_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_790_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_791_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_792_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_793_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_794_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_795_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_796_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_797_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_798_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_799_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_800_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_801_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_802_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_803_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_804_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_805_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_806_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_807_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_808_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_809_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_810_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
& ~ ( P @ M6 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_811_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_812_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_813_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_814_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_815_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_816_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_817_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_818_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_819_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_820_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_821_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_822_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_823_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_824_length__induct,axiom,
! [P: list_set_a > $o,Xs: list_set_a] :
( ! [Xs2: list_set_a] :
( ! [Ys2: list_set_a] :
( ( ord_less_nat @ ( size_size_list_set_a @ Ys2 ) @ ( size_size_list_set_a @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_825_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_826_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_827_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_828_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_829_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_830_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_831_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_832_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_833_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_834_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_835_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_836_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_837_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_838_empty__neutral_I2_J,axiom,
! [X2: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ X2 @ zero_z5079479921072680283_set_a )
= X2 ) ).
% empty_neutral(2)
thf(fact_839_empty__neutral_I1_J,axiom,
! [X2: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ X2 )
= X2 ) ).
% empty_neutral(1)
thf(fact_840_Multiset_Odiff__cancel,axiom,
! [A2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A2 @ A2 )
= zero_z5079479921072680283_set_a ) ).
% Multiset.diff_cancel
thf(fact_841_diff__empty,axiom,
! [M3: multiset_set_a] :
( ( ( minus_706656509937749387_set_a @ M3 @ zero_z5079479921072680283_set_a )
= M3 )
& ( ( minus_706656509937749387_set_a @ zero_z5079479921072680283_set_a @ M3 )
= zero_z5079479921072680283_set_a ) ) ).
% diff_empty
thf(fact_842_sum__squares__gt__zero__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X2 != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_843_sum__squares__gt__zero__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X2 != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_844_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_845_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_846_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_847_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_848_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_849_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_850_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_851_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_852_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_853_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_854_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_855_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_856_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_857_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_858_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_859_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_860_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_861_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_862_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_863_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_864_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_865_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_866_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_867_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_868_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_869_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_870_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_871_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_872_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_873_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_874_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_875_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_876_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_877_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_878_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_879_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_880_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_881_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_882_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_883_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_884_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_885_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_886_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_887_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_888_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_889_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_890_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_891_add__neg__neg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le5765082015083327056_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% add_neg_neg
thf(fact_892_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_893_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_894_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_895_add__pos__pos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_896_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_897_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_898_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_899_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_900_pos__add__strict,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le5765082015083327056_set_a @ B @ C )
=> ( ord_le5765082015083327056_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_901_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_902_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_903_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_904_add__less__zeroD,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X2 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X2 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_905_add__less__zeroD,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X2 @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_906_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_907_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_908_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_909_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_910_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_911_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_912_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_913_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_914_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_915_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_916_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_917_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_918_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_919_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_920_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_921_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_922_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_923_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_924_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_925_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_926_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_927_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_928_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_929_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_930_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_931_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_932_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_933_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_934_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_935_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_936_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_937_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_938_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_939_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_940_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_941_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_942_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_943_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_944_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_945_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_946_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_947_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_948_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_949_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_950_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_951_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_952_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_953_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_954_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_955_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_956_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_957_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_958_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_959_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_960_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_961_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_962_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_963_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_964_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_965_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_966_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_967_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_968_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_969_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_970_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_971_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_972_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_973_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_974_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_975_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_976_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_977_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_978_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_979_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_980_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_981_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_982_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_983_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_984_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_985_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_986_add__strict__increasing2,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le5765082015083327056_set_a @ B @ C )
=> ( ord_le5765082015083327056_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_987_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_988_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_989_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_990_add__strict__increasing,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le7905258569527593284_set_a @ B @ C )
=> ( ord_le5765082015083327056_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_991_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_992_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_993_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_994_add__pos__nonneg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_995_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_996_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_997_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_998_add__nonpos__neg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le5765082015083327056_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% add_nonpos_neg
thf(fact_999_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1000_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1001_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_1002_add__nonneg__pos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1003_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1004_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1005_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1006_add__neg__nonpos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% add_neg_nonpos
thf(fact_1007_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1008_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1009_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_1010_not__sum__squares__lt__zero,axiom,
! [X2: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_1011_not__sum__squares__lt__zero,axiom,
! [X2: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_1012_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1013_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1014_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_1015_less__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_1016_less__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_1017_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_1018_length__pos__if__in__set,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1019_length__pos__if__in__set,axiom,
! [X2: mat_int,Xs: list_mat_int] :
( ( member_mat_int @ X2 @ ( set_mat_int2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6799427505661098007at_int @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1020_length__pos__if__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1021_length__pos__if__in__set,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1022_length__pos__if__in__set,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_set_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1023_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1024_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1025_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1026_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1027_card__1__singletonE,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= one_one_nat )
=> ~ ! [X3: a] :
( A2
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_1028_card__1__singletonE,axiom,
! [A2: set_int] :
( ( ( finite_card_int @ A2 )
= one_one_nat )
=> ~ ! [X3: int] :
( A2
!= ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% card_1_singletonE
thf(fact_1029_card__1__singletonE,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= one_one_nat )
=> ~ ! [X3: nat] :
( A2
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_1030_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1031_mult__less__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1032_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1033_mult__less__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1034_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1035_mult__less__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1036_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1037_mult__less__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1038_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1039_mult__le__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1040_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1041_mult__le__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1042_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1043_mult__le__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1044_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1045_mult__le__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1046_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1047_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1048_card__Diff1__le,axiom,
! [A2: set_int,X2: int] : ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A2 ) ) ).
% card_Diff1_le
thf(fact_1049_card__Diff1__le,axiom,
! [A2: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1050_card__Diff1__le,axiom,
! [A2: set_a,X2: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_1051_convex__bound__lt,axiom,
! [X2: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X2 @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1052_convex__bound__lt,axiom,
! [X2: real,A: real,Y: real,U: real,V: real] :
( ( ord_less_real @ X2 @ A )
=> ( ( ord_less_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1053_proper__inc__mat__def,axiom,
( incide294466202882093137at_int
= ( ^ [M7: mat_int] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_row_int @ M7 ) )
& ( ord_less_nat @ zero_zero_nat @ ( dim_col_int @ M7 ) ) ) ) ) ).
% proper_inc_mat_def
thf(fact_1054_card__Diff__singleton__if,axiom,
! [X2: list_a,A2: set_list_a] :
( ( ( member_list_a @ X2 @ A2 )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) )
= ( minus_minus_nat @ ( finite_card_list_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_list_a @ X2 @ A2 )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) )
= ( finite_card_list_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1055_card__Diff__singleton__if,axiom,
! [X2: set_a,A2: set_set_a] :
( ( ( member_set_a @ X2 @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_set_a @ X2 @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) )
= ( finite_card_set_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1056_card__Diff__singleton__if,axiom,
! [X2: mat_int,A2: set_mat_int] :
( ( ( member_mat_int @ X2 @ A2 )
=> ( ( finite_card_mat_int @ ( minus_7323808874771660652at_int @ A2 @ ( insert_mat_int @ X2 @ bot_bot_set_mat_int ) ) )
= ( minus_minus_nat @ ( finite_card_mat_int @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_mat_int @ X2 @ A2 )
=> ( ( finite_card_mat_int @ ( minus_7323808874771660652at_int @ A2 @ ( insert_mat_int @ X2 @ bot_bot_set_mat_int ) ) )
= ( finite_card_mat_int @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1057_card__Diff__singleton__if,axiom,
! [X2: int,A2: set_int] :
( ( ( member_int @ X2 @ A2 )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
= ( minus_minus_nat @ ( finite_card_int @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_int @ X2 @ A2 )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
= ( finite_card_int @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1058_card__Diff__singleton__if,axiom,
! [X2: nat,A2: set_nat] :
( ( ( member_nat @ X2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1059_card__Diff__singleton__if,axiom,
! [X2: a,A2: set_a] :
( ( ( member_a @ X2 @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X2 @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1060_card__Diff__singleton,axiom,
! [X2: list_a,A2: set_list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) )
= ( minus_minus_nat @ ( finite_card_list_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1061_card__Diff__singleton,axiom,
! [X2: set_a,A2: set_set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1062_card__Diff__singleton,axiom,
! [X2: mat_int,A2: set_mat_int] :
( ( member_mat_int @ X2 @ A2 )
=> ( ( finite_card_mat_int @ ( minus_7323808874771660652at_int @ A2 @ ( insert_mat_int @ X2 @ bot_bot_set_mat_int ) ) )
= ( minus_minus_nat @ ( finite_card_mat_int @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1063_card__Diff__singleton,axiom,
! [X2: int,A2: set_int] :
( ( member_int @ X2 @ A2 )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
= ( minus_minus_nat @ ( finite_card_int @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1064_card__Diff__singleton,axiom,
! [X2: nat,A2: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1065_card__Diff__singleton,axiom,
! [X2: a,A2: set_a] :
( ( member_a @ X2 @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1066_card__insert__le__m1,axiom,
! [N: nat,Y: set_int,X2: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( insert_int @ X2 @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1067_card__insert__le__m1,axiom,
! [N: nat,Y: set_a,X2: a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( insert_a @ X2 @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1068_card__insert__le__m1,axiom,
! [N: nat,Y: set_nat,X2: 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 @ X2 @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1069_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1070_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1071_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_1072_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1073_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1074_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_1075_sum__squares__le__zero__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1076_sum__squares__le__zero__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1077_all__cols__non__empty,axiom,
! [J: nat] :
( ( ord_less_nat @ J @ ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( incide6851923868969248411ol_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ J ) ) ).
% all_cols_non_empty
thf(fact_1078_incidence__mat__rep__num,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( incide7000514267430604580um_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ I )
= r ) ) ).
% incidence_mat_rep_num
thf(fact_1079_lift__mat__elems,axiom,
ord_less_eq_set_real @ ( elements_mat_real @ ( matrix1811533390045330843t_real @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) ).
% lift_mat_elems
thf(fact_1080_lift__mat__elems,axiom,
ord_less_eq_set_nat @ ( elements_mat_nat @ ( matrix326359094246023743nt_nat @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ).
% lift_mat_elems
thf(fact_1081_lift__mat__elems,axiom,
ord_less_eq_set_int @ ( elements_mat_int @ ( matrix323868623736973467nt_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) ).
% lift_mat_elems
thf(fact_1082_delete__point__blocks__wf,axiom,
! [B: set_a,P2: a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P2 ) ) )
=> ( ord_less_eq_set_a @ B @ ( minus_minus_set_a @ ( set_a2 @ v_s ) @ ( insert_a @ P2 @ bot_bot_set_a ) ) ) ) ).
% delete_point_blocks_wf
thf(fact_1083_insert__Diff__single,axiom,
! [A: int,A2: set_int] :
( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( insert_int @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_1084_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_1085_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_1086_elems01,axiom,
ord_less_eq_set_int @ ( elements_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) ).
% elems01
thf(fact_1087_psubsetI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_1088_psubsetI,axiom,
! [A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_int @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_1089_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_1090_subset__antisym,axiom,
! [A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_1091_subsetI,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( member_list_a @ X3 @ B2 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1092_subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1093_subsetI,axiom,
! [A2: set_mat_int,B2: set_mat_int] :
( ! [X3: mat_int] :
( ( member_mat_int @ X3 @ A2 )
=> ( member_mat_int @ X3 @ B2 ) )
=> ( ord_le5299038897506728741at_int @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1094_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1095_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1096_subsetI,axiom,
! [A2: set_int,B2: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( member_int @ X3 @ B2 ) )
=> ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1097_insert__absorb2,axiom,
! [X2: a,A2: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A2 ) )
= ( insert_a @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_1098_insert__absorb2,axiom,
! [X2: int,A2: set_int] :
( ( insert_int @ X2 @ ( insert_int @ X2 @ A2 ) )
= ( insert_int @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_1099_insert__absorb2,axiom,
! [X2: nat,A2: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A2 ) )
= ( insert_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_1100_insert__iff,axiom,
! [A: int,B: int,A2: set_int] :
( ( member_int @ A @ ( insert_int @ B @ A2 ) )
= ( ( A = B )
| ( member_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1101_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1102_insert__iff,axiom,
! [A: list_a,B: list_a,A2: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
= ( ( A = B )
| ( member_list_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1103_insert__iff,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
= ( ( A = B )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1104_insert__iff,axiom,
! [A: mat_int,B: mat_int,A2: set_mat_int] :
( ( member_mat_int @ A @ ( insert_mat_int @ B @ A2 ) )
= ( ( A = B )
| ( member_mat_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1105_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_1106_insertCI,axiom,
! [A: int,B2: set_int,B: int] :
( ( ~ ( member_int @ A @ B2 )
=> ( A = B ) )
=> ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1107_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1108_insertCI,axiom,
! [A: list_a,B2: set_list_a,B: list_a] :
( ( ~ ( member_list_a @ A @ B2 )
=> ( A = B ) )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1109_insertCI,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B2 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1110_insertCI,axiom,
! [A: mat_int,B2: set_mat_int,B: mat_int] :
( ( ~ ( member_mat_int @ A @ B2 )
=> ( A = B ) )
=> ( member_mat_int @ A @ ( insert_mat_int @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1111_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_1112_Diff__idemp,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_1113_Diff__iff,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
= ( ( member_list_a @ C @ A2 )
& ~ ( member_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1114_Diff__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
& ~ ( member_set_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1115_Diff__iff,axiom,
! [C: mat_int,A2: set_mat_int,B2: set_mat_int] :
( ( member_mat_int @ C @ ( minus_7323808874771660652at_int @ A2 @ B2 ) )
= ( ( member_mat_int @ C @ A2 )
& ~ ( member_mat_int @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1116_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1117_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1118_DiffI,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A2 )
=> ( ~ ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1119_DiffI,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1120_DiffI,axiom,
! [C: mat_int,A2: set_mat_int,B2: set_mat_int] :
( ( member_mat_int @ C @ A2 )
=> ( ~ ( member_mat_int @ C @ B2 )
=> ( member_mat_int @ C @ ( minus_7323808874771660652at_int @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1121_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1122_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1123_blocks__nempty__alt,axiom,
! [X5: set_a] :
( ( member_set_a @ X5 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( X5 != bot_bot_set_a ) ) ).
% blocks_nempty_alt
thf(fact_1124_blocks__nempty,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( Bl != bot_bot_set_a ) ) ).
% blocks_nempty
thf(fact_1125_delete__invalid__block__eq,axiom,
! [B: set_a] :
( ~ ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B )
= ( mset_set_a @ b_s ) ) ) ).
% delete_invalid_block_eq
thf(fact_1126_block__set__nempty__imp__block__ex,axiom,
( ( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a )
=> ? [Bl2: set_a] : ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% block_set_nempty_imp_block_ex
thf(fact_1127_delete__point__strong__block__not__in,axiom,
! [P2: a,Bl: set_a] :
( ( member_a @ P2 @ Bl )
=> ~ ( member_set_a @ Bl @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P2 ) ) ) ) ).
% delete_point_strong_block_not_in
thf(fact_1128_delete__point__strong__block__in__orig,axiom,
! [Bl: set_a,P2: a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P2 ) ) )
=> ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% delete_point_strong_block_in_orig
thf(fact_1129_delete__point__strong__block__in__iff,axiom,
! [Bl: set_a,P2: a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P2 ) ) )
= ( ~ ( member_a @ P2 @ Bl ) ) ) ) ).
% delete_point_strong_block_in_iff
thf(fact_1130_delete__point__strong__block__in,axiom,
! [P2: a,Bl: set_a] :
( ~ ( member_a @ P2 @ Bl )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_set_a @ Bl @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P2 ) ) ) ) ) ).
% delete_point_strong_block_in
thf(fact_1131_delete__point__p__not__in__bl__blocks,axiom,
! [P2: a] :
( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ~ ( member_a @ P2 @ Bl2 ) )
=> ( ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P2 )
= ( mset_set_a @ b_s ) ) ) ).
% delete_point_p_not_in_bl_blocks
thf(fact_1132_transpose__entries,axiom,
ord_less_eq_set_int @ ( elements_mat_int @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) @ ( insert_int @ zero_zero_int @ ( insert_int @ one_one_int @ bot_bot_set_int ) ) ).
% transpose_entries
thf(fact_1133_del__block__b_I2_J,axiom,
! [Bl: set_a] :
( ~ ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% del_block_b(2)
thf(fact_1134_delete__point__blocks__sub,axiom,
! [B: set_a,P2: a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P2 ) ) )
=> ~ ! [Bl2: set_a] :
~ ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
& ( ord_less_eq_set_a @ B @ Bl2 ) ) ) ).
% delete_point_blocks_sub
thf(fact_1135_wf__invalid__point,axiom,
! [X2: a,B: set_a] :
( ~ ( member_a @ X2 @ ( set_a2 @ v_s ) )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ~ ( member_a @ X2 @ B ) ) ) ).
% wf_invalid_point
thf(fact_1136_del__invalid__add__block__eq,axiom,
! [Bl: set_a] :
( ~ ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl ) @ Bl )
= ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ Bl ) ) ) ).
% del_invalid_add_block_eq
thf(fact_1137_del__add__block__inv,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl ) @ Bl )
= ( mset_set_a @ b_s ) ) ) ).
% del_add_block_inv
thf(fact_1138_block__size__gt__0,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ Bl ) ) ) ).
% block_size_gt_0
thf(fact_1139_complete__block__size__eq__points,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ( finite_card_a @ Bl )
= ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( Bl
= ( set_a2 @ v_s ) ) ) ) ).
% complete_block_size_eq_points
thf(fact_1140_wf__list,axiom,
! [B: set_a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ord_less_eq_set_a @ B @ ( set_a2 @ v_s ) ) ) ).
% wf_list
thf(fact_1141_block__complement__inv,axiom,
! [Bl: set_a,Bl22: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl )
= Bl22 )
=> ( ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl22 )
= Bl ) ) ) ).
% block_complement_inv
thf(fact_1142_del__block__b_I1_J,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl ) )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) @ one_one_nat ) ) ) ).
% del_block_b(1)
thf(fact_1143_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_1144_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_1145_subset__empty,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
= ( A2 = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_1146_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_1147_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_1148_empty__subsetI,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% empty_subsetI
thf(fact_1149_singletonI,axiom,
! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_1150_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_1151_singletonI,axiom,
! [A: mat_int] : ( member_mat_int @ A @ ( insert_mat_int @ A @ bot_bot_set_mat_int ) ) ).
% singletonI
thf(fact_1152_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_1153_singletonI,axiom,
! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% singletonI
thf(fact_1154_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1155_insert__subset,axiom,
! [X2: list_a,A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X2 @ A2 ) @ B2 )
= ( ( member_list_a @ X2 @ B2 )
& ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1156_insert__subset,axiom,
! [X2: set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ A2 ) @ B2 )
= ( ( member_set_a @ X2 @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1157_insert__subset,axiom,
! [X2: mat_int,A2: set_mat_int,B2: set_mat_int] :
( ( ord_le5299038897506728741at_int @ ( insert_mat_int @ X2 @ A2 ) @ B2 )
= ( ( member_mat_int @ X2 @ B2 )
& ( ord_le5299038897506728741at_int @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1158_insert__subset,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( ( member_nat @ X2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1159_insert__subset,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A2 ) @ B2 )
= ( ( member_a @ X2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1160_insert__subset,axiom,
! [X2: int,A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ ( insert_int @ X2 @ A2 ) @ B2 )
= ( ( member_int @ X2 @ B2 )
& ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1161_incomplete__block__proper__subset,axiom,
! [Bl: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) )
=> ( ord_less_set_a @ Bl @ ( set_a2 @ v_s ) ) ) ).
% incomplete_block_proper_subset
thf(fact_1162_incomplete__alt__in,axiom,
! [Bl: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) )
=> ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% incomplete_alt_in
thf(fact_1163_incomplete__alt__size,axiom,
! [Bl: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) )
=> ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ).
% incomplete_alt_size
thf(fact_1164_block__size__lt__v,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ).
% block_size_lt_v
thf(fact_1165_Diff__empty,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
= A2 ) ).
% Diff_empty
thf(fact_1166_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_1167_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_1168_empty__Diff,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
= bot_bot_set_int ) ).
% empty_Diff
thf(fact_1169_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_1170_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_1171_Diff__cancel,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ A2 @ A2 )
= bot_bot_set_int ) ).
% Diff_cancel
thf(fact_1172_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_1173_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_1174_insert__Diff1,axiom,
! [X2: int,B2: set_int,A2: set_int] :
( ( member_int @ X2 @ B2 )
=> ( ( minus_minus_set_int @ ( insert_int @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1175_insert__Diff1,axiom,
! [X2: list_a,B2: set_list_a,A2: set_list_a] :
( ( member_list_a @ X2 @ B2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X2 @ A2 ) @ B2 )
= ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1176_insert__Diff1,axiom,
! [X2: set_a,B2: set_set_a,A2: set_set_a] :
( ( member_set_a @ X2 @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1177_insert__Diff1,axiom,
! [X2: mat_int,B2: set_mat_int,A2: set_mat_int] :
( ( member_mat_int @ X2 @ B2 )
=> ( ( minus_7323808874771660652at_int @ ( insert_mat_int @ X2 @ A2 ) @ B2 )
= ( minus_7323808874771660652at_int @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1178_insert__Diff1,axiom,
! [X2: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1179_insert__Diff1,axiom,
! [X2: a,B2: set_a,A2: set_a] :
( ( member_a @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1180_Diff__insert0,axiom,
! [X2: int,A2: set_int,B2: set_int] :
( ~ ( member_int @ X2 @ A2 )
=> ( ( minus_minus_set_int @ A2 @ ( insert_int @ X2 @ B2 ) )
= ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1181_Diff__insert0,axiom,
! [X2: list_a,A2: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ X2 @ A2 )
=> ( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X2 @ B2 ) )
= ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1182_Diff__insert0,axiom,
! [X2: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X2 @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X2 @ B2 ) )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1183_Diff__insert0,axiom,
! [X2: mat_int,A2: set_mat_int,B2: set_mat_int] :
( ~ ( member_mat_int @ X2 @ A2 )
=> ( ( minus_7323808874771660652at_int @ A2 @ ( insert_mat_int @ X2 @ B2 ) )
= ( minus_7323808874771660652at_int @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1184_Diff__insert0,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1185_Diff__insert0,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1186_complete__block__all__subsets,axiom,
! [Bl: set_a,Ps: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ( finite_card_a @ Bl )
= ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
=> ( ord_less_eq_set_a @ Ps @ Bl ) ) ) ) ).
% complete_block_all_subsets
thf(fact_1187_index__zero__iff,axiom,
( ( lambda = zero_zero_nat )
= ( ! [X: set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( finite_card_a @ X )
= one_one_nat ) ) ) ) ).
% index_zero_iff
thf(fact_1188_remove__invalid__point__block,axiom,
! [P2: a,Bl: set_a] :
( ~ ( member_a @ P2 @ ( set_a2 @ v_s ) )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( minus_minus_set_a @ Bl @ ( insert_a @ P2 @ bot_bot_set_a ) )
= Bl ) ) ) ).
% remove_invalid_point_block
thf(fact_1189_block__comp__incomplete,axiom,
! [Bl: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl ) ) ) ) ).
% block_comp_incomplete
thf(fact_1190_block__comp__incomplete__nempty,axiom,
! [Bl: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) )
=> ( ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl )
!= bot_bot_set_a ) ) ).
% block_comp_incomplete_nempty
thf(fact_1191_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1192_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1193_singleton__insert__inj__eq_H,axiom,
! [A: int,A2: set_int,B: int] :
( ( ( insert_int @ A @ A2 )
= ( insert_int @ B @ bot_bot_set_int ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1194_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1195_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1196_singleton__insert__inj__eq,axiom,
! [B: int,A: int,A2: set_int] :
( ( ( insert_int @ B @ bot_bot_set_int )
= ( insert_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1197_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1198_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1199_Diff__eq__empty__iff,axiom,
! [A2: set_int,B2: set_int] :
( ( ( minus_minus_set_int @ A2 @ B2 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1200_set__mset__eq__empty__iff,axiom,
! [M3: multiset_set_a] :
( ( ( set_mset_set_a @ M3 )
= bot_bot_set_set_a )
= ( M3 = zero_z5079479921072680283_set_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_1201_set__mset__eq__empty__iff,axiom,
! [M3: multiset_a] :
( ( ( set_mset_a @ M3 )
= bot_bot_set_a )
= ( M3 = zero_zero_multiset_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_1202_set__mset__eq__empty__iff,axiom,
! [M3: multiset_int] :
( ( ( set_mset_int @ M3 )
= bot_bot_set_int )
= ( M3 = zero_z3170743180189231877et_int ) ) ).
% set_mset_eq_empty_iff
thf(fact_1203_set__mset__eq__empty__iff,axiom,
! [M3: multiset_nat] :
( ( ( set_mset_nat @ M3 )
= bot_bot_set_nat )
= ( M3 = zero_z7348594199698428585et_nat ) ) ).
% set_mset_eq_empty_iff
thf(fact_1204_set__mset__empty,axiom,
( ( set_mset_set_a @ zero_z5079479921072680283_set_a )
= bot_bot_set_set_a ) ).
% set_mset_empty
thf(fact_1205_set__mset__empty,axiom,
( ( set_mset_a @ zero_zero_multiset_a )
= bot_bot_set_a ) ).
% set_mset_empty
thf(fact_1206_set__mset__empty,axiom,
( ( set_mset_int @ zero_z3170743180189231877et_int )
= bot_bot_set_int ) ).
% set_mset_empty
thf(fact_1207_set__mset__empty,axiom,
( ( set_mset_nat @ zero_z7348594199698428585et_nat )
= bot_bot_set_nat ) ).
% set_mset_empty
thf(fact_1208_set__mset__mset,axiom,
! [Xs: list_a] :
( ( set_mset_a @ ( mset_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_mset_mset
thf(fact_1209_set__mset__mset,axiom,
! [Xs: list_set_a] :
( ( set_mset_set_a @ ( mset_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_mset_mset
thf(fact_1210_in__multiset__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ ( mset_nat @ Xs ) ) )
= ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_1211_in__multiset__in__set,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_mset_a @ ( mset_a @ Xs ) ) )
= ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_1212_in__multiset__in__set,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_mset_set_a @ ( mset_set_a @ Xs ) ) )
= ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_1213_incomplete__alt__imp,axiom,
! [Bl: set_a] :
( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ) ).
% incomplete_alt_imp
thf(fact_1214_design__support__def,axiom,
( ( design5397942185814921632port_a @ ( mset_set_a @ b_s ) )
= ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ).
% design_support_def
thf(fact_1215_incidence__alt__def,axiom,
! [P2: a,B: set_a] :
( ( member_a @ P2 @ ( set_a2 @ v_s ) )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design3210447939978979927dent_a @ ( mset_set_a @ b_s ) @ P2 @ B )
= ( member_a @ P2 @ B ) ) ) ) ).
% incidence_alt_def
thf(fact_1216_reals__Archimedean3,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ! [Y3: real] :
? [N3: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X2 ) ) ) ).
% reals_Archimedean3
thf(fact_1217_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y5: real] :
( ( ord_less_real @ X @ Y5 )
| ( X = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_1218_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N2: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1219_n__inter__num__zero,axiom,
! [B1: set_a,B22: set_a] :
( ( member_set_a @ B1 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_set_a @ B22 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design735257067508376852mber_a @ B1 @ zero_zero_nat @ B22 )
= one_one_nat ) ) ) ).
% n_inter_num_zero
thf(fact_1220_str__del__block__del__point,axiom,
! [X2: a] :
( ~ ( member_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design4241783006516448631lock_a @ ( mset_set_a @ b_s ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ X2 ) ) ) ).
% str_del_block_del_point
thf(fact_1221_incidence__mat__des__two__index,axiom,
! [I1: nat,I22: nat] :
( ( ord_less_nat @ I1 @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( ord_less_nat @ I22 @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( I1 != I22 )
=> ( ( incide1709061621920695677ex_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( insert_nat @ I1 @ ( insert_nat @ I22 @ bot_bot_set_nat ) ) )
= lambda ) ) ) ) ).
% incidence_mat_des_two_index
thf(fact_1222_complement__rep__number,axiom,
( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) )
=> ( block_6028206285060069402sign_a @ ( set_a2 @ v_s ) @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) @ ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) @ r ) ) ) ).
% complement_rep_number
thf(fact_1223_complement__blocks__wf,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) )
=> ( ord_less_eq_set_a @ Bl @ ( set_a2 @ v_s ) ) ) ).
% complement_blocks_wf
thf(fact_1224_obtain__comp__block__orig,axiom,
! [Bl1: set_a] :
( ( member_set_a @ Bl1 @ ( set_mset_set_a @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) )
=> ~ ! [Bl23: set_a] :
( ( member_set_a @ Bl23 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( Bl1
!= ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl23 ) ) ) ) ).
% obtain_comp_block_orig
thf(fact_1225_complement__same__b,axiom,
( ( size_s6566526139600085008_set_a @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% complement_same_b
thf(fact_1226_replication__numbers__non__empty,axiom,
( ( ( set_a2 @ v_s )
!= bot_bot_set_a )
=> ( ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) )
!= bot_bot_set_nat ) ) ).
% replication_numbers_non_empty
thf(fact_1227_remove__complete__blocks__set__pbd,axiom,
! [X2: nat,A2: multiset_set_a] :
( ( ord_less_nat @ X2 @ lambda )
=> ( ( ( size_s6566526139600085008_set_a @ A2 )
= X2 )
=> ( ( subset_mset_set_a @ A2 @ ( mset_set_a @ b_s ) )
=> ( ! [A5: set_a] :
( ( member_set_a @ A5 @ ( set_mset_set_a @ A2 ) )
=> ( A5
= ( set_a2 @ v_s ) ) )
=> ( block_5355636846524985331ance_a @ ( set_a2 @ v_s ) @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ A2 ) @ ( minus_minus_nat @ lambda @ X2 ) ) ) ) ) ) ).
% remove_complete_blocks_set_pbd
thf(fact_1228_rep__numbers__constant,axiom,
( ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) )
= ( insert_nat @ r @ bot_bot_set_nat ) ) ).
% rep_numbers_constant
thf(fact_1229_replication__number__single,axiom,
is_singleton_nat @ ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% replication_number_single
thf(fact_1230_block__sizes__non__empty,axiom,
( ( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ) ) ) ).
% block_sizes_non_empty
thf(fact_1231_sys__block__sizes__in,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_nat @ ( finite_card_a @ Bl ) @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ) ) ).
% sys_block_sizes_in
thf(fact_1232_sys__block__sizes__obtain__bl,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
& ( ( finite_card_a @ X3 )
= X2 ) ) ) ).
% sys_block_sizes_obtain_bl
thf(fact_1233_block__sizes__non__empty__set,axiom,
( ( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a )
=> ( ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) )
!= bot_bot_set_nat ) ) ).
% block_sizes_non_empty_set
thf(fact_1234_complement__proper__design,axiom,
( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) )
=> ( design7287791228148780576sign_a @ ( set_a2 @ v_s ) @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ).
% complement_proper_design
thf(fact_1235_proper__design__axioms,axiom,
design7287791228148780576sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).
% proper_design_axioms
thf(fact_1236_del__block__proper,axiom,
! [Bl: set_a] :
( ( ord_less_nat @ one_one_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( design7287791228148780576sign_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl ) ) ) ).
% del_block_proper
thf(fact_1237_proper__designI,axiom,
( ( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
!= zero_zero_nat )
=> ( design7287791228148780576sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ).
% proper_designI
thf(fact_1238_multiple__rep__number,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( block_6028206285060069402sign_a @ ( set_a2 @ v_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ ( times_times_nat @ r @ N ) ) ) ).
% multiple_rep_number
thf(fact_1239_strong__del__block__des,axiom,
! [B: set_a] :
( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ~ ( ord_less_set_a @ Bl2 @ B ) )
=> ( design_design_a @ ( minus_minus_set_a @ ( set_a2 @ v_s ) @ B ) @ ( design4241783006516448631lock_a @ ( mset_set_a @ b_s ) @ B ) ) ) ).
% strong_del_block_des
thf(fact_1240_multiple__1__same,axiom,
( ( repeat_mset_set_a @ one_one_nat @ ( mset_set_a @ b_s ) )
= ( mset_set_a @ b_s ) ) ).
% multiple_1_same
thf(fact_1241_multiple__block__in__original,axiom,
! [B: set_a,N: nat] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) )
=> ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% multiple_block_in_original
thf(fact_1242_wf__design,axiom,
design_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).
% wf_design
thf(fact_1243_multiple__block__in,axiom,
! [N: nat,B: set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ) ) ).
% multiple_block_in
thf(fact_1244_multiple__block__sizes__same,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) )
= ( design1769254222028858111izes_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ) ).
% multiple_block_sizes_same
thf(fact_1245_repeat__mset__block__point__rel,axiom,
! [B: set_a,N: nat,X2: a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) )
=> ( ( member_a @ X2 @ B )
=> ( member_a @ X2 @ ( set_a2 @ v_s ) ) ) ) ).
% repeat_mset_block_point_rel
thf(fact_1246_multiple__is__design,axiom,
! [N: nat] : ( design_design_a @ ( set_a2 @ v_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ).
% multiple_is_design
thf(fact_1247_delete__block__design,axiom,
! [Bl: set_a] : ( design_design_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl ) ) ).
% delete_block_design
thf(fact_1248_add__point__design,axiom,
! [P2: a] : ( design_design_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P2 ) @ ( mset_set_a @ b_s ) ) ).
% add_point_design
thf(fact_1249_multiple__blocks__gt,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) @ ( size_s6566526139600085008_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ) ).
% multiple_blocks_gt
thf(fact_1250_strong__del__point__design,axiom,
! [P2: a] : ( design_design_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P2 ) @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P2 ) ) ).
% strong_del_point_design
thf(fact_1251_multiple__proper__design,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( design7287791228148780576sign_a @ ( set_a2 @ v_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ).
% multiple_proper_design
thf(fact_1252_multiple__orig__sub__system,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( sub_su7923802003039619913stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( set_a2 @ v_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ).
% multiple_orig_sub_system
thf(fact_1253_multiple__orig__sub__des,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( sub_sub_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( set_a2 @ v_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ).
% multiple_orig_sub_des
thf(fact_1254_add__block__design__cond,axiom,
! [Bl: set_a] :
( ( ord_less_eq_set_a @ Bl @ ( set_a2 @ v_s ) )
=> ( ( Bl != bot_bot_set_a )
=> ( design_design_a @ ( set_a2 @ v_s ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ Bl ) ) ) ) ).
% add_block_design_cond
thf(fact_1255_complement__design,axiom,
( ! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) )
=> ( design_design_a @ ( set_a2 @ v_s ) @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ).
% complement_design
thf(fact_1256_multiple__blocks__num,axiom,
! [N: nat] :
( ( size_s6566526139600085008_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) )
= ( times_times_nat @ N @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% multiple_blocks_num
thf(fact_1257_sub__designI,axiom,
! [U2: set_a,A6: multiset_set_a] :
( ( design_design_a @ U2 @ A6 )
=> ( ( sub_su7923802003039619913stem_a @ U2 @ A6 @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) )
=> ( sub_sub_design_a @ U2 @ A6 @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ).
% sub_designI
thf(fact_1258_sub__designII,axiom,
! [U2: set_a,A6: multiset_set_a] :
( ( design_design_a @ U2 @ A6 )
=> ( ( sub_su7923802003039619913stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ U2 @ A6 )
=> ( sub_sub_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ U2 @ A6 ) ) ) ).
% sub_designII
thf(fact_1259_designI,axiom,
( ! [B5: set_a] :
( ( member_set_a @ B5 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( B5 != bot_bot_set_a ) )
=> ( ( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a )
=> ( ( ( set_a2 @ v_s )
!= bot_bot_set_a )
=> ( design_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ) ).
% designI
thf(fact_1260_multiple__not__simple,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a )
=> ~ ( design1338723777345758283stem_a @ ( set_a2 @ v_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ) ).
% multiple_not_simple
thf(fact_1261_wf__design__iff,axiom,
! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) )
= ( ( ord_less_eq_set_a @ Bl @ ( set_a2 @ v_s ) )
& ( finite_finite_a @ ( set_a2 @ v_s ) )
& ( Bl != bot_bot_set_a ) ) ) ) ).
% wf_design_iff
thf(fact_1262_simple__not__multiplicity,axiom,
! [B: set_a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ one_one_nat @ ( count_set_a @ ( mset_set_a @ b_s ) @ B ) )
=> ~ ( design1338723777345758283stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ).
% simple_not_multiplicity
thf(fact_1263_finite__sets,axiom,
finite_finite_a @ ( set_a2 @ v_s ) ).
% finite_sets
thf(fact_1264_finite__blocks,axiom,
! [B: set_a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( finite_finite_a @ B ) ) ).
% finite_blocks
thf(fact_1265_block__original__count__le,axiom,
! [N: nat,B: set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( count_set_a @ ( mset_set_a @ b_s ) @ B ) @ ( count_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ B ) ) ) ).
% block_original_count_le
thf(fact_1266_count__complete__lt__balance,axiom,
ord_less_eq_nat @ ( count_set_a @ ( mset_set_a @ b_s ) @ ( set_a2 @ v_s ) ) @ lambda ).
% count_complete_lt_balance
thf(fact_1267_multiple__block__multiplicity,axiom,
! [N: nat,Bl: set_a] :
( ( count_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ Bl )
= ( times_times_nat @ ( count_set_a @ ( mset_set_a @ b_s ) @ Bl ) @ N ) ) ).
% multiple_block_multiplicity
thf(fact_1268_remove__all__complete__blocks__pbd,axiom,
( ( ord_less_nat @ ( count_set_a @ ( mset_set_a @ b_s ) @ ( set_a2 @ v_s ) ) @ lambda )
=> ( block_5355636846524985331ance_a @ ( set_a2 @ v_s ) @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( replicate_mset_set_a @ ( count_set_a @ ( mset_set_a @ b_s ) @ ( set_a2 @ v_s ) ) @ ( set_a2 @ v_s ) ) ) @ ( minus_minus_nat @ lambda @ ( count_set_a @ ( mset_set_a @ b_s ) @ ( set_a2 @ v_s ) ) ) ) ) ).
% remove_all_complete_blocks_pbd
thf(fact_1269_add__block__sub__des,axiom,
! [B: set_a] :
( ( finite_finite_a @ B )
=> ( ( B != bot_bot_set_a )
=> ( sub_sub_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( sup_sup_set_a @ ( set_a2 @ v_s ) @ B ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B ) ) ) ) ).
% add_block_sub_des
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
= ( plus_plus_mat_int @ ( smult_mat_int @ ( semiri1314217659103216013at_int @ lambda ) @ ( matrix8485685120660989714at_int @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) @ ( smult_mat_int @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ r @ lambda ) ) @ ( one_mat_int @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ) ) ).
%------------------------------------------------------------------------------