TPTP Problem File: SLH0395^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Fishers_Inequality/0038_Fishers_Inequality/prob_00167_009682__28203298_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1623 ( 609 unt; 348 typ; 0 def)
% Number of atoms : 3164 (1180 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 11313 ( 173 ~; 39 |; 145 &;9801 @)
% ( 0 <=>;1155 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 42 ( 41 usr)
% Number of type conns : 591 ( 591 >; 0 *; 0 +; 0 <<)
% Number of symbols : 308 ( 307 usr; 22 con; 0-4 aty)
% Number of variables : 2517 ( 41 ^;2439 !; 37 ?;2517 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:49:44.323
%------------------------------------------------------------------------------
% Could-be-implicit typings (41)
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Matrix__Ovec_It__Int__Oint_J_J_J,type,
multiset_set_vec_int: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Matrix__Ovec_It__Int__Oint_J_J_J,type,
list_set_vec_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Matrix__Ovec_It__Int__Oint_J_J_J,type,
set_set_vec_int: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
multiset_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_list_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
list_set_set_nat: $tType ).
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__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
multiset_set_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Int__Oint_J_J,type,
multiset_set_int: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Ovec_It__Real__Oreal_J_J,type,
list_vec_real: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Omat_It__Real__Oreal_J_J,type,
list_mat_real: $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__Ovec_It__Int__Oint_J_J,type,
list_vec_int: $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__Ovec_It__Int__Oint_J_J,type,
set_vec_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__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
list_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $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__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__Matrix__Ovec_It__Int__Oint_J,type,
vec_int: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Int__Oint_J,type,
mat_int: $tType ).
thf(ty_n_t__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 (307)
thf(sy_c_Design__Basics_Odesign_001t__Matrix__Ovec_It__Int__Oint_J,type,
design3193161975243185419ec_int: set_vec_int > multiset_set_vec_int > $o ).
thf(sy_c_Design__Basics_Odesign_001t__Nat__Onat,type,
design_design_nat: set_nat > multiset_set_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_Ofinite__incidence__system_001t__Matrix__Ovec_It__Int__Oint_J,type,
design5943305819988106723ec_int: set_vec_int > multiset_set_vec_int > $o ).
thf(sy_c_Design__Basics_Ofinite__incidence__system_001t__Nat__Onat,type,
design5426232790142929158em_nat: set_nat > multiset_set_nat > $o ).
thf(sy_c_Design__Basics_Ofinite__incidence__system_001tf__a,type,
design9187838744727572296stem_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Basics_Oincidence__system_001t__Matrix__Ovec_It__Int__Oint_J,type,
design752387947253681460ec_int: set_vec_int > multiset_set_vec_int > $o ).
thf(sy_c_Design__Basics_Oincidence__system_001t__Nat__Onat,type,
design3753904077504641269em_nat: set_nat > multiset_set_nat > $o ).
thf(sy_c_Design__Basics_Oincidence__system_001tf__a,type,
design1863209521793301785stem_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Basics_Oincidence__system_Oblock__complement_001t__Nat__Onat,type,
design2875492832550762736nt_nat: set_nat > set_nat > set_nat ).
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_001t__Nat__Onat,type,
design5569578106646884273ks_nat: set_nat > multiset_set_nat > multiset_set_nat ).
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_001t__Nat__Onat,type,
design4862117536649126062rt_nat: multiset_set_nat > set_set_nat ).
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_001t__Nat__Onat,type,
design8502206366797944887nt_nat: multiset_set_nat > nat > set_nat > $o ).
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_Opoint__indices_001t__Nat__Onat,type,
design1227534709319296284es_nat: set_nat > multiset_set_nat > nat > set_nat ).
thf(sy_c_Design__Basics_Oincidence__system_Opoint__indices_001tf__a,type,
design328527185268214962ices_a: set_a > multiset_set_a > nat > set_nat ).
thf(sy_c_Design__Basics_Oincidence__system_Oreplication__numbers_001t__Nat__Onat,type,
design3853898657598026467rs_nat: set_nat > multiset_set_nat > set_nat ).
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_001t__Nat__Onat,type,
design8152002643121538447es_nat: multiset_set_nat > 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_Ointersection__number_001t__Nat__Onat,type,
design7485525362727208274er_nat: set_nat > set_nat > nat ).
thf(sy_c_Design__Basics_Ointersection__number_001tf__a,type,
design7842873109100088828mber_a: set_a > set_a > 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_Opoint__replication__number_001t__Nat__Onat,type,
design3571518413069006949er_nat: multiset_set_nat > nat > nat ).
thf(sy_c_Design__Basics_Opoint__replication__number_001tf__a,type,
design6637022207325878697mber_a: multiset_set_a > a > nat ).
thf(sy_c_Design__Basics_Opoints__index_001t__Nat__Onat,type,
design6574611146354332593ex_nat: multiset_set_nat > set_nat > nat ).
thf(sy_c_Design__Basics_Opoints__index_001tf__a,type,
design254580327166089565ndex_a: multiset_set_a > set_a > nat ).
thf(sy_c_Design__Basics_Oproper__design_001t__Matrix__Ovec_It__Int__Oint_J,type,
design4550196916521492667ec_int: set_vec_int > multiset_set_vec_int > $o ).
thf(sy_c_Design__Basics_Oproper__design_001t__Nat__Onat,type,
design435815215503836206gn_nat: set_nat > multiset_set_nat > $o ).
thf(sy_c_Design__Basics_Oproper__design_001t__Set__Oset_Itf__a_J,type,
design8920408637772541824_set_a: set_set_a > multiset_set_set_a > $o ).
thf(sy_c_Design__Basics_Oproper__design_001tf__a,type,
design7287791228148780576sign_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__block_001t__Nat__Onat,type,
design4725324266511619850ck_nat: multiset_set_nat > set_nat > multiset_set_nat ).
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_001t__Nat__Onat,type,
design8239173135376323853nt_nat: set_nat > nat > set_nat ).
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_001t__Nat__Onat,type,
design5698312687278145166ks_nat: multiset_set_nat > nat > set_set_nat > multiset_set_nat ).
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_001t__Nat__Onat,type,
design755385109423264192ck_nat: multiset_set_nat > set_nat > multiset_set_nat ).
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_001t__Nat__Onat,type,
design4269233978287968195nt_nat: set_nat > nat > set_nat ).
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_001t__Nat__Onat,type,
design4832208198062110345ks_nat: multiset_set_nat > nat > multiset_set_nat ).
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_001t__Nat__Onat,type,
design3550126062406151447ck_nat: multiset_set_nat > set_nat > multiset_set_nat ).
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_001t__Nat__Onat,type,
design3278834155446248416ks_nat: multiset_set_nat > nat > multiset_set_nat ).
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_Dual__Systems_Odual__blocks_001tf__a,type,
dual_dual_blocks_a: set_a > list_set_a > multiset_set_nat ).
thf(sy_c_Dual__Systems_Oordered__incidence__system_Odual__blocks__ordered_001tf__a,type,
dual_o8653602421375082564ered_a: list_a > list_set_a > list_set_nat ).
thf(sy_c_Dual__Systems_Oordered__incidence__system_Ois__dual_001t__Nat__Onat_001t__Nat__Onat,type,
dual_o7421254750609082426at_nat: list_nat > list_set_nat > list_nat > list_set_nat > $o ).
thf(sy_c_Dual__Systems_Oordered__incidence__system_Ois__dual_001t__Nat__Onat_001tf__a,type,
dual_o2119597811525285140_nat_a: list_nat > list_set_nat > list_a > list_set_a > $o ).
thf(sy_c_Dual__Systems_Oordered__incidence__system_Ois__dual_001tf__a_001t__Nat__Onat,type,
dual_o7311038347201774518_a_nat: list_a > list_set_a > list_nat > list_set_nat > $o ).
thf(sy_c_Dual__Systems_Oordered__incidence__system_Ois__dual_001tf__a_001tf__a,type,
dual_o5859382014055506072al_a_a: list_a > list_set_a > list_a > list_set_a > $o ).
thf(sy_c_Finite__Set_Ocard_001t__Matrix__Ovec_It__Int__Oint_J,type,
finite_card_vec_int: set_vec_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_It__Nat__Onat_J,type,
finite_card_set_nat: set_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_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
finite1091814263879798189et_nat: set_list_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
finite1971793804006318733_set_a: set_list_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Matrix__Ovec_It__Int__Oint_J,type,
finite8375754479472695874ec_int: set_vec_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
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__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
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_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_It__Nat__Onat_J_J,type,
plus_p8712254050562127327et_nat: multiset_set_nat > multiset_set_nat > multiset_set_nat ).
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__Matrix__Omat_It__Real__Oreal_J,type,
times_times_mat_real: mat_real > mat_real > mat_real ).
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__Set__Oset_It__Nat__Onat_J_J,type,
zero_z3157962936165190495et_nat: multiset_set_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__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_Incidence__Matrices_Oinc__mat__of_001t__Matrix__Ovec_It__Int__Oint_J_001t__Int__Oint,type,
incide2612794423365091098nt_int: list_vec_int > list_set_vec_int > mat_int ).
thf(sy_c_Incidence__Matrices_Oinc__mat__of_001t__Nat__Onat_001t__Int__Oint,type,
incide1177982898701834729at_int: list_nat > list_set_nat > mat_int ).
thf(sy_c_Incidence__Matrices_Oinc__mat__of_001t__Set__Oset_It__Nat__Onat_J_001t__Int__Oint,type,
incide8356258661762153887at_int: list_set_nat > list_set_set_nat > mat_int ).
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_Oinc__vec__of_001t__Nat__Onat_001t__Int__Oint,type,
incide8105300093900788317at_int: list_nat > set_nat > vec_int ).
thf(sy_c_Incidence__Matrices_Oinc__vec__of_001tf__a_001t__Int__Oint,type,
incide4681487840832596491_a_int: list_a > set_a > vec_int ).
thf(sy_c_Incidence__Matrices_Ois__incidence__matrix_001t__Int__Oint_001t__Matrix__Ovec_It__Int__Oint_J,type,
incide3638075654752512162ec_int: mat_int > set_vec_int > multiset_set_vec_int > $o ).
thf(sy_c_Incidence__Matrices_Ois__incidence__matrix_001t__Int__Oint_001t__Nat__Onat,type,
incide8019310070229209863nt_nat: mat_int > set_nat > multiset_set_nat > $o ).
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__block__size_001t__Int__Oint,type,
incide7086334917823153261ze_int: mat_int > nat > nat ).
thf(sy_c_Incidence__Matrices_Omat__inter__num_001t__Int__Oint,type,
incide903612199826827877um_int: mat_int > nat > nat > nat ).
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_Onon__empty__col_001t__Real__Oreal,type,
incide8049862060206209947l_real: mat_real > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__design_001t__Matrix__Ovec_It__Int__Oint_J,type,
incide7599675121277616863ec_int: list_vec_int > list_set_vec_int > $o ).
thf(sy_c_Incidence__Matrices_Oordered__design_001t__Nat__Onat,type,
incide8999572217031194378gn_nat: list_nat > list_set_nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__design_001tf__a,type,
incide2848671379600480836sign_a: list_a > list_set_a > $o ).
thf(sy_c_Incidence__Matrices_Oordered__incidence__system_001t__Int__Oint,type,
incide6996049454332333349em_int: list_int > list_set_int > $o ).
thf(sy_c_Incidence__Matrices_Oordered__incidence__system_001t__Matrix__Ovec_It__Int__Oint_J,type,
incide9132343510602215008ec_int: list_vec_int > list_set_vec_int > $o ).
thf(sy_c_Incidence__Matrices_Oordered__incidence__system_001t__Nat__Onat,type,
incide6998539924841383625em_nat: list_nat > list_set_nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__incidence__system_001t__Set__Oset_It__Nat__Onat_J,type,
incide937065307065650815et_nat: list_set_nat > list_set_set_nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__incidence__system_001t__Set__Oset_Itf__a_J,type,
incide2166342132139297189_set_a: list_set_a > list_set_set_a > $o ).
thf(sy_c_Incidence__Matrices_Oordered__incidence__system_001tf__a,type,
incide1624170830610365509stem_a: list_a > list_set_a > $o ).
thf(sy_c_Incidence__Matrices_Oordered__proper__design_001t__Matrix__Ovec_It__Int__Oint_J,type,
incide8030590239390523023ec_int: list_vec_int > list_set_vec_int > $o ).
thf(sy_c_Incidence__Matrices_Oordered__proper__design_001t__Nat__Onat,type,
incide1001368407746664282gn_nat: list_nat > list_set_nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__proper__design_001tf__a,type,
incide3676903341588786676sign_a: list_a > list_set_a > $o ).
thf(sy_c_Incidence__Matrices_Oordered__simple__design_001t__Nat__Onat,type,
incide8427317466731264060gn_nat: list_nat > list_set_nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__simple__design_001tf__a,type,
incide371748008924627346sign_a: list_a > list_set_a > $o ).
thf(sy_c_Incidence__Matrices_Oproper__inc__mat_001t__Int__Oint,type,
incide294466202882093137at_int: mat_int > $o ).
thf(sy_c_Incidence__Matrices_Oproper__inc__mat_001t__Real__Oreal,type,
incide8174291662766164561t_real: mat_real > $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_Omap__col__to__block_001t__Int__Oint,type,
incide3973235006681262014ck_int: vec_int > set_nat ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix__int,type,
incide8301514189696901506ix_int: mat_int > $o ).
thf(sy_c_Incidence__Matrices_Ozero__one__matrix__ring__1_001t__Int__Oint,type,
incide6080938071136783841_1_int: mat_int > $o ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Matrix__Ovec_It__Int__Oint_J_J,type,
sup_sup_set_vec_int: set_vec_int > set_vec_int > set_vec_int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
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_Odistinct_001t__Int__Oint,type,
distinct_int: list_int > $o ).
thf(sy_c_List_Odistinct_001t__Matrix__Ovec_It__Int__Oint_J,type,
distinct_vec_int: list_vec_int > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_It__Nat__Onat_J,type,
distinct_set_nat: list_set_nat > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_Itf__a_J,type,
distinct_set_a: list_set_a > $o ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__Matrix__Ovec_It__Int__Oint_J,type,
nil_vec_int: list_vec_int ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Matrix__Ovec_It__Int__Oint_J_J,type,
nil_set_vec_int: list_set_vec_int ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Nat__Onat_J,type,
nil_set_nat: list_set_nat ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_Itf__a_J,type,
nil_set_a: list_set_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Omap_001t__Matrix__Ovec_It__Int__Oint_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_vec_int_set_nat: ( vec_int > set_nat ) > list_vec_int > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
map_set_a_set_a: ( set_a > set_a ) > list_set_a > list_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__Matrix__Omat_It__Int__Oint_J,type,
set_mat_int2: list_mat_int > set_mat_int ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Omat_It__Real__Oreal_J,type,
set_mat_real2: list_mat_real > set_mat_real ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Ovec_It__Int__Oint_J,type,
set_vec_int2: list_vec_int > set_vec_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_It__Nat__Onat_J,type,
set_set_nat2: list_set_nat > set_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_List_Onth_001t__Matrix__Ovec_It__Int__Oint_J,type,
nth_vec_int: list_vec_int > nat > vec_int ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
nth_set_nat: list_set_nat > nat > set_nat ).
thf(sy_c_List_Onth_001t__Set__Oset_Itf__a_J,type,
nth_set_a: list_set_a > nat > set_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Ounion_001t__Matrix__Ovec_It__Int__Oint_J,type,
union_vec_int: list_vec_int > list_vec_int > list_vec_int ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001tf__a,type,
union_a: list_a > list_a > list_a ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Int__Oint,type,
carrier_mat_int: nat > nat > set_mat_int ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Nat__Onat,type,
carrier_mat_nat: nat > nat > set_mat_nat ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Real__Oreal,type,
carrier_mat_real: nat > nat > set_mat_real ).
thf(sy_c_Matrix_Ocol_001t__Int__Oint,type,
col_int: mat_int > nat > vec_int ).
thf(sy_c_Matrix_Ocols_001t__Int__Oint,type,
cols_int: mat_int > list_vec_int ).
thf(sy_c_Matrix_Ocols_001t__Real__Oreal,type,
cols_real: mat_real > list_vec_real ).
thf(sy_c_Matrix_Ocomm__monoid__add__class_Osum__mat_001t__Int__Oint,type,
comm_m4053738856622352096at_int: mat_int > int ).
thf(sy_c_Matrix_Ocomm__monoid__add__class_Osum__mat_001t__Nat__Onat,type,
comm_m4056229327131402372at_nat: mat_nat > nat ).
thf(sy_c_Matrix_Ocomm__monoid__add__class_Osum__mat_001t__Real__Oreal,type,
comm_m8678487124704766304t_real: mat_real > real ).
thf(sy_c_Matrix_Odim__col_001t__Int__Oint,type,
dim_col_int: mat_int > nat ).
thf(sy_c_Matrix_Odim__col_001t__Real__Oreal,type,
dim_col_real: mat_real > nat ).
thf(sy_c_Matrix_Odim__row_001t__Int__Oint,type,
dim_row_int: mat_int > nat ).
thf(sy_c_Matrix_Odim__row_001t__Real__Oreal,type,
dim_row_real: mat_real > nat ).
thf(sy_c_Matrix_Odim__vec_001t__Int__Oint,type,
dim_vec_int: vec_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_Omap__mat_001t__Int__Oint_001t__Int__Oint,type,
map_mat_int_int: ( int > int ) > mat_int > mat_int ).
thf(sy_c_Matrix_Omap__mat_001t__Int__Oint_001t__Real__Oreal,type,
map_mat_int_real: ( int > real ) > mat_int > mat_real ).
thf(sy_c_Matrix_Omap__mat_001t__Real__Oreal_001t__Int__Oint,type,
map_mat_real_int: ( real > int ) > mat_real > mat_int ).
thf(sy_c_Matrix_Omap__mat_001t__Real__Oreal_001t__Real__Oreal,type,
map_mat_real_real: ( real > real ) > mat_real > mat_real ).
thf(sy_c_Matrix_Opow__mat_001t__Int__Oint,type,
pow_mat_int: mat_int > nat > mat_int ).
thf(sy_c_Matrix_Opow__mat_001t__Real__Oreal,type,
pow_mat_real: mat_real > nat > mat_real ).
thf(sy_c_Matrix_Orow_001t__Int__Oint,type,
row_int: mat_int > nat > vec_int ).
thf(sy_c_Matrix_Orows_001t__Int__Oint,type,
rows_int: mat_int > list_vec_int ).
thf(sy_c_Matrix_Orows_001t__Real__Oreal,type,
rows_real: mat_real > list_vec_real ).
thf(sy_c_Matrix_Osimilar__mat__wit_001t__Int__Oint,type,
similar_mat_wit_int: mat_int > mat_int > mat_int > mat_int > $o ).
thf(sy_c_Matrix_Osimilar__mat__wit_001t__Real__Oreal,type,
similar_mat_wit_real: mat_real > mat_real > mat_real > mat_real > $o ).
thf(sy_c_Matrix_Otranspose__mat_001t__Int__Oint,type,
transpose_mat_int: mat_int > mat_int ).
thf(sy_c_Matrix_Otranspose__mat_001t__Real__Oreal,type,
transpose_mat_real: mat_real > mat_real ).
thf(sy_c_Matrix_Ovec__index_001t__Int__Oint,type,
vec_index_int: vec_int > nat > int ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001t__Int__Oint_001t__Int__Oint,type,
matrix4192408864422109198nt_int: ( int > int ) > $o ).
thf(sy_c_Matrix__Vector__Extras_Oinj__on__01__hom_001t__Int__Oint_001t__Real__Oreal,type,
matrix8987650032624085902t_real: ( int > real ) > $o ).
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_More__Matrix_Otake__cols_001t__Int__Oint,type,
more_take_cols_int: mat_int > list_nat > mat_int ).
thf(sy_c_More__Matrix_Otake__cols_001t__Real__Oreal,type,
more_take_cols_real: mat_real > list_nat > mat_real ).
thf(sy_c_Multiset_Omset_001t__Set__Oset_It__Int__Oint_J,type,
mset_set_int: list_set_int > multiset_set_int ).
thf(sy_c_Multiset_Omset_001t__Set__Oset_It__Matrix__Ovec_It__Int__Oint_J_J,type,
mset_set_vec_int: list_set_vec_int > multiset_set_vec_int ).
thf(sy_c_Multiset_Omset_001t__Set__Oset_It__Nat__Onat_J,type,
mset_set_nat: list_set_nat > multiset_set_nat ).
thf(sy_c_Multiset_Omset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
mset_set_set_a: list_set_set_a > multiset_set_set_a ).
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_Oset__mset_001t__Set__Oset_It__Int__Oint_J,type,
set_mset_set_int: multiset_set_int > set_set_int ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_It__Matrix__Ovec_It__Int__Oint_J_J,type,
set_mset_set_vec_int: multiset_set_vec_int > set_set_vec_int ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_It__Nat__Onat_J,type,
set_mset_set_nat: multiset_set_nat > set_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_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__Matrix__Omat_It__Real__Oreal_J_J,type,
size_s7383440822540753431t_real: list_mat_real > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Ovec_It__Int__Oint_J_J,type,
size_s6547088122282965643ec_int: list_vec_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_It__Nat__Onat_J_J,type,
size_s3254054031482475050et_nat: list_set_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_It__Matrix__Ovec_It__Int__Oint_J_J_J,type,
size_s3706287384392134315ec_int: multiset_set_vec_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
size_s7462436076474991978et_nat: multiset_set_nat > 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_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Matrix__Ovec_It__Int__Oint_J_J,type,
bot_bot_set_vec_int: set_vec_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__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_It__Nat__Onat_J_J,type,
ord_le6127582356673144618et_nat: multiset_set_nat > multiset_set_nat > $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__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $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_It__Nat__Onat_J_J,type,
ord_le4034546139768944438et_nat: multiset_set_nat > multiset_set_nat > $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__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__Matrix__Omat_It__Real__Oreal_J_J,type,
ord_le6763538137481655845t_real: set_mat_real > set_mat_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_It__Int__Oint_J_J,type,
ord_le5046699514128596377ec_int: set_vec_int > set_vec_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__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $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_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
real_V1803761363581548252l_real: real > real ).
thf(sy_c_Ring__Hom_Oab__semigroup__add__hom_001t__Int__Oint_001t__Real__Oreal,type,
ring_a779443308300532965t_real: ( int > real ) > $o ).
thf(sy_c_Ring__Hom_Oab__semigroup__mult__hom_001t__Int__Oint_001t__Int__Oint,type,
ring_a7291918497645896362nt_int: ( int > int ) > $o ).
thf(sy_c_Ring__Hom_Oab__semigroup__mult__hom_001t__Int__Oint_001t__Real__Oreal,type,
ring_a4547970760173101098t_real: ( int > real ) > $o ).
thf(sy_c_Ring__Hom_Ocomm__monoid__add__hom__0_001t__Int__Oint_001t__Int__Oint,type,
ring_c6059480546188650321nt_int: ( int > int ) > $o ).
thf(sy_c_Ring__Hom_Ocomm__monoid__add__hom__0_001t__Int__Oint_001t__Real__Oreal,type,
ring_c3298759929950863697t_real: ( int > real ) > $o ).
thf(sy_c_Ring__Hom_Ocomm__monoid__mult__hom__1_001t__Int__Oint_001t__Int__Oint,type,
ring_c5842864267467415849nt_int: ( int > int ) > $o ).
thf(sy_c_Ring__Hom_Ocomm__monoid__mult__hom__1_001t__Int__Oint_001t__Real__Oreal,type,
ring_c8180009538971569961t_real: ( int > real ) > $o ).
thf(sy_c_Ring__Hom_Ocomm__semiring__hom_001t__Int__Oint_001t__Int__Oint,type,
ring_c1049078972110686252nt_int: ( int > int ) > $o ).
thf(sy_c_Ring__Hom_Ocomm__semiring__hom_001t__Int__Oint_001t__Real__Oreal,type,
ring_c7120671943487988908t_real: ( int > real ) > $o ).
thf(sy_c_Ring__Hom_Ocomm__semiring__hom_001t__Real__Oreal_001t__Int__Oint,type,
ring_c275196434560039724al_int: ( real > int ) > $o ).
thf(sy_c_Ring__Hom_Ocomm__semiring__hom_001t__Real__Oreal_001t__Real__Oreal,type,
ring_c5154114807373371308l_real: ( real > real ) > $o ).
thf(sy_c_Ring__Hom_Omonoid__add__hom__0_001t__Int__Oint_001t__Int__Oint,type,
ring_m4348472603105607453nt_int: ( int > int ) > $o ).
thf(sy_c_Ring__Hom_Omonoid__add__hom__0_001t__Int__Oint_001t__Real__Oreal,type,
ring_m8610364392332473117t_real: ( int > real ) > $o ).
thf(sy_c_Ring__Hom_Omonoid__mult__hom__1_001t__Int__Oint_001t__Int__Oint,type,
ring_m4647122448705125469nt_int: ( int > int ) > $o ).
thf(sy_c_Ring__Hom_Omonoid__mult__hom__1_001t__Int__Oint_001t__Real__Oreal,type,
ring_m3572329010447437917t_real: ( int > real ) > $o ).
thf(sy_c_Ring__Hom_Osemiring__hom_001t__Int__Oint_001t__Int__Oint,type,
ring_s4547151187181212152nt_int: ( int > int ) > $o ).
thf(sy_c_Ring__Hom_Osemiring__hom_001t__Int__Oint_001t__Nat__Onat,type,
ring_s4549641657690262428nt_nat: ( int > nat ) > $o ).
thf(sy_c_Ring__Hom_Osemiring__hom_001t__Int__Oint_001t__Real__Oreal,type,
ring_s3477937289336866936t_real: ( int > real ) > $o ).
thf(sy_c_Ring__Hom_Osemiring__hom_001t__Nat__Onat_001t__Int__Oint,type,
ring_s3547797475280157596at_int: ( nat > int ) > $o ).
thf(sy_c_Ring__Hom_Osemiring__hom_001t__Nat__Onat_001t__Nat__Onat,type,
ring_s3550287945789207872at_nat: ( nat > nat ) > $o ).
thf(sy_c_Ring__Hom_Osemiring__hom_001t__Nat__Onat_001t__Real__Oreal,type,
ring_s1291015714643845148t_real: ( nat > real ) > $o ).
thf(sy_c_Ring__Hom_Osemiring__hom_001t__Real__Oreal_001t__Int__Oint,type,
ring_s5855833817263693560al_int: ( real > int ) > $o ).
thf(sy_c_Ring__Hom_Osemiring__hom_001t__Real__Oreal_001t__Nat__Onat,type,
ring_s5858324287772743836al_nat: ( real > nat ) > $o ).
thf(sy_c_Ring__Hom_Osemiring__hom_001t__Real__Oreal_001t__Real__Oreal,type,
ring_s1386325929209763192l_real: ( real > real ) > $o ).
thf(sy_c_Ring__Hom__Matrix_Oreal__embedding__class_Oreal__of_001t__Int__Oint,type,
ring_H3062062880446257660of_int: int > real ).
thf(sy_c_Ring__Hom__Matrix_Oreal__embedding__class_Oreal__of_001t__Real__Oreal,type,
ring_H569718986840772220f_real: real > real ).
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__Matrix__Omat_It__Real__Oreal_J,type,
collect_mat_real: ( mat_real > $o ) > set_mat_real ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_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__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
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_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
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_001t__Nat__Onat,type,
sub_su5953739893325741765em_nat: set_nat > multiset_set_nat > set_nat > multiset_set_nat > $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_It__Nat__Onat_J_J,type,
member_list_set_nat: list_set_nat > set_list_set_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__Matrix__Ovec_It__Int__Oint_J,type,
member_vec_int: vec_int > set_vec_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Matrix__Ovec_It__Int__Oint_J_J,type,
member_set_vec_int: set_vec_int > set_set_vec_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_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_060V_062s,type,
v_s: list_a ).
% Relevant facts (1274)
thf(fact_0_of__int__hom_Osemiring__hom__axioms,axiom,
ring_s4547151187181212152nt_int @ ring_1_of_int_int ).
% of_int_hom.semiring_hom_axioms
thf(fact_1_of__int__hom_Osemiring__hom__axioms,axiom,
ring_s3477937289336866936t_real @ ring_1_of_int_real ).
% of_int_hom.semiring_hom_axioms
thf(fact_2_map__mat__transpose,axiom,
! [F: int > int,A: mat_int] :
( ( transpose_mat_int @ ( map_mat_int_int @ F @ A ) )
= ( map_mat_int_int @ F @ ( transpose_mat_int @ A ) ) ) ).
% map_mat_transpose
thf(fact_3_map__mat__transpose,axiom,
! [F: real > int,A: mat_real] :
( ( transpose_mat_int @ ( map_mat_real_int @ F @ A ) )
= ( map_mat_real_int @ F @ ( transpose_mat_real @ A ) ) ) ).
% map_mat_transpose
thf(fact_4_map__mat__transpose,axiom,
! [F: int > real,A: mat_int] :
( ( transpose_mat_real @ ( map_mat_int_real @ F @ A ) )
= ( map_mat_int_real @ F @ ( transpose_mat_int @ A ) ) ) ).
% map_mat_transpose
thf(fact_5_map__mat__transpose,axiom,
! [F: real > real,A: mat_real] :
( ( transpose_mat_real @ ( map_mat_real_real @ F @ A ) )
= ( map_mat_real_real @ F @ ( transpose_mat_real @ A ) ) ) ).
% map_mat_transpose
thf(fact_6_semiring__hom_Omat__hom__mult,axiom,
! [Hom: int > int,A: mat_int,Nr: nat,N: nat,B: mat_int,Nc: nat] :
( ( ring_s4547151187181212152nt_int @ Hom )
=> ( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( map_mat_int_int @ Hom @ ( times_times_mat_int @ A @ B ) )
= ( times_times_mat_int @ ( map_mat_int_int @ Hom @ A ) @ ( map_mat_int_int @ Hom @ B ) ) ) ) ) ) ).
% semiring_hom.mat_hom_mult
thf(fact_7_semiring__hom_Omat__hom__mult,axiom,
! [Hom: int > real,A: mat_int,Nr: nat,N: nat,B: mat_int,Nc: nat] :
( ( ring_s3477937289336866936t_real @ Hom )
=> ( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( map_mat_int_real @ Hom @ ( times_times_mat_int @ A @ B ) )
= ( times_times_mat_real @ ( map_mat_int_real @ Hom @ A ) @ ( map_mat_int_real @ Hom @ B ) ) ) ) ) ) ).
% semiring_hom.mat_hom_mult
thf(fact_8_semiring__hom_Omat__hom__mult,axiom,
! [Hom: real > int,A: mat_real,Nr: nat,N: nat,B: mat_real,Nc: nat] :
( ( ring_s5855833817263693560al_int @ Hom )
=> ( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ N ) )
=> ( ( member_mat_real @ B @ ( carrier_mat_real @ N @ Nc ) )
=> ( ( map_mat_real_int @ Hom @ ( times_times_mat_real @ A @ B ) )
= ( times_times_mat_int @ ( map_mat_real_int @ Hom @ A ) @ ( map_mat_real_int @ Hom @ B ) ) ) ) ) ) ).
% semiring_hom.mat_hom_mult
thf(fact_9_semiring__hom_Omat__hom__mult,axiom,
! [Hom: real > real,A: mat_real,Nr: nat,N: nat,B: mat_real,Nc: nat] :
( ( ring_s1386325929209763192l_real @ Hom )
=> ( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ N ) )
=> ( ( member_mat_real @ B @ ( carrier_mat_real @ N @ Nc ) )
=> ( ( map_mat_real_real @ Hom @ ( times_times_mat_real @ A @ B ) )
= ( times_times_mat_real @ ( map_mat_real_real @ Hom @ A ) @ ( map_mat_real_real @ Hom @ B ) ) ) ) ) ) ).
% semiring_hom.mat_hom_mult
thf(fact_10_mat__is__proper,axiom,
incide294466202882093137at_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ).
% mat_is_proper
thf(fact_11_is__dualD1,axiom,
! [Vs: list_a,Bs: list_set_a] :
( ( dual_o5859382014055506072al_a_a @ v_s @ b_s @ Vs @ Bs )
=> ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ).
% is_dualD1
thf(fact_12_is__dualD1,axiom,
! [Vs: list_nat,Bs: list_set_nat] :
( ( dual_o7311038347201774518_a_nat @ v_s @ b_s @ Vs @ Bs )
=> ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ).
% is_dualD1
thf(fact_13_ordered__design__axioms,axiom,
incide2848671379600480836sign_a @ v_s @ b_s ).
% ordered_design_axioms
thf(fact_14_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_15_ordered__proper__design__axioms,axiom,
incide3676903341588786676sign_a @ v_s @ b_s ).
% ordered_proper_design_axioms
thf(fact_16_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
= ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_mult
thf(fact_17_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
= ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_mult
thf(fact_18_of__int__hom_Ohom__mult,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_real @ ( times_times_int @ X @ Y ) )
= ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) ) ) ).
% of_int_hom.hom_mult
thf(fact_19_of__int__hom_Ohom__mult,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_int @ ( times_times_int @ X @ Y ) )
= ( times_times_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) ) ) ).
% of_int_hom.hom_mult
thf(fact_20_zero__one__matrix__axioms,axiom,
incide4964164200581851450ix_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ).
% zero_one_matrix_axioms
thf(fact_21_transpose__mat__eq,axiom,
! [A: mat_int,B: mat_int] :
( ( ( transpose_mat_int @ A )
= ( transpose_mat_int @ B ) )
= ( A = B ) ) ).
% transpose_mat_eq
thf(fact_22_transpose__mat__eq,axiom,
! [A: mat_real,B: mat_real] :
( ( ( transpose_mat_real @ A )
= ( transpose_mat_real @ B ) )
= ( A = B ) ) ).
% transpose_mat_eq
thf(fact_23_Matrix_Otranspose__transpose,axiom,
! [A: mat_int] :
( ( transpose_mat_int @ ( transpose_mat_int @ A ) )
= A ) ).
% Matrix.transpose_transpose
thf(fact_24_Matrix_Otranspose__transpose,axiom,
! [A: mat_real] :
( ( transpose_mat_real @ ( transpose_mat_real @ A ) )
= A ) ).
% Matrix.transpose_transpose
thf(fact_25_Ring__Hom_Oof__int__hom_Oeq__iff,axiom,
! [X: int,Y: int] :
( ( ( ring_1_of_int_real @ X )
= ( ring_1_of_int_real @ Y ) )
= ( X = Y ) ) ).
% Ring_Hom.of_int_hom.eq_iff
thf(fact_26_Ring__Hom_Oof__int__hom_Oeq__iff,axiom,
! [X: int,Y: int] :
( ( ( ring_1_of_int_int @ X )
= ( ring_1_of_int_int @ Y ) )
= ( X = Y ) ) ).
% Ring_Hom.of_int_hom.eq_iff
thf(fact_27_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_real @ W )
= ( ring_1_of_int_real @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_28_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_int @ W )
= ( ring_1_of_int_int @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_29_Matrix__Vector__Extras_Oof__int__hom_Oeq__iff,axiom,
! [X: int,Univ: set_int,Y: int] :
( ( member_int @ X @ Univ )
=> ( ( member_int @ Y @ Univ )
=> ( ( ( ring_1_of_int_real @ X )
= ( ring_1_of_int_real @ Y ) )
= ( X = Y ) ) ) ) ).
% Matrix_Vector_Extras.of_int_hom.eq_iff
thf(fact_30_Matrix__Vector__Extras_Oof__int__hom_Oeq__iff,axiom,
! [X: int,Univ: set_int,Y: int] :
( ( member_int @ X @ Univ )
=> ( ( member_int @ Y @ Univ )
=> ( ( ( ring_1_of_int_int @ X )
= ( ring_1_of_int_int @ Y ) )
= ( X = Y ) ) ) ) ).
% Matrix_Vector_Extras.of_int_hom.eq_iff
thf(fact_31_assoc__mult__mat,axiom,
! [A: mat_int,N_1: nat,N_2: nat,B: mat_int,N_3: nat,C: mat_int,N_4: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N_1 @ N_2 ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N_2 @ N_3 ) )
=> ( ( member_mat_int @ C @ ( carrier_mat_int @ N_3 @ N_4 ) )
=> ( ( times_times_mat_int @ ( times_times_mat_int @ A @ B ) @ C )
= ( times_times_mat_int @ A @ ( times_times_mat_int @ B @ C ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_32_assoc__mult__mat,axiom,
! [A: mat_real,N_1: nat,N_2: nat,B: mat_real,N_3: nat,C: mat_real,N_4: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N_1 @ N_2 ) )
=> ( ( member_mat_real @ B @ ( carrier_mat_real @ N_2 @ N_3 ) )
=> ( ( member_mat_real @ C @ ( carrier_mat_real @ N_3 @ N_4 ) )
=> ( ( times_times_mat_real @ ( times_times_mat_real @ A @ B ) @ C )
= ( times_times_mat_real @ A @ ( times_times_mat_real @ B @ C ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_33_transpose__carrier__mat,axiom,
! [A: mat_int,Nc: nat,Nr: nat] :
( ( member_mat_int @ ( transpose_mat_int @ A ) @ ( carrier_mat_int @ Nc @ Nr ) )
= ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_34_transpose__carrier__mat,axiom,
! [A: mat_real,Nc: nat,Nr: nat] :
( ( member_mat_real @ ( transpose_mat_real @ A ) @ ( carrier_mat_real @ Nc @ Nr ) )
= ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_35_map__carrier__mat,axiom,
! [F: int > int,A: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_int @ ( map_mat_int_int @ F @ A ) @ ( carrier_mat_int @ Nr @ Nc ) )
= ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ Nc ) ) ) ).
% map_carrier_mat
thf(fact_36_map__carrier__mat,axiom,
! [F: real > int,A: mat_real,Nr: nat,Nc: nat] :
( ( member_mat_int @ ( map_mat_real_int @ F @ A ) @ ( carrier_mat_int @ Nr @ Nc ) )
= ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ Nc ) ) ) ).
% map_carrier_mat
thf(fact_37_map__carrier__mat,axiom,
! [F: int > real,A: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_real @ ( map_mat_int_real @ F @ A ) @ ( carrier_mat_real @ Nr @ Nc ) )
= ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ Nc ) ) ) ).
% map_carrier_mat
thf(fact_38_map__carrier__mat,axiom,
! [F: real > real,A: mat_real,Nr: nat,Nc: nat] :
( ( member_mat_real @ ( map_mat_real_real @ F @ A ) @ ( carrier_mat_real @ Nr @ Nc ) )
= ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ Nc ) ) ) ).
% map_carrier_mat
thf(fact_39_mult__carrier__mat,axiom,
! [A: mat_int,Nr: nat,N: nat,B: mat_int,Nc: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N @ Nc ) )
=> ( member_mat_int @ ( times_times_mat_int @ A @ B ) @ ( carrier_mat_int @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_40_mult__carrier__mat,axiom,
! [A: mat_real,Nr: nat,N: nat,B: mat_real,Nc: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ N ) )
=> ( ( member_mat_real @ B @ ( carrier_mat_real @ N @ Nc ) )
=> ( member_mat_real @ ( times_times_mat_real @ A @ B ) @ ( carrier_mat_real @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_41_of__int__hom_Omat__hom__mult,axiom,
! [A: mat_int,Nr: nat,N: nat,B: mat_int,Nc: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( map_mat_int_int @ ring_1_of_int_int @ ( times_times_mat_int @ A @ B ) )
= ( times_times_mat_int @ ( map_mat_int_int @ ring_1_of_int_int @ A ) @ ( map_mat_int_int @ ring_1_of_int_int @ B ) ) ) ) ) ).
% of_int_hom.mat_hom_mult
thf(fact_42_of__int__hom_Omat__hom__mult,axiom,
! [A: mat_int,Nr: nat,N: nat,B: mat_int,Nc: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( map_mat_int_real @ ring_1_of_int_real @ ( times_times_mat_int @ A @ B ) )
= ( times_times_mat_real @ ( map_mat_int_real @ ring_1_of_int_real @ A ) @ ( map_mat_int_real @ ring_1_of_int_real @ B ) ) ) ) ) ).
% of_int_hom.mat_hom_mult
thf(fact_43_of__int__hom_Oinjectivity__lim,axiom,
! [X: int,Univ: set_int,Y: int] :
( ( member_int @ X @ Univ )
=> ( ( member_int @ Y @ Univ )
=> ( ( ( ring_1_of_int_real @ X )
= ( ring_1_of_int_real @ Y ) )
=> ( X = Y ) ) ) ) ).
% of_int_hom.injectivity_lim
thf(fact_44_of__int__hom_Oinjectivity__lim,axiom,
! [X: int,Univ: set_int,Y: int] :
( ( member_int @ X @ Univ )
=> ( ( member_int @ Y @ Univ )
=> ( ( ( ring_1_of_int_int @ X )
= ( ring_1_of_int_int @ Y ) )
=> ( X = Y ) ) ) ) ).
% of_int_hom.injectivity_lim
thf(fact_45_of__int__hom_Oinjectivity,axiom,
! [X: int,Y: int] :
( ( ( ring_1_of_int_real @ X )
= ( ring_1_of_int_real @ Y ) )
=> ( X = Y ) ) ).
% of_int_hom.injectivity
thf(fact_46_of__int__hom_Oinjectivity,axiom,
! [X: int,Y: int] :
( ( ( ring_1_of_int_int @ X )
= ( ring_1_of_int_int @ Y ) )
=> ( X = Y ) ) ).
% of_int_hom.injectivity
thf(fact_47_transpose__mult,axiom,
! [A: mat_int,Nr: nat,N: nat,B: mat_int,Nc: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( transpose_mat_int @ ( times_times_mat_int @ A @ B ) )
= ( times_times_mat_int @ ( transpose_mat_int @ B ) @ ( transpose_mat_int @ A ) ) ) ) ) ).
% transpose_mult
thf(fact_48_transpose__mult,axiom,
! [A: mat_real,Nr: nat,N: nat,B: mat_real,Nc: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ N ) )
=> ( ( member_mat_real @ B @ ( carrier_mat_real @ N @ Nc ) )
=> ( ( transpose_mat_real @ ( times_times_mat_real @ A @ B ) )
= ( times_times_mat_real @ ( transpose_mat_real @ B ) @ ( transpose_mat_real @ A ) ) ) ) ) ).
% transpose_mult
thf(fact_49_mult__of__int__commute,axiom,
! [X: int,Y: int] :
( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
= ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).
% mult_of_int_commute
thf(fact_50_mult__of__int__commute,axiom,
! [X: int,Y: real] :
( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
= ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).
% mult_of_int_commute
thf(fact_51_Matrix__Vector__Extras_Oof__int__hom_Omat__hom__inj,axiom,
! [A: mat_int,B: mat_int] :
( ( ( map_mat_int_real @ ring_1_of_int_real @ A )
= ( map_mat_int_real @ ring_1_of_int_real @ B ) )
=> ( A = B ) ) ).
% Matrix_Vector_Extras.of_int_hom.mat_hom_inj
thf(fact_52_Matrix__Vector__Extras_Oof__int__hom_Omat__hom__inj,axiom,
! [A: mat_int,B: mat_int] :
( ( ( map_mat_int_int @ ring_1_of_int_int @ A )
= ( map_mat_int_int @ ring_1_of_int_int @ B ) )
=> ( A = B ) ) ).
% Matrix_Vector_Extras.of_int_hom.mat_hom_inj
thf(fact_53_lift__mat__is__0__1,axiom,
incide4964164200581851450ix_int @ ( matrix323868623736973467nt_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ).
% lift_mat_is_0_1
thf(fact_54_is__dual__def,axiom,
! [Vs: list_nat,Bs: list_set_nat] :
( ( dual_o7311038347201774518_a_nat @ v_s @ b_s @ Vs @ Bs )
= ( ( incide6998539924841383625em_nat @ Vs @ Bs )
& ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ).
% is_dual_def
thf(fact_55_is__dual__def,axiom,
! [Vs: list_a,Bs: list_set_a] :
( ( dual_o5859382014055506072al_a_a @ v_s @ b_s @ Vs @ Bs )
= ( ( incide1624170830610365509stem_a @ Vs @ Bs )
& ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ).
% is_dual_def
thf(fact_56_is__dualI,axiom,
! [Vs: list_nat,Bs: list_set_nat] :
( ( incide6998539924841383625em_nat @ Vs @ Bs )
=> ( ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( dual_o7311038347201774518_a_nat @ v_s @ b_s @ Vs @ Bs ) ) ) ).
% is_dualI
thf(fact_57_is__dualI,axiom,
! [Vs: list_a,Bs: list_set_a] :
( ( incide1624170830610365509stem_a @ Vs @ Bs )
=> ( ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( dual_o5859382014055506072al_a_a @ v_s @ b_s @ Vs @ Bs ) ) ) ).
% is_dualI
thf(fact_58_ordered__proper__design_Omat__is__proper,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide1001368407746664282gn_nat @ V_s @ B_s )
=> ( incide294466202882093137at_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ).
% ordered_proper_design.mat_is_proper
thf(fact_59_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_60_is__dualD2,axiom,
! [Vs: list_nat,Bs: list_set_nat] :
( ( dual_o7311038347201774518_a_nat @ v_s @ b_s @ Vs @ Bs )
=> ( incide6998539924841383625em_nat @ Vs @ Bs ) ) ).
% is_dualD2
thf(fact_61_is__dualD2,axiom,
! [Vs: list_a,Bs: list_set_a] :
( ( dual_o5859382014055506072al_a_a @ v_s @ b_s @ Vs @ Bs )
=> ( incide1624170830610365509stem_a @ Vs @ Bs ) ) ).
% is_dualD2
thf(fact_62_proper__inc__mat__map,axiom,
! [M: mat_real,F: real > real] :
( ( incide8174291662766164561t_real @ M )
=> ( incide8174291662766164561t_real @ ( map_mat_real_real @ F @ M ) ) ) ).
% proper_inc_mat_map
thf(fact_63_proper__inc__mat__map,axiom,
! [M: mat_real,F: real > int] :
( ( incide8174291662766164561t_real @ M )
=> ( incide294466202882093137at_int @ ( map_mat_real_int @ F @ M ) ) ) ).
% proper_inc_mat_map
thf(fact_64_proper__inc__mat__map,axiom,
! [M: mat_int,F: int > real] :
( ( incide294466202882093137at_int @ M )
=> ( incide8174291662766164561t_real @ ( map_mat_int_real @ F @ M ) ) ) ).
% proper_inc_mat_map
thf(fact_65_proper__inc__mat__map,axiom,
! [M: mat_int,F: int > int] :
( ( incide294466202882093137at_int @ M )
=> ( incide294466202882093137at_int @ ( map_mat_int_int @ F @ M ) ) ) ).
% proper_inc_mat_map
thf(fact_66_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_67_zero__one__matrix__ring__1_Ointro,axiom,
! [M: mat_int] :
( ( incide4964164200581851450ix_int @ M )
=> ( incide6080938071136783841_1_int @ M ) ) ).
% zero_one_matrix_ring_1.intro
thf(fact_68_zero__one__matrix__ring__1_Oaxioms,axiom,
! [M: mat_int] :
( ( incide6080938071136783841_1_int @ M )
=> ( incide4964164200581851450ix_int @ M ) ) ).
% zero_one_matrix_ring_1.axioms
thf(fact_69_zero__one__matrix__ring__1__def,axiom,
incide6080938071136783841_1_int = incide4964164200581851450ix_int ).
% zero_one_matrix_ring_1_def
thf(fact_70_inc__mat__of__01__mat,axiom,
! [Vs2: list_a,Bs2: list_set_a] : ( incide6080938071136783841_1_int @ ( incide7016682120514301311_a_int @ Vs2 @ Bs2 ) ) ).
% inc_mat_of_01_mat
thf(fact_71_inc__mat__of__01__mat,axiom,
! [Vs2: list_nat,Bs2: list_set_nat] : ( incide6080938071136783841_1_int @ ( incide1177982898701834729at_int @ Vs2 @ Bs2 ) ) ).
% inc_mat_of_01_mat
thf(fact_72_ordered__incidence__system__axioms,axiom,
incide1624170830610365509stem_a @ v_s @ b_s ).
% ordered_incidence_system_axioms
thf(fact_73_ordered__design_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide2848671379600480836sign_a @ V_s @ B_s )
=> ( incide1624170830610365509stem_a @ V_s @ B_s ) ) ).
% ordered_design.axioms(1)
thf(fact_74_ordered__design_Oaxioms_I1_J,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide8999572217031194378gn_nat @ V_s @ B_s )
=> ( incide6998539924841383625em_nat @ V_s @ B_s ) ) ).
% ordered_design.axioms(1)
thf(fact_75_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_76_ordered__incidence__system_Ois__dualI,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_nat,Bs: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( incide6998539924841383625em_nat @ Vs @ Bs )
=> ( ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) )
=> ( dual_o7421254750609082426at_nat @ V_s @ B_s @ Vs @ Bs ) ) ) ) ).
% ordered_incidence_system.is_dualI
thf(fact_77_ordered__incidence__system_Ois__dualI,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_nat,Bs: list_set_nat] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( incide6998539924841383625em_nat @ Vs @ Bs )
=> ( ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) )
=> ( dual_o7311038347201774518_a_nat @ V_s @ B_s @ Vs @ Bs ) ) ) ) ).
% ordered_incidence_system.is_dualI
thf(fact_78_ordered__incidence__system_Ois__dualI,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_a,Bs: list_set_a] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( incide1624170830610365509stem_a @ Vs @ Bs )
=> ( ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) )
=> ( dual_o2119597811525285140_nat_a @ V_s @ B_s @ Vs @ Bs ) ) ) ) ).
% ordered_incidence_system.is_dualI
thf(fact_79_ordered__incidence__system_Ois__dualI,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_a,Bs: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( incide1624170830610365509stem_a @ Vs @ Bs )
=> ( ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) )
=> ( dual_o5859382014055506072al_a_a @ V_s @ B_s @ Vs @ Bs ) ) ) ) ).
% ordered_incidence_system.is_dualI
thf(fact_80_ordered__incidence__system_Ois__dualD1,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_nat,Bs: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dual_o7421254750609082426at_nat @ V_s @ B_s @ Vs @ Bs )
=> ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ).
% ordered_incidence_system.is_dualD1
thf(fact_81_ordered__incidence__system_Ois__dualD1,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_nat,Bs: list_set_nat] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dual_o7311038347201774518_a_nat @ V_s @ B_s @ Vs @ Bs )
=> ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ).
% ordered_incidence_system.is_dualD1
thf(fact_82_ordered__incidence__system_Ois__dualD1,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_a,Bs: list_set_a] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dual_o2119597811525285140_nat_a @ V_s @ B_s @ Vs @ Bs )
=> ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ).
% ordered_incidence_system.is_dualD1
thf(fact_83_ordered__incidence__system_Ois__dualD1,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_a,Bs: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dual_o5859382014055506072al_a_a @ V_s @ B_s @ Vs @ Bs )
=> ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ).
% ordered_incidence_system.is_dualD1
thf(fact_84_ordered__incidence__system_Ois__dual__def,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_nat,Bs: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dual_o7421254750609082426at_nat @ V_s @ B_s @ Vs @ Bs )
= ( ( incide6998539924841383625em_nat @ Vs @ Bs )
& ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ) ).
% ordered_incidence_system.is_dual_def
thf(fact_85_ordered__incidence__system_Ois__dual__def,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_nat,Bs: list_set_nat] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dual_o7311038347201774518_a_nat @ V_s @ B_s @ Vs @ Bs )
= ( ( incide6998539924841383625em_nat @ Vs @ Bs )
& ( ( incide1177982898701834729at_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ) ).
% ordered_incidence_system.is_dual_def
thf(fact_86_ordered__incidence__system_Ois__dual__def,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_a,Bs: list_set_a] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dual_o2119597811525285140_nat_a @ V_s @ B_s @ Vs @ Bs )
= ( ( incide1624170830610365509stem_a @ Vs @ Bs )
& ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ) ).
% ordered_incidence_system.is_dual_def
thf(fact_87_ordered__incidence__system_Ois__dual__def,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_a,Bs: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dual_o5859382014055506072al_a_a @ V_s @ B_s @ Vs @ Bs )
= ( ( incide1624170830610365509stem_a @ Vs @ Bs )
& ( ( incide7016682120514301311_a_int @ Vs @ Bs )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ) ).
% ordered_incidence_system.is_dual_def
thf(fact_88_mem__Collect__eq,axiom,
! [A2: mat_int,P: mat_int > $o] :
( ( member_mat_int @ A2 @ ( collect_mat_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_89_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_90_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_92_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
! [A2: mat_real,P: mat_real > $o] :
( ( member_mat_real @ A2 @ ( collect_mat_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_94_Collect__mem__eq,axiom,
! [A: set_mat_int] :
( ( collect_mat_int
@ ^ [X2: mat_int] : ( member_mat_int @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_95_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_96_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_97_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_98_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_99_Collect__mem__eq,axiom,
! [A: set_mat_real] :
( ( collect_mat_real
@ ^ [X2: mat_real] : ( member_mat_real @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_100_ordered__incidence__system_Ois__dualD2,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_nat,Bs: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dual_o7421254750609082426at_nat @ V_s @ B_s @ Vs @ Bs )
=> ( incide6998539924841383625em_nat @ Vs @ Bs ) ) ) ).
% ordered_incidence_system.is_dualD2
thf(fact_101_ordered__incidence__system_Ois__dualD2,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_nat,Bs: list_set_nat] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dual_o7311038347201774518_a_nat @ V_s @ B_s @ Vs @ Bs )
=> ( incide6998539924841383625em_nat @ Vs @ Bs ) ) ) ).
% ordered_incidence_system.is_dualD2
thf(fact_102_ordered__incidence__system_Ois__dualD2,axiom,
! [V_s: list_nat,B_s: list_set_nat,Vs: list_a,Bs: list_set_a] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dual_o2119597811525285140_nat_a @ V_s @ B_s @ Vs @ Bs )
=> ( incide1624170830610365509stem_a @ Vs @ Bs ) ) ) ).
% ordered_incidence_system.is_dualD2
thf(fact_103_ordered__incidence__system_Ois__dualD2,axiom,
! [V_s: list_a,B_s: list_set_a,Vs: list_a,Bs: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dual_o5859382014055506072al_a_a @ V_s @ B_s @ Vs @ Bs )
=> ( incide1624170830610365509stem_a @ Vs @ Bs ) ) ) ).
% ordered_incidence_system.is_dualD2
thf(fact_104_ordered__incidence__system_Ois__dual_Ocong,axiom,
dual_o7311038347201774518_a_nat = dual_o7311038347201774518_a_nat ).
% ordered_incidence_system.is_dual.cong
thf(fact_105_ordered__incidence__system_Ois__dual_Ocong,axiom,
dual_o2119597811525285140_nat_a = dual_o2119597811525285140_nat_a ).
% ordered_incidence_system.is_dual.cong
thf(fact_106_ordered__incidence__system_Ois__dual_Ocong,axiom,
dual_o5859382014055506072al_a_a = dual_o5859382014055506072al_a_a ).
% ordered_incidence_system.is_dual.cong
thf(fact_107_semiring__hom_Omat__hom__pow,axiom,
! [Hom: int > real,A: mat_int,N: nat,K: nat] :
( ( ring_s3477937289336866936t_real @ Hom )
=> ( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( map_mat_int_real @ Hom @ ( pow_mat_int @ A @ K ) )
= ( pow_mat_real @ ( map_mat_int_real @ Hom @ A ) @ K ) ) ) ) ).
% semiring_hom.mat_hom_pow
thf(fact_108_semiring__hom_Omat__hom__pow,axiom,
! [Hom: real > real,A: mat_real,N: nat,K: nat] :
( ( ring_s1386325929209763192l_real @ Hom )
=> ( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( map_mat_real_real @ Hom @ ( pow_mat_real @ A @ K ) )
= ( pow_mat_real @ ( map_mat_real_real @ Hom @ A ) @ K ) ) ) ) ).
% semiring_hom.mat_hom_pow
thf(fact_109_semiring__hom_Omat__hom__pow,axiom,
! [Hom: real > int,A: mat_real,N: nat,K: nat] :
( ( ring_s5855833817263693560al_int @ Hom )
=> ( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( map_mat_real_int @ Hom @ ( pow_mat_real @ A @ K ) )
= ( pow_mat_int @ ( map_mat_real_int @ Hom @ A ) @ K ) ) ) ) ).
% semiring_hom.mat_hom_pow
thf(fact_110_semiring__hom_Omat__hom__pow,axiom,
! [Hom: int > int,A: mat_int,N: nat,K: nat] :
( ( ring_s4547151187181212152nt_int @ Hom )
=> ( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( map_mat_int_int @ Hom @ ( pow_mat_int @ A @ K ) )
= ( pow_mat_int @ ( map_mat_int_int @ Hom @ A ) @ K ) ) ) ) ).
% semiring_hom.mat_hom_pow
thf(fact_111_semiring__hom_Ohom__sum__mat,axiom,
! [Hom: int > real,A: mat_int] :
( ( ring_s3477937289336866936t_real @ Hom )
=> ( ( Hom @ ( comm_m4053738856622352096at_int @ A ) )
= ( comm_m8678487124704766304t_real @ ( map_mat_int_real @ Hom @ A ) ) ) ) ).
% semiring_hom.hom_sum_mat
thf(fact_112_semiring__hom_Ohom__sum__mat,axiom,
! [Hom: real > real,A: mat_real] :
( ( ring_s1386325929209763192l_real @ Hom )
=> ( ( Hom @ ( comm_m8678487124704766304t_real @ A ) )
= ( comm_m8678487124704766304t_real @ ( map_mat_real_real @ Hom @ A ) ) ) ) ).
% semiring_hom.hom_sum_mat
thf(fact_113_semiring__hom_Ohom__sum__mat,axiom,
! [Hom: real > int,A: mat_real] :
( ( ring_s5855833817263693560al_int @ Hom )
=> ( ( Hom @ ( comm_m8678487124704766304t_real @ A ) )
= ( comm_m4053738856622352096at_int @ ( map_mat_real_int @ Hom @ A ) ) ) ) ).
% semiring_hom.hom_sum_mat
thf(fact_114_semiring__hom_Ohom__sum__mat,axiom,
! [Hom: int > int,A: mat_int] :
( ( ring_s4547151187181212152nt_int @ Hom )
=> ( ( Hom @ ( comm_m4053738856622352096at_int @ A ) )
= ( comm_m4053738856622352096at_int @ ( map_mat_int_int @ Hom @ A ) ) ) ) ).
% semiring_hom.hom_sum_mat
thf(fact_115_real__embedding_Omat__hom__mult,axiom,
! [A: mat_real,Nr: nat,N: nat,B: mat_real,Nc: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ N ) )
=> ( ( member_mat_real @ B @ ( carrier_mat_real @ N @ Nc ) )
=> ( ( map_mat_real_real @ ring_H569718986840772220f_real @ ( times_times_mat_real @ A @ B ) )
= ( times_times_mat_real @ ( map_mat_real_real @ ring_H569718986840772220f_real @ A ) @ ( map_mat_real_real @ ring_H569718986840772220f_real @ B ) ) ) ) ) ).
% real_embedding.mat_hom_mult
thf(fact_116_real__embedding_Omat__hom__mult,axiom,
! [A: mat_int,Nr: nat,N: nat,B: mat_int,Nc: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( map_mat_int_real @ ring_H3062062880446257660of_int @ ( times_times_mat_int @ A @ B ) )
= ( times_times_mat_real @ ( map_mat_int_real @ ring_H3062062880446257660of_int @ A ) @ ( map_mat_int_real @ ring_H3062062880446257660of_int @ B ) ) ) ) ) ).
% real_embedding.mat_hom_mult
thf(fact_117_distinct,axiom,
distinct_a @ v_s ).
% distinct
thf(fact_118_zero__one__matrix__int__def,axiom,
incide8301514189696901506ix_int = incide6080938071136783841_1_int ).
% zero_one_matrix_int_def
thf(fact_119_zero__one__matrix__int_Oaxioms,axiom,
! [M: mat_int] :
( ( incide8301514189696901506ix_int @ M )
=> ( incide6080938071136783841_1_int @ M ) ) ).
% zero_one_matrix_int.axioms
thf(fact_120_real__embedding_Ohom__sum__mat,axiom,
! [A: mat_real] :
( ( ring_H569718986840772220f_real @ ( comm_m8678487124704766304t_real @ A ) )
= ( comm_m8678487124704766304t_real @ ( map_mat_real_real @ ring_H569718986840772220f_real @ A ) ) ) ).
% real_embedding.hom_sum_mat
thf(fact_121_real__embedding_Ohom__sum__mat,axiom,
! [A: mat_int] :
( ( ring_H3062062880446257660of_int @ ( comm_m4053738856622352096at_int @ A ) )
= ( comm_m8678487124704766304t_real @ ( map_mat_int_real @ ring_H3062062880446257660of_int @ A ) ) ) ).
% real_embedding.hom_sum_mat
thf(fact_122_real__embedding_Osemiring__hom__axioms,axiom,
ring_s1386325929209763192l_real @ ring_H569718986840772220f_real ).
% real_embedding.semiring_hom_axioms
thf(fact_123_real__embedding_Osemiring__hom__axioms,axiom,
ring_s3477937289336866936t_real @ ring_H3062062880446257660of_int ).
% real_embedding.semiring_hom_axioms
thf(fact_124_real__mult,axiom,
! [X: real,Y: real] :
( ( ring_H569718986840772220f_real @ ( times_times_real @ X @ Y ) )
= ( times_times_real @ ( ring_H569718986840772220f_real @ X ) @ ( ring_H569718986840772220f_real @ Y ) ) ) ).
% real_mult
thf(fact_125_real__mult,axiom,
! [X: int,Y: int] :
( ( ring_H3062062880446257660of_int @ ( times_times_int @ X @ Y ) )
= ( times_times_real @ ( ring_H3062062880446257660of_int @ X ) @ ( ring_H3062062880446257660of_int @ Y ) ) ) ).
% real_mult
thf(fact_126_real__embedding_Omat__hom__pow,axiom,
! [A: mat_real,N: nat,K: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( map_mat_real_real @ ring_H569718986840772220f_real @ ( pow_mat_real @ A @ K ) )
= ( pow_mat_real @ ( map_mat_real_real @ ring_H569718986840772220f_real @ A ) @ K ) ) ) ).
% real_embedding.mat_hom_pow
thf(fact_127_real__embedding_Omat__hom__pow,axiom,
! [A: mat_int,N: nat,K: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( map_mat_int_real @ ring_H3062062880446257660of_int @ ( pow_mat_int @ A @ K ) )
= ( pow_mat_real @ ( map_mat_int_real @ ring_H3062062880446257660of_int @ A ) @ K ) ) ) ).
% real_embedding.mat_hom_pow
thf(fact_128_real__of__int__def,axiom,
ring_H3062062880446257660of_int = ring_1_of_int_real ).
% real_of_int_def
thf(fact_129_ordered__incidence__system_Odistinct,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( distinct_a @ V_s ) ) ).
% ordered_incidence_system.distinct
thf(fact_130_ordered__incidence__system_Odistinct,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( distinct_nat @ V_s ) ) ).
% ordered_incidence_system.distinct
thf(fact_131_pow__carrier__mat,axiom,
! [A: mat_int,N: nat,K: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( member_mat_int @ ( pow_mat_int @ A @ K ) @ ( carrier_mat_int @ N @ N ) ) ) ).
% pow_carrier_mat
thf(fact_132_pow__carrier__mat,axiom,
! [A: mat_real,N: nat,K: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( member_mat_real @ ( pow_mat_real @ A @ K ) @ ( carrier_mat_real @ N @ N ) ) ) ).
% pow_carrier_mat
thf(fact_133_of__int__hom_Ohom__sum__mat,axiom,
! [A: mat_int] :
( ( ring_1_of_int_real @ ( comm_m4053738856622352096at_int @ A ) )
= ( comm_m8678487124704766304t_real @ ( map_mat_int_real @ ring_1_of_int_real @ A ) ) ) ).
% of_int_hom.hom_sum_mat
thf(fact_134_of__int__hom_Ohom__sum__mat,axiom,
! [A: mat_int] :
( ( ring_1_of_int_int @ ( comm_m4053738856622352096at_int @ A ) )
= ( comm_m4053738856622352096at_int @ ( map_mat_int_int @ ring_1_of_int_int @ A ) ) ) ).
% of_int_hom.hom_sum_mat
thf(fact_135_of__int__hom_Omat__hom__pow,axiom,
! [A: mat_int,N: nat,K: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( map_mat_int_real @ ring_1_of_int_real @ ( pow_mat_int @ A @ K ) )
= ( pow_mat_real @ ( map_mat_int_real @ ring_1_of_int_real @ A ) @ K ) ) ) ).
% of_int_hom.mat_hom_pow
thf(fact_136_of__int__hom_Omat__hom__pow,axiom,
! [A: mat_int,N: nat,K: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( map_mat_int_int @ ring_1_of_int_int @ ( pow_mat_int @ A @ K ) )
= ( pow_mat_int @ ( map_mat_int_int @ ring_1_of_int_int @ A ) @ K ) ) ) ).
% of_int_hom.mat_hom_pow
thf(fact_137_zero__one__matrix__int_Ointro,axiom,
! [M: mat_int] :
( ( incide6080938071136783841_1_int @ M )
=> ( incide8301514189696901506ix_int @ M ) ) ).
% zero_one_matrix_int.intro
thf(fact_138_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ Xs ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_139_real__embedding_Oab__semigroup__mult__hom__axioms,axiom,
ring_a4547970760173101098t_real @ ring_H3062062880446257660of_int ).
% real_embedding.ab_semigroup_mult_hom_axioms
thf(fact_140_of__int__hom_Ocomm__monoid__mult__hom__1__axioms,axiom,
ring_c8180009538971569961t_real @ ring_1_of_int_real ).
% of_int_hom.comm_monoid_mult_hom_1_axioms
thf(fact_141_of__int__hom_Ocomm__monoid__mult__hom__1__axioms,axiom,
ring_c5842864267467415849nt_int @ ring_1_of_int_int ).
% of_int_hom.comm_monoid_mult_hom_1_axioms
thf(fact_142_of__int__hom_Ocomm__monoid__add__hom__0__axioms,axiom,
ring_c3298759929950863697t_real @ ring_1_of_int_real ).
% of_int_hom.comm_monoid_add_hom_0_axioms
thf(fact_143_of__int__hom_Ocomm__monoid__add__hom__0__axioms,axiom,
ring_c6059480546188650321nt_int @ ring_1_of_int_int ).
% of_int_hom.comm_monoid_add_hom_0_axioms
thf(fact_144_of__int__hom_Oab__semigroup__mult__hom__axioms,axiom,
ring_a4547970760173101098t_real @ ring_1_of_int_real ).
% of_int_hom.ab_semigroup_mult_hom_axioms
thf(fact_145_of__int__hom_Oab__semigroup__mult__hom__axioms,axiom,
ring_a7291918497645896362nt_int @ ring_1_of_int_int ).
% of_int_hom.ab_semigroup_mult_hom_axioms
thf(fact_146_helper,axiom,
! [X: int,Y: int,Z: int] :
( ( times_times_int @ X @ ( times_times_int @ Y @ Z ) )
= ( times_times_int @ ( times_times_int @ Y @ X ) @ Z ) ) ).
% helper
thf(fact_147_helper,axiom,
! [X: real,Y: real,Z: real] :
( ( times_times_real @ X @ ( times_times_real @ Y @ Z ) )
= ( times_times_real @ ( times_times_real @ Y @ X ) @ Z ) ) ).
% helper
thf(fact_148_mult_Oleft__commute,axiom,
! [B2: int,A2: int,C2: int] :
( ( times_times_int @ B2 @ ( times_times_int @ A2 @ C2 ) )
= ( times_times_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_149_mult_Oleft__commute,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( times_times_nat @ B2 @ ( times_times_nat @ A2 @ C2 ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_150_mult_Oleft__commute,axiom,
! [B2: real,A2: real,C2: real] :
( ( times_times_real @ B2 @ ( times_times_real @ A2 @ C2 ) )
= ( times_times_real @ A2 @ ( times_times_real @ B2 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_151_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_152_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_153_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_154_mult_Oassoc,axiom,
! [A2: int,B2: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
= ( times_times_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) ) ).
% mult.assoc
thf(fact_155_mult_Oassoc,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% mult.assoc
thf(fact_156_mult_Oassoc,axiom,
! [A2: real,B2: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B2 ) @ C2 )
= ( times_times_real @ A2 @ ( times_times_real @ B2 @ C2 ) ) ) ).
% mult.assoc
thf(fact_157_ideal_Oscale__left__commute,axiom,
! [A2: int,B2: int,X: int] :
( ( times_times_int @ A2 @ ( times_times_int @ B2 @ X ) )
= ( times_times_int @ B2 @ ( times_times_int @ A2 @ X ) ) ) ).
% ideal.scale_left_commute
thf(fact_158_ideal_Oscale__left__commute,axiom,
! [A2: real,B2: real,X: real] :
( ( times_times_real @ A2 @ ( times_times_real @ B2 @ X ) )
= ( times_times_real @ B2 @ ( times_times_real @ A2 @ X ) ) ) ).
% ideal.scale_left_commute
thf(fact_159_class__cring_Ofactors__equal,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( A2 = B2 )
=> ( ( C2 = D )
=> ( ( times_times_int @ A2 @ C2 )
= ( times_times_int @ B2 @ D ) ) ) ) ).
% class_cring.factors_equal
thf(fact_160_class__cring_Ofactors__equal,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( A2 = B2 )
=> ( ( C2 = D )
=> ( ( times_times_real @ A2 @ C2 )
= ( times_times_real @ B2 @ D ) ) ) ) ).
% class_cring.factors_equal
thf(fact_161_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: int,B2: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
= ( times_times_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_162_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_163_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: real,B2: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B2 ) @ C2 )
= ( times_times_real @ A2 @ ( times_times_real @ B2 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_164_ideal_Oscale__scale,axiom,
! [A2: int,B2: int,X: int] :
( ( times_times_int @ A2 @ ( times_times_int @ B2 @ X ) )
= ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ X ) ) ).
% ideal.scale_scale
thf(fact_165_ideal_Oscale__scale,axiom,
! [A2: real,B2: real,X: real] :
( ( times_times_real @ A2 @ ( times_times_real @ B2 @ X ) )
= ( times_times_real @ ( times_times_real @ A2 @ B2 ) @ X ) ) ).
% ideal.scale_scale
thf(fact_166_distinct__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_a @ ( union_a @ Xs @ Ys ) )
= ( distinct_a @ Ys ) ) ).
% distinct_union
thf(fact_167_distinct__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_168_of__real__hom_Ohom__sum__mat,axiom,
! [A: mat_real] :
( ( real_V1803761363581548252l_real @ ( comm_m8678487124704766304t_real @ A ) )
= ( comm_m8678487124704766304t_real @ ( map_mat_real_real @ real_V1803761363581548252l_real @ A ) ) ) ).
% of_real_hom.hom_sum_mat
thf(fact_169_lift__01__mat__distinct__cols,axiom,
( ( distinct_vec_int @ ( cols_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( distinct_vec_int @ ( cols_int @ ( matrix323868623736973467nt_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ).
% lift_01_mat_distinct_cols
thf(fact_170_of__real__hom_Omat__hom__pow,axiom,
! [A: mat_real,N: nat,K: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( map_mat_real_real @ real_V1803761363581548252l_real @ ( pow_mat_real @ A @ K ) )
= ( pow_mat_real @ ( map_mat_real_real @ real_V1803761363581548252l_real @ A ) @ K ) ) ) ).
% of_real_hom.mat_hom_pow
thf(fact_171_of__real__hom_Osemiring__hom__axioms,axiom,
ring_s1386325929209763192l_real @ real_V1803761363581548252l_real ).
% of_real_hom.semiring_hom_axioms
thf(fact_172_Matrix__Vector__Extras_Oof__real__hom_Omat__hom__inj,axiom,
! [A: mat_real,B: mat_real] :
( ( ( map_mat_real_real @ real_V1803761363581548252l_real @ A )
= ( map_mat_real_real @ real_V1803761363581548252l_real @ B ) )
=> ( A = B ) ) ).
% Matrix_Vector_Extras.of_real_hom.mat_hom_inj
thf(fact_173_of__int__hom_Omonoid__mult__hom__1__axioms,axiom,
ring_m3572329010447437917t_real @ ring_1_of_int_real ).
% of_int_hom.monoid_mult_hom_1_axioms
thf(fact_174_of__int__hom_Omonoid__mult__hom__1__axioms,axiom,
ring_m4647122448705125469nt_int @ ring_1_of_int_int ).
% of_int_hom.monoid_mult_hom_1_axioms
thf(fact_175_of__int__hom_Omonoid__add__hom__0__axioms,axiom,
ring_m8610364392332473117t_real @ ring_1_of_int_real ).
% of_int_hom.monoid_add_hom_0_axioms
thf(fact_176_of__int__hom_Omonoid__add__hom__0__axioms,axiom,
ring_m4348472603105607453nt_int @ ring_1_of_int_int ).
% of_int_hom.monoid_add_hom_0_axioms
thf(fact_177_zero__one__matrix_Olift__01__mat__distinct__cols,axiom,
! [Matrix: mat_int] :
( ( incide4964164200581851450ix_int @ Matrix )
=> ( ( distinct_vec_int @ ( cols_int @ Matrix ) )
=> ( distinct_vec_int @ ( cols_int @ ( matrix323868623736973467nt_int @ Matrix ) ) ) ) ) ).
% zero_one_matrix.lift_01_mat_distinct_cols
thf(fact_178_of__real__hom_Omat__hom__mult,axiom,
! [A: mat_real,Nr: nat,N: nat,B: mat_real,Nc: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ N ) )
=> ( ( member_mat_real @ B @ ( carrier_mat_real @ N @ Nc ) )
=> ( ( map_mat_real_real @ real_V1803761363581548252l_real @ ( times_times_mat_real @ A @ B ) )
= ( times_times_mat_real @ ( map_mat_real_real @ real_V1803761363581548252l_real @ A ) @ ( map_mat_real_real @ real_V1803761363581548252l_real @ B ) ) ) ) ) ).
% of_real_hom.mat_hom_mult
thf(fact_179_of__real__of__int__eq,axiom,
! [Z: int] :
( ( real_V1803761363581548252l_real @ ( ring_1_of_int_real @ Z ) )
= ( ring_1_of_int_real @ Z ) ) ).
% of_real_of_int_eq
thf(fact_180_of__real__mult,axiom,
! [X: real,Y: real] :
( ( real_V1803761363581548252l_real @ ( times_times_real @ X @ Y ) )
= ( times_times_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% of_real_mult
thf(fact_181_ordered__simple__design_Olift__01__distinct__cols__N,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide371748008924627346sign_a @ V_s @ B_s )
=> ( distinct_vec_int @ ( cols_int @ ( matrix323868623736973467nt_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ) ).
% ordered_simple_design.lift_01_distinct_cols_N
thf(fact_182_ordered__simple__design_Olift__01__distinct__cols__N,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide8427317466731264060gn_nat @ V_s @ B_s )
=> ( distinct_vec_int @ ( cols_int @ ( matrix323868623736973467nt_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ) ).
% ordered_simple_design.lift_01_distinct_cols_N
thf(fact_183_ordered__simple__design_Odistinct__cols__N,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide371748008924627346sign_a @ V_s @ B_s )
=> ( distinct_vec_int @ ( cols_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) ) ).
% ordered_simple_design.distinct_cols_N
thf(fact_184_ordered__simple__design_Odistinct__cols__N,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide8427317466731264060gn_nat @ V_s @ B_s )
=> ( distinct_vec_int @ ( cols_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) ) ).
% ordered_simple_design.distinct_cols_N
thf(fact_185_cols__transpose,axiom,
! [A: mat_int] :
( ( cols_int @ ( transpose_mat_int @ A ) )
= ( rows_int @ A ) ) ).
% cols_transpose
thf(fact_186_cols__transpose,axiom,
! [A: mat_real] :
( ( cols_real @ ( transpose_mat_real @ A ) )
= ( rows_real @ A ) ) ).
% cols_transpose
thf(fact_187_rows__transpose,axiom,
! [A: mat_int] :
( ( rows_int @ ( transpose_mat_int @ A ) )
= ( cols_int @ A ) ) ).
% rows_transpose
thf(fact_188_rows__transpose,axiom,
! [A: mat_real] :
( ( rows_real @ ( transpose_mat_real @ A ) )
= ( cols_real @ A ) ) ).
% rows_transpose
thf(fact_189_real__embedding_Oab__semigroup__add__hom__axioms,axiom,
ring_a779443308300532965t_real @ ring_H3062062880446257660of_int ).
% real_embedding.ab_semigroup_add_hom_axioms
thf(fact_190_ordered__simple__design_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide371748008924627346sign_a @ V_s @ B_s )
=> ( incide2848671379600480836sign_a @ V_s @ B_s ) ) ).
% ordered_simple_design.axioms(1)
thf(fact_191_is__incidence__mat__alt,axiom,
! [N2: mat_int] :
( ( incide3487307869859529607_int_a @ N2 @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) )
= ( ? [Vs3: list_a,Bs3: list_set_a] :
( ( ( set_a2 @ Vs3 )
= ( set_a2 @ v_s ) )
& ( ( mset_set_a @ Bs3 )
= ( mset_set_a @ b_s ) )
& ( incide1624170830610365509stem_a @ Vs3 @ Bs3 )
& ( N2
= ( incide7016682120514301311_a_int @ Vs3 @ Bs3 ) ) ) ) ) ).
% is_incidence_mat_alt
thf(fact_192_comm__semiring__hom__def,axiom,
ring_c7120671943487988908t_real = ring_s3477937289336866936t_real ).
% comm_semiring_hom_def
thf(fact_193_comm__semiring__hom__def,axiom,
ring_c5154114807373371308l_real = ring_s1386325929209763192l_real ).
% comm_semiring_hom_def
thf(fact_194_comm__semiring__hom__def,axiom,
ring_c275196434560039724al_int = ring_s5855833817263693560al_int ).
% comm_semiring_hom_def
thf(fact_195_comm__semiring__hom__def,axiom,
ring_c1049078972110686252nt_int = ring_s4547151187181212152nt_int ).
% comm_semiring_hom_def
thf(fact_196_comm__semiring__hom_Oaxioms,axiom,
! [Hom: int > real] :
( ( ring_c7120671943487988908t_real @ Hom )
=> ( ring_s3477937289336866936t_real @ Hom ) ) ).
% comm_semiring_hom.axioms
thf(fact_197_comm__semiring__hom_Oaxioms,axiom,
! [Hom: real > real] :
( ( ring_c5154114807373371308l_real @ Hom )
=> ( ring_s1386325929209763192l_real @ Hom ) ) ).
% comm_semiring_hom.axioms
thf(fact_198_comm__semiring__hom_Oaxioms,axiom,
! [Hom: real > int] :
( ( ring_c275196434560039724al_int @ Hom )
=> ( ring_s5855833817263693560al_int @ Hom ) ) ).
% comm_semiring_hom.axioms
thf(fact_199_comm__semiring__hom_Oaxioms,axiom,
! [Hom: int > int] :
( ( ring_c1049078972110686252nt_int @ Hom )
=> ( ring_s4547151187181212152nt_int @ Hom ) ) ).
% comm_semiring_hom.axioms
thf(fact_200_comm__semiring__hom_Ointro,axiom,
! [Hom: int > real] :
( ( ring_s3477937289336866936t_real @ Hom )
=> ( ring_c7120671943487988908t_real @ Hom ) ) ).
% comm_semiring_hom.intro
thf(fact_201_comm__semiring__hom_Ointro,axiom,
! [Hom: real > real] :
( ( ring_s1386325929209763192l_real @ Hom )
=> ( ring_c5154114807373371308l_real @ Hom ) ) ).
% comm_semiring_hom.intro
thf(fact_202_comm__semiring__hom_Ointro,axiom,
! [Hom: real > int] :
( ( ring_s5855833817263693560al_int @ Hom )
=> ( ring_c275196434560039724al_int @ Hom ) ) ).
% comm_semiring_hom.intro
thf(fact_203_comm__semiring__hom_Ointro,axiom,
! [Hom: int > int] :
( ( ring_s4547151187181212152nt_int @ Hom )
=> ( ring_c1049078972110686252nt_int @ Hom ) ) ).
% comm_semiring_hom.intro
thf(fact_204_take__cols__mat__mul,axiom,
! [A: mat_int,Nr: nat,N: nat,B: mat_int,Nc: nat,Inds: list_nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ N ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ N @ Nc ) )
=> ( ( times_times_mat_int @ A @ ( more_take_cols_int @ B @ Inds ) )
= ( more_take_cols_int @ ( times_times_mat_int @ A @ B ) @ Inds ) ) ) ) ).
% take_cols_mat_mul
thf(fact_205_take__cols__mat__mul,axiom,
! [A: mat_real,Nr: nat,N: nat,B: mat_real,Nc: nat,Inds: list_nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ N ) )
=> ( ( member_mat_real @ B @ ( carrier_mat_real @ N @ Nc ) )
=> ( ( times_times_mat_real @ A @ ( more_take_cols_real @ B @ Inds ) )
= ( more_take_cols_real @ ( times_times_mat_real @ A @ B ) @ Inds ) ) ) ) ).
% take_cols_mat_mul
thf(fact_206_similar__mat__wit__pow__id,axiom,
! [A: mat_int,B: mat_int,P: mat_int,Q: mat_int,K: nat] :
( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( ( pow_mat_int @ A @ K )
= ( times_times_mat_int @ ( times_times_mat_int @ P @ ( pow_mat_int @ B @ K ) ) @ Q ) ) ) ).
% similar_mat_wit_pow_id
thf(fact_207_similar__mat__wit__pow__id,axiom,
! [A: mat_real,B: mat_real,P: mat_real,Q: mat_real,K: nat] :
( ( similar_mat_wit_real @ A @ B @ P @ Q )
=> ( ( pow_mat_real @ A @ K )
= ( times_times_mat_real @ ( times_times_mat_real @ P @ ( pow_mat_real @ B @ K ) ) @ Q ) ) ) ).
% similar_mat_wit_pow_id
thf(fact_208_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_209_similar__mat__witD2_I7_J,axiom,
! [A: mat_int,N: nat,M2: nat,B: mat_int,P: mat_int,Q: mat_int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ M2 ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( member_mat_int @ Q @ ( carrier_mat_int @ N @ N ) ) ) ) ).
% similar_mat_witD2(7)
thf(fact_210_similar__mat__witD2_I7_J,axiom,
! [A: mat_real,N: nat,M2: nat,B: mat_real,P: mat_real,Q: mat_real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ M2 ) )
=> ( ( similar_mat_wit_real @ A @ B @ P @ Q )
=> ( member_mat_real @ Q @ ( carrier_mat_real @ N @ N ) ) ) ) ).
% similar_mat_witD2(7)
thf(fact_211_similar__mat__witD2_I6_J,axiom,
! [A: mat_int,N: nat,M2: nat,B: mat_int,P: mat_int,Q: mat_int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ M2 ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( member_mat_int @ P @ ( carrier_mat_int @ N @ N ) ) ) ) ).
% similar_mat_witD2(6)
thf(fact_212_similar__mat__witD2_I6_J,axiom,
! [A: mat_real,N: nat,M2: nat,B: mat_real,P: mat_real,Q: mat_real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ M2 ) )
=> ( ( similar_mat_wit_real @ A @ B @ P @ Q )
=> ( member_mat_real @ P @ ( carrier_mat_real @ N @ N ) ) ) ) ).
% similar_mat_witD2(6)
thf(fact_213_similar__mat__witD2_I5_J,axiom,
! [A: mat_int,N: nat,M2: nat,B: mat_int,P: mat_int,Q: mat_int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ M2 ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( member_mat_int @ B @ ( carrier_mat_int @ N @ N ) ) ) ) ).
% similar_mat_witD2(5)
thf(fact_214_similar__mat__witD2_I5_J,axiom,
! [A: mat_real,N: nat,M2: nat,B: mat_real,P: mat_real,Q: mat_real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ M2 ) )
=> ( ( similar_mat_wit_real @ A @ B @ P @ Q )
=> ( member_mat_real @ B @ ( carrier_mat_real @ N @ N ) ) ) ) ).
% similar_mat_witD2(5)
thf(fact_215_similar__mat__witD2_I4_J,axiom,
! [A: mat_int,N: nat,M2: nat,B: mat_int,P: mat_int,Q: mat_int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ M2 ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) ) ) ) ).
% similar_mat_witD2(4)
thf(fact_216_similar__mat__witD2_I4_J,axiom,
! [A: mat_real,N: nat,M2: nat,B: mat_real,P: mat_real,Q: mat_real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ M2 ) )
=> ( ( similar_mat_wit_real @ A @ B @ P @ Q )
=> ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) ) ) ) ).
% similar_mat_witD2(4)
thf(fact_217_similar__mat__wit__trans,axiom,
! [A: mat_int,B: mat_int,P: mat_int,Q: mat_int,C: mat_int,P2: mat_int,Q2: mat_int] :
( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( ( similar_mat_wit_int @ B @ C @ P2 @ Q2 )
=> ( similar_mat_wit_int @ A @ C @ ( times_times_mat_int @ P @ P2 ) @ ( times_times_mat_int @ Q2 @ Q ) ) ) ) ).
% similar_mat_wit_trans
thf(fact_218_similar__mat__wit__trans,axiom,
! [A: mat_real,B: mat_real,P: mat_real,Q: mat_real,C: mat_real,P2: mat_real,Q2: mat_real] :
( ( similar_mat_wit_real @ A @ B @ P @ Q )
=> ( ( similar_mat_wit_real @ B @ C @ P2 @ Q2 )
=> ( similar_mat_wit_real @ A @ C @ ( times_times_mat_real @ P @ P2 ) @ ( times_times_mat_real @ Q2 @ Q ) ) ) ) ).
% similar_mat_wit_trans
thf(fact_219_take__cols__carrier__mat,axiom,
! [A: mat_int,Nr: nat,Nc: nat,Inds: list_nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ Nc ) )
=> ~ ! [N3: nat] :
~ ( member_mat_int @ ( more_take_cols_int @ A @ Inds ) @ ( carrier_mat_int @ Nr @ N3 ) ) ) ).
% take_cols_carrier_mat
thf(fact_220_take__cols__carrier__mat,axiom,
! [A: mat_real,Nr: nat,Nc: nat,Inds: list_nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ Nc ) )
=> ~ ! [N3: nat] :
~ ( member_mat_real @ ( more_take_cols_real @ A @ Inds ) @ ( carrier_mat_real @ Nr @ N3 ) ) ) ).
% take_cols_carrier_mat
thf(fact_221_of__int__hom_Ocomm__semiring__hom__axioms,axiom,
ring_c7120671943487988908t_real @ ring_1_of_int_real ).
% of_int_hom.comm_semiring_hom_axioms
thf(fact_222_of__int__hom_Ocomm__semiring__hom__axioms,axiom,
ring_c1049078972110686252nt_int @ ring_1_of_int_int ).
% of_int_hom.comm_semiring_hom_axioms
thf(fact_223_real__embedding_Ocomm__semiring__hom__axioms,axiom,
ring_c7120671943487988908t_real @ ring_H3062062880446257660of_int ).
% real_embedding.comm_semiring_hom_axioms
thf(fact_224_similar__mat__witD2_I3_J,axiom,
! [A: mat_int,N: nat,M2: nat,B: mat_int,P: mat_int,Q: mat_int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ M2 ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( A
= ( times_times_mat_int @ ( times_times_mat_int @ P @ B ) @ Q ) ) ) ) ).
% similar_mat_witD2(3)
thf(fact_225_similar__mat__witD2_I3_J,axiom,
! [A: mat_real,N: nat,M2: nat,B: mat_real,P: mat_real,Q: mat_real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ M2 ) )
=> ( ( similar_mat_wit_real @ A @ B @ P @ Q )
=> ( A
= ( times_times_mat_real @ ( times_times_mat_real @ P @ B ) @ Q ) ) ) ) ).
% similar_mat_witD2(3)
thf(fact_226_add__point__existing__blocks,axiom,
! [Bs4: set_set_a,P3: a] :
( ! [Bl: set_a] :
( ( member_set_a @ Bl @ Bs4 )
=> ( member_a @ P3 @ Bl ) )
=> ( ( design2935547469388721088ocks_a @ ( mset_set_a @ b_s ) @ P3 @ Bs4 )
= ( mset_set_a @ b_s ) ) ) ).
% add_point_existing_blocks
thf(fact_227_del__invalid__point__blocks,axiom,
! [P3: a] :
( ~ ( member_a @ P3 @ ( set_a2 @ v_s ) )
=> ( ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P3 )
= ( mset_set_a @ b_s ) ) ) ).
% del_invalid_point_blocks
thf(fact_228_delete__invalid__pt__strong__eq,axiom,
! [P3: a] :
( ~ ( member_a @ P3 @ ( set_a2 @ v_s ) )
=> ( ( mset_set_a @ b_s )
= ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P3 ) ) ) ).
% delete_invalid_pt_strong_eq
thf(fact_229_proper__design__axioms,axiom,
design7287791228148780576sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).
% proper_design_axioms
thf(fact_230_wf__invalid__point,axiom,
! [X: a,B2: set_a] :
( ~ ( member_a @ X @ ( set_a2 @ v_s ) )
=> ( ( member_set_a @ B2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ~ ( member_a @ X @ B2 ) ) ) ).
% wf_invalid_point
thf(fact_231_finite__incidence__system__axioms,axiom,
design9187838744727572296stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).
% finite_incidence_system_axioms
thf(fact_232_incidence__system__axioms,axiom,
design1863209521793301785stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).
% incidence_system_axioms
thf(fact_233_del__invalid__point,axiom,
! [P3: a] :
( ~ ( member_a @ P3 @ ( set_a2 @ v_s ) )
=> ( ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P3 )
= ( set_a2 @ v_s ) ) ) ).
% del_invalid_point
thf(fact_234_constant__rep__point__pair,axiom,
! [X1: a,X22: a] :
( ( member_a @ X1 @ ( set_a2 @ v_s ) )
=> ( ( member_a @ X22 @ ( set_a2 @ v_s ) )
=> ( ( X1 != X22 )
=> ( ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X1 )
= ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X22 ) ) ) ) ) ).
% constant_rep_point_pair
thf(fact_235_delete__point__strong__block__in,axiom,
! [P3: a,Bl2: set_a] :
( ~ ( member_a @ P3 @ Bl2 )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P3 ) ) ) ) ) ).
% delete_point_strong_block_in
thf(fact_236_delete__point__strong__block__in__iff,axiom,
! [Bl2: set_a,P3: a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P3 ) ) )
= ( ~ ( member_a @ P3 @ Bl2 ) ) ) ) ).
% delete_point_strong_block_in_iff
thf(fact_237_delete__point__strong__block__in__orig,axiom,
! [Bl2: set_a,P3: a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P3 ) ) )
=> ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% delete_point_strong_block_in_orig
thf(fact_238_delete__point__strong__block__not__in,axiom,
! [P3: a,Bl2: set_a] :
( ( member_a @ P3 @ Bl2 )
=> ~ ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P3 ) ) ) ) ).
% delete_point_strong_block_not_in
thf(fact_239_delete__point__p__not__in__bl__blocks,axiom,
! [P3: a] :
( ! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ~ ( member_a @ P3 @ Bl ) )
=> ( ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P3 )
= ( mset_set_a @ b_s ) ) ) ).
% delete_point_p_not_in_bl_blocks
thf(fact_240_constant__rep__alt,axiom,
! [X1: a,R2: nat,X: a] :
( ( member_a @ X1 @ ( set_a2 @ v_s ) )
=> ( ( ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X1 )
= R2 )
=> ( ( member_a @ X @ ( set_a2 @ v_s ) )
=> ( ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X )
= R2 ) ) ) ) ).
% constant_rep_alt
thf(fact_241_strong__del__point__incidence__wf,axiom,
! [P3: a] : ( design1863209521793301785stem_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P3 ) ) ).
% strong_del_point_incidence_wf
thf(fact_242_strong__del__point__finite,axiom,
! [P3: a] : ( design9187838744727572296stem_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P3 ) ) ).
% strong_del_point_finite
thf(fact_243_delete__point__wf,axiom,
! [P3: a] : ( design1863209521793301785stem_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P3 ) ) ).
% delete_point_wf
thf(fact_244_delete__point__finite,axiom,
! [P3: a] : ( design9187838744727572296stem_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P3 ) ) ).
% delete_point_finite
thf(fact_245_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_246_incidence__alt__def,axiom,
! [P3: a,B2: set_a] :
( ( member_a @ P3 @ ( set_a2 @ v_s ) )
=> ( ( member_set_a @ B2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design3210447939978979927dent_a @ ( mset_set_a @ b_s ) @ P3 @ B2 )
= ( member_a @ P3 @ B2 ) ) ) ) ).
% incidence_alt_def
thf(fact_247_ordered__proper__design_Oaxioms_I2_J,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide8030590239390523023ec_int @ V_s @ B_s )
=> ( design4550196916521492667ec_int @ ( set_vec_int2 @ V_s ) @ ( mset_set_vec_int @ B_s ) ) ) ).
% ordered_proper_design.axioms(2)
thf(fact_248_ordered__proper__design_Oaxioms_I2_J,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide1001368407746664282gn_nat @ V_s @ B_s )
=> ( design435815215503836206gn_nat @ ( set_nat2 @ V_s ) @ ( mset_set_nat @ B_s ) ) ) ).
% ordered_proper_design.axioms(2)
thf(fact_249_ordered__proper__design_Oaxioms_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide3676903341588786676sign_a @ V_s @ B_s )
=> ( design7287791228148780576sign_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) ) ) ).
% ordered_proper_design.axioms(2)
thf(fact_250_inc__sys__orderedI,axiom,
! [V: set_vec_int,B: multiset_set_vec_int,Vs2: list_vec_int,Bs2: list_set_vec_int] :
( ( design752387947253681460ec_int @ V @ B )
=> ( ( distinct_vec_int @ Vs2 )
=> ( ( ( set_vec_int2 @ Vs2 )
= V )
=> ( ( ( mset_set_vec_int @ Bs2 )
= B )
=> ( incide9132343510602215008ec_int @ Vs2 @ Bs2 ) ) ) ) ) ).
% inc_sys_orderedI
thf(fact_251_inc__sys__orderedI,axiom,
! [V: set_a,B: multiset_set_a,Vs2: list_a,Bs2: list_set_a] :
( ( design1863209521793301785stem_a @ V @ B )
=> ( ( distinct_a @ Vs2 )
=> ( ( ( set_a2 @ Vs2 )
= V )
=> ( ( ( mset_set_a @ Bs2 )
= B )
=> ( incide1624170830610365509stem_a @ Vs2 @ Bs2 ) ) ) ) ) ).
% inc_sys_orderedI
thf(fact_252_inc__sys__orderedI,axiom,
! [V: set_nat,B: multiset_set_nat,Vs2: list_nat,Bs2: list_set_nat] :
( ( design3753904077504641269em_nat @ V @ B )
=> ( ( distinct_nat @ Vs2 )
=> ( ( ( set_nat2 @ Vs2 )
= V )
=> ( ( ( mset_set_nat @ Bs2 )
= B )
=> ( incide6998539924841383625em_nat @ Vs2 @ Bs2 ) ) ) ) ) ).
% inc_sys_orderedI
thf(fact_253_ordered__incidence__sysII,axiom,
! [V2: set_vec_int,B4: multiset_set_vec_int,V_s: list_vec_int,B_s: list_set_vec_int] :
( ( design5943305819988106723ec_int @ V2 @ B4 )
=> ( ( ( set_vec_int2 @ V_s )
= V2 )
=> ( ( distinct_vec_int @ V_s )
=> ( ( ( mset_set_vec_int @ B_s )
= B4 )
=> ( incide9132343510602215008ec_int @ V_s @ B_s ) ) ) ) ) ).
% ordered_incidence_sysII
thf(fact_254_ordered__incidence__sysII,axiom,
! [V2: set_a,B4: multiset_set_a,V_s: list_a,B_s: list_set_a] :
( ( design9187838744727572296stem_a @ V2 @ B4 )
=> ( ( ( set_a2 @ V_s )
= V2 )
=> ( ( distinct_a @ V_s )
=> ( ( ( mset_set_a @ B_s )
= B4 )
=> ( incide1624170830610365509stem_a @ V_s @ B_s ) ) ) ) ) ).
% ordered_incidence_sysII
thf(fact_255_ordered__incidence__sysII,axiom,
! [V2: set_nat,B4: multiset_set_nat,V_s: list_nat,B_s: list_set_nat] :
( ( design5426232790142929158em_nat @ V2 @ B4 )
=> ( ( ( set_nat2 @ V_s )
= V2 )
=> ( ( distinct_nat @ V_s )
=> ( ( ( mset_set_nat @ B_s )
= B4 )
=> ( incide6998539924841383625em_nat @ V_s @ B_s ) ) ) ) ) ).
% ordered_incidence_sysII
thf(fact_256_ordered__proper__design__def,axiom,
( incide8030590239390523023ec_int
= ( ^ [V_s2: list_vec_int,B_s2: list_set_vec_int] :
( ( incide7599675121277616863ec_int @ V_s2 @ B_s2 )
& ( design4550196916521492667ec_int @ ( set_vec_int2 @ V_s2 ) @ ( mset_set_vec_int @ B_s2 ) ) ) ) ) ).
% ordered_proper_design_def
thf(fact_257_ordered__proper__design__def,axiom,
( incide1001368407746664282gn_nat
= ( ^ [V_s2: list_nat,B_s2: list_set_nat] :
( ( incide8999572217031194378gn_nat @ V_s2 @ B_s2 )
& ( design435815215503836206gn_nat @ ( set_nat2 @ V_s2 ) @ ( mset_set_nat @ B_s2 ) ) ) ) ) ).
% ordered_proper_design_def
thf(fact_258_ordered__proper__design__def,axiom,
( incide3676903341588786676sign_a
= ( ^ [V_s2: list_a,B_s2: list_set_a] :
( ( incide2848671379600480836sign_a @ V_s2 @ B_s2 )
& ( design7287791228148780576sign_a @ ( set_a2 @ V_s2 ) @ ( mset_set_a @ B_s2 ) ) ) ) ) ).
% ordered_proper_design_def
thf(fact_259_ordered__proper__design_Ointro,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide7599675121277616863ec_int @ V_s @ B_s )
=> ( ( design4550196916521492667ec_int @ ( set_vec_int2 @ V_s ) @ ( mset_set_vec_int @ B_s ) )
=> ( incide8030590239390523023ec_int @ V_s @ B_s ) ) ) ).
% ordered_proper_design.intro
thf(fact_260_ordered__proper__design_Ointro,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide8999572217031194378gn_nat @ V_s @ B_s )
=> ( ( design435815215503836206gn_nat @ ( set_nat2 @ V_s ) @ ( mset_set_nat @ B_s ) )
=> ( incide1001368407746664282gn_nat @ V_s @ B_s ) ) ) ).
% ordered_proper_design.intro
thf(fact_261_ordered__proper__design_Ointro,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide2848671379600480836sign_a @ V_s @ B_s )
=> ( ( design7287791228148780576sign_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) )
=> ( incide3676903341588786676sign_a @ V_s @ B_s ) ) ) ).
% ordered_proper_design.intro
thf(fact_262_incidence__system_Ois__incidence__mat__alt,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N2: mat_int] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( incide3487307869859529607_int_a @ N2 @ Point_set @ Block_collection )
= ( ? [Vs3: list_a,Bs3: list_set_a] :
( ( ( set_a2 @ Vs3 )
= Point_set )
& ( ( mset_set_a @ Bs3 )
= Block_collection )
& ( incide1624170830610365509stem_a @ Vs3 @ Bs3 )
& ( N2
= ( incide7016682120514301311_a_int @ Vs3 @ Bs3 ) ) ) ) ) ) ).
% incidence_system.is_incidence_mat_alt
thf(fact_263_incidence__system_Ois__incidence__mat__alt,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,N2: mat_int] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( incide8019310070229209863nt_nat @ N2 @ Point_set @ Block_collection )
= ( ? [Vs3: list_nat,Bs3: list_set_nat] :
( ( ( set_nat2 @ Vs3 )
= Point_set )
& ( ( mset_set_nat @ Bs3 )
= Block_collection )
& ( incide6998539924841383625em_nat @ Vs3 @ Bs3 )
& ( N2
= ( incide1177982898701834729at_int @ Vs3 @ Bs3 ) ) ) ) ) ) ).
% incidence_system.is_incidence_mat_alt
thf(fact_264_ordered__incidence__system_Ois__incidence__mat__true,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s @ B_s )
=> ( incide3638075654752512162ec_int @ ( incide2612794423365091098nt_int @ V_s @ B_s ) @ ( set_vec_int2 @ V_s ) @ ( mset_set_vec_int @ B_s ) ) ) ).
% ordered_incidence_system.is_incidence_mat_true
thf(fact_265_ordered__incidence__system_Ois__incidence__mat__true,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( incide3487307869859529607_int_a @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) ) ) ).
% ordered_incidence_system.is_incidence_mat_true
thf(fact_266_ordered__incidence__system_Ois__incidence__mat__true,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( incide8019310070229209863nt_nat @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( set_nat2 @ V_s ) @ ( mset_set_nat @ B_s ) ) ) ).
% ordered_incidence_system.is_incidence_mat_true
thf(fact_267_inc__vec__eq__iff__blocks,axiom,
! [Bl2: set_a,Bl3: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_set_a @ Bl3 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ( incide4681487840832596491_a_int @ v_s @ Bl2 )
= ( incide4681487840832596491_a_int @ v_s @ Bl3 ) )
= ( Bl2 = Bl3 ) ) ) ) ).
% inc_vec_eq_iff_blocks
thf(fact_268_strong__del__point__sub__des,axiom,
! [P3: a] : ( sub_sub_design_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P3 ) @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% strong_del_point_sub_des
thf(fact_269_add__point__blocks__wf,axiom,
! [P3: a,Bs4: set_set_a] : ( design1863209521793301785stem_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( design2935547469388721088ocks_a @ ( mset_set_a @ b_s ) @ P3 @ Bs4 ) ) ).
% add_point_blocks_wf
thf(fact_270_add__point__to__blocks__finite,axiom,
! [P3: a,Bs4: set_set_a] : ( design9187838744727572296stem_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( design2935547469388721088ocks_a @ ( mset_set_a @ b_s ) @ P3 @ Bs4 ) ) ).
% add_point_to_blocks_finite
thf(fact_271_obtain__point__with__rep,axiom,
! [R: nat] :
( ( member_nat @ R @ ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ v_s ) )
& ( ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X3 )
= R ) ) ) ).
% obtain_point_with_rep
thf(fact_272_point__rep__number__in__set,axiom,
! [X: a] :
( ( member_a @ X @ ( set_a2 @ v_s ) )
=> ( member_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) @ ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ).
% point_rep_number_in_set
thf(fact_273_add__point__wf,axiom,
! [P3: a] : ( design1863209521793301785stem_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( mset_set_a @ b_s ) ) ).
% add_point_wf
thf(fact_274_add__point__finite,axiom,
! [P3: a] : ( design9187838744727572296stem_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( mset_set_a @ b_s ) ) ).
% add_point_finite
thf(fact_275_strong__del__point__sub__sys,axiom,
! [P3: a] : ( sub_su7923802003039619913stem_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P3 ) @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% strong_del_point_sub_sys
thf(fact_276_add__delete__point__inv,axiom,
! [P3: a] :
( ~ ( member_a @ P3 @ ( set_a2 @ v_s ) )
=> ( ( design108908007054065099oint_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P3 ) @ P3 )
= ( set_a2 @ v_s ) ) ) ).
% add_delete_point_inv
thf(fact_277_add__point__sub__sys,axiom,
! [P3: a] : ( sub_su7923802003039619913stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( mset_set_a @ b_s ) ) ).
% add_point_sub_sys
thf(fact_278_add__point__sub__des,axiom,
! [P3: a] : ( sub_sub_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( mset_set_a @ b_s ) ) ).
% add_point_sub_des
thf(fact_279_add__existing__point,axiom,
! [P3: a] :
( ( member_a @ P3 @ ( set_a2 @ v_s ) )
=> ( ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P3 )
= ( set_a2 @ v_s ) ) ) ).
% add_existing_point
thf(fact_280_ordered__incidence__system_Oinc__vec__eq__iff__blocks,axiom,
! [V_s: list_a,B_s: list_set_a,Bl2: set_a,Bl3: set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ B_s ) ) )
=> ( ( member_set_a @ Bl3 @ ( set_mset_set_a @ ( mset_set_a @ B_s ) ) )
=> ( ( ( incide4681487840832596491_a_int @ V_s @ Bl2 )
= ( incide4681487840832596491_a_int @ V_s @ Bl3 ) )
= ( Bl2 = Bl3 ) ) ) ) ) ).
% ordered_incidence_system.inc_vec_eq_iff_blocks
thf(fact_281_ordered__incidence__system_Oinc__vec__eq__iff__blocks,axiom,
! [V_s: list_nat,B_s: list_set_nat,Bl2: set_nat,Bl3: set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( mset_set_nat @ B_s ) ) )
=> ( ( member_set_nat @ Bl3 @ ( set_mset_set_nat @ ( mset_set_nat @ B_s ) ) )
=> ( ( ( incide8105300093900788317at_int @ V_s @ Bl2 )
= ( incide8105300093900788317at_int @ V_s @ Bl3 ) )
= ( Bl2 = Bl3 ) ) ) ) ) ).
% ordered_incidence_system.inc_vec_eq_iff_blocks
thf(fact_282_replication__number__single,axiom,
is_singleton_nat @ ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% replication_number_single
thf(fact_283_complement__finite,axiom,
design9187838744727572296stem_a @ ( set_a2 @ v_s ) @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% complement_finite
thf(fact_284_delete__block__sub__des,axiom,
! [B2: set_a] : ( sub_sub_design_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B2 ) @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% delete_block_sub_des
thf(fact_285_delete__block__sub__sys,axiom,
! [B2: set_a] : ( sub_su7923802003039619913stem_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B2 ) @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% delete_block_sub_sys
thf(fact_286_delete__block__fin__incidence__sys,axiom,
! [B2: set_a] : ( design9187838744727572296stem_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B2 ) ) ).
% delete_block_fin_incidence_sys
thf(fact_287_delete__block__wf,axiom,
! [B2: set_a] : ( design1863209521793301785stem_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B2 ) ) ).
% delete_block_wf
thf(fact_288_constant__rep__point__not__0,axiom,
! [X: a] :
( ( member_a @ X @ ( set_a2 @ v_s ) )
=> ( ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X )
!= zero_zero_nat ) ) ).
% constant_rep_point_not_0
thf(fact_289_blocks__list__nempty,axiom,
( ( design7287791228148780576sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) )
=> ( b_s != nil_set_a ) ) ).
% blocks_list_nempty
thf(fact_290_delete__invalid__block__eq,axiom,
! [B2: set_a] :
( ~ ( member_set_a @ B2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B2 )
= ( mset_set_a @ b_s ) ) ) ).
% delete_invalid_block_eq
thf(fact_291_ideal_Oscale__zero__left,axiom,
! [X: int] :
( ( times_times_int @ zero_zero_int @ X )
= zero_zero_int ) ).
% ideal.scale_zero_left
thf(fact_292_ideal_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% ideal.scale_zero_left
thf(fact_293_ideal_Oscale__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% ideal.scale_zero_right
thf(fact_294_ideal_Oscale__zero__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% ideal.scale_zero_right
thf(fact_295_mult__hom_Ohom__zero,axiom,
! [C2: int] :
( ( times_times_int @ C2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_296_mult__hom_Ohom__zero,axiom,
! [C2: nat] :
( ( times_times_nat @ C2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_297_mult__hom_Ohom__zero,axiom,
! [C2: real] :
( ( times_times_real @ C2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_298_of__int__hom_Ohom__zero,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_hom.hom_zero
thf(fact_299_of__int__hom_Ohom__zero,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_hom.hom_zero
thf(fact_300_of__int__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( ring_1_of_int_real @ X )
= zero_zero_real )
= ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0_iff
thf(fact_301_of__int__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( ring_1_of_int_int @ X )
= zero_zero_int )
= ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0_iff
thf(fact_302_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_303_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_304_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= zero_zero_real )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_305_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_306_real__zero,axiom,
( ( ring_H3062062880446257660of_int @ zero_zero_int )
= zero_zero_real ) ).
% real_zero
thf(fact_307_point__rep__non__existance,axiom,
! [X: a] :
( ~ ( member_a @ X @ ( set_a2 @ v_s ) )
=> ( ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X )
= zero_zero_nat ) ) ).
% point_rep_non_existance
thf(fact_308_complement__wf,axiom,
design1863209521793301785stem_a @ ( set_a2 @ v_s ) @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% complement_wf
thf(fact_309_of__int__hom_Ohom__0,axiom,
! [X: int] :
( ( ( ring_1_of_int_real @ X )
= zero_zero_real )
=> ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0
thf(fact_310_of__int__hom_Ohom__0,axiom,
! [X: int] :
( ( ( ring_1_of_int_int @ X )
= zero_zero_int )
=> ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0
thf(fact_311_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_312_zero__reorient,axiom,
! [X: multiset_set_a] :
( ( zero_z5079479921072680283_set_a = X )
= ( X = zero_z5079479921072680283_set_a ) ) ).
% zero_reorient
thf(fact_313_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_314_zero__reorient,axiom,
! [X: multiset_set_nat] :
( ( zero_z3157962936165190495et_nat = X )
= ( X = zero_z3157962936165190495et_nat ) ) ).
% zero_reorient
thf(fact_315_distinct_Osimps_I1_J,axiom,
distinct_set_a @ nil_set_a ).
% distinct.simps(1)
thf(fact_316_distinct_Osimps_I1_J,axiom,
distinct_a @ nil_a ).
% distinct.simps(1)
thf(fact_317_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
% distinct.simps(1)
thf(fact_318_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) )
= zero_zero_int ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_319_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) )
= zero_zero_real ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_320_of__real__hom_Ohom__mult__eq__zero,axiom,
! [X: real,Y: real] :
( ( ( times_times_real @ X @ Y )
= zero_zero_real )
=> ( ( times_times_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) )
= zero_zero_real ) ) ).
% of_real_hom.hom_mult_eq_zero
thf(fact_321_real__embedding_Ohom__mult__eq__zero,axiom,
! [X: real,Y: real] :
( ( ( times_times_real @ X @ Y )
= zero_zero_real )
=> ( ( times_times_real @ ( ring_H569718986840772220f_real @ X ) @ ( ring_H569718986840772220f_real @ Y ) )
= zero_zero_real ) ) ).
% real_embedding.hom_mult_eq_zero
thf(fact_322_real__embedding_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_times_real @ ( ring_H3062062880446257660of_int @ X ) @ ( ring_H3062062880446257660of_int @ Y ) )
= zero_zero_real ) ) ).
% real_embedding.hom_mult_eq_zero
thf(fact_323_semiring__hom_Ohom__mult__eq__zero,axiom,
! [Hom: int > int,X: int,Y: int] :
( ( ring_s4547151187181212152nt_int @ Hom )
=> ( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_times_int @ ( Hom @ X ) @ ( Hom @ Y ) )
= zero_zero_int ) ) ) ).
% semiring_hom.hom_mult_eq_zero
thf(fact_324_semiring__hom_Ohom__mult__eq__zero,axiom,
! [Hom: int > nat,X: int,Y: int] :
( ( ring_s4549641657690262428nt_nat @ Hom )
=> ( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_times_nat @ ( Hom @ X ) @ ( Hom @ Y ) )
= zero_zero_nat ) ) ) ).
% semiring_hom.hom_mult_eq_zero
thf(fact_325_semiring__hom_Ohom__mult__eq__zero,axiom,
! [Hom: int > real,X: int,Y: int] :
( ( ring_s3477937289336866936t_real @ Hom )
=> ( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_times_real @ ( Hom @ X ) @ ( Hom @ Y ) )
= zero_zero_real ) ) ) ).
% semiring_hom.hom_mult_eq_zero
thf(fact_326_semiring__hom_Ohom__mult__eq__zero,axiom,
! [Hom: nat > int,X: nat,Y: nat] :
( ( ring_s3547797475280157596at_int @ Hom )
=> ( ( ( times_times_nat @ X @ Y )
= zero_zero_nat )
=> ( ( times_times_int @ ( Hom @ X ) @ ( Hom @ Y ) )
= zero_zero_int ) ) ) ).
% semiring_hom.hom_mult_eq_zero
thf(fact_327_semiring__hom_Ohom__mult__eq__zero,axiom,
! [Hom: nat > nat,X: nat,Y: nat] :
( ( ring_s3550287945789207872at_nat @ Hom )
=> ( ( ( times_times_nat @ X @ Y )
= zero_zero_nat )
=> ( ( times_times_nat @ ( Hom @ X ) @ ( Hom @ Y ) )
= zero_zero_nat ) ) ) ).
% semiring_hom.hom_mult_eq_zero
thf(fact_328_semiring__hom_Ohom__mult__eq__zero,axiom,
! [Hom: nat > real,X: nat,Y: nat] :
( ( ring_s1291015714643845148t_real @ Hom )
=> ( ( ( times_times_nat @ X @ Y )
= zero_zero_nat )
=> ( ( times_times_real @ ( Hom @ X ) @ ( Hom @ Y ) )
= zero_zero_real ) ) ) ).
% semiring_hom.hom_mult_eq_zero
thf(fact_329_semiring__hom_Ohom__mult__eq__zero,axiom,
! [Hom: real > int,X: real,Y: real] :
( ( ring_s5855833817263693560al_int @ Hom )
=> ( ( ( times_times_real @ X @ Y )
= zero_zero_real )
=> ( ( times_times_int @ ( Hom @ X ) @ ( Hom @ Y ) )
= zero_zero_int ) ) ) ).
% semiring_hom.hom_mult_eq_zero
thf(fact_330_semiring__hom_Ohom__mult__eq__zero,axiom,
! [Hom: real > nat,X: real,Y: real] :
( ( ring_s5858324287772743836al_nat @ Hom )
=> ( ( ( times_times_real @ X @ Y )
= zero_zero_real )
=> ( ( times_times_nat @ ( Hom @ X ) @ ( Hom @ Y ) )
= zero_zero_nat ) ) ) ).
% semiring_hom.hom_mult_eq_zero
thf(fact_331_semiring__hom_Ohom__mult__eq__zero,axiom,
! [Hom: real > real,X: real,Y: real] :
( ( ring_s1386325929209763192l_real @ Hom )
=> ( ( ( times_times_real @ X @ Y )
= zero_zero_real )
=> ( ( times_times_real @ ( Hom @ X ) @ ( Hom @ Y ) )
= zero_zero_real ) ) ) ).
% semiring_hom.hom_mult_eq_zero
thf(fact_332_ordered__incidence__system_Opoints__list__nempty,axiom,
! [V_s: list_set_a,B_s: list_set_set_a] :
( ( incide2166342132139297189_set_a @ V_s @ B_s )
=> ( ( design8920408637772541824_set_a @ ( set_set_a2 @ V_s ) @ ( mset_set_set_a @ B_s ) )
=> ( V_s != nil_set_a ) ) ) ).
% ordered_incidence_system.points_list_nempty
thf(fact_333_ordered__incidence__system_Opoints__list__nempty,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s @ B_s )
=> ( ( design4550196916521492667ec_int @ ( set_vec_int2 @ V_s ) @ ( mset_set_vec_int @ B_s ) )
=> ( V_s != nil_vec_int ) ) ) ).
% ordered_incidence_system.points_list_nempty
thf(fact_334_ordered__incidence__system_Opoints__list__nempty,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( design7287791228148780576sign_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) )
=> ( V_s != nil_a ) ) ) ).
% ordered_incidence_system.points_list_nempty
thf(fact_335_ordered__incidence__system_Opoints__list__nempty,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( design435815215503836206gn_nat @ ( set_nat2 @ V_s ) @ ( mset_set_nat @ B_s ) )
=> ( V_s != nil_nat ) ) ) ).
% ordered_incidence_system.points_list_nempty
thf(fact_336_ordered__incidence__system_Oblocks__list__nempty,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s @ B_s )
=> ( ( design4550196916521492667ec_int @ ( set_vec_int2 @ V_s ) @ ( mset_set_vec_int @ B_s ) )
=> ( B_s != nil_set_vec_int ) ) ) ).
% ordered_incidence_system.blocks_list_nempty
thf(fact_337_ordered__incidence__system_Oblocks__list__nempty,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( design7287791228148780576sign_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) )
=> ( B_s != nil_set_a ) ) ) ).
% ordered_incidence_system.blocks_list_nempty
thf(fact_338_ordered__incidence__system_Oblocks__list__nempty,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( design435815215503836206gn_nat @ ( set_nat2 @ V_s ) @ ( mset_set_nat @ B_s ) )
=> ( B_s != nil_set_nat ) ) ) ).
% ordered_incidence_system.blocks_list_nempty
thf(fact_339_point__in__block__rep__min__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( set_a2 @ v_s ) )
=> ? [Bl: set_a] :
( ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
& ( member_a @ X @ Bl ) )
= ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) ) ) ) ).
% point_in_block_rep_min_iff
thf(fact_340_complement__blocks__wf,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) )
=> ( ord_less_eq_set_a @ Bl2 @ ( set_a2 @ v_s ) ) ) ).
% complement_blocks_wf
thf(fact_341_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 ) ) ) )
=> ~ ! [Bl22: set_a] :
( ( member_set_a @ Bl22 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( Bl1
!= ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl22 ) ) ) ) ).
% obtain_comp_block_orig
thf(fact_342_rep__number__non__zero__system__point,axiom,
! [X: a] :
( ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) )
=> ( member_a @ X @ ( set_a2 @ v_s ) ) ) ).
% rep_number_non_zero_system_point
thf(fact_343_points__list__nempty,axiom,
( ( design7287791228148780576sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) )
=> ( v_s != nil_a ) ) ).
% points_list_nempty
thf(fact_344_sys__block__sizes__uniform__single,axiom,
is_singleton_nat @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ).
% sys_block_sizes_uniform_single
thf(fact_345_del__invalid__add__block__eq,axiom,
! [Bl2: set_a] :
( ~ ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) @ Bl2 )
= ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) ) ) ).
% del_invalid_add_block_eq
thf(fact_346_del__add__block__inv,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) @ Bl2 )
= ( mset_set_a @ b_s ) ) ) ).
% del_add_block_inv
thf(fact_347_design__blocks__nempty,axiom,
( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a ) ).
% design_blocks_nempty
thf(fact_348_block__set__nempty__imp__block__ex,axiom,
( ( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a )
=> ? [Bl: set_a] : ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% block_set_nempty_imp_block_ex
thf(fact_349_blocks__list__empty__iff,axiom,
( ( b_s = nil_set_a )
= ( ( mset_set_a @ b_s )
= zero_z5079479921072680283_set_a ) ) ).
% blocks_list_empty_iff
thf(fact_350_block__complement__subset__points,axiom,
! [Ps: set_a,Bl2: set_a] :
( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 ) )
=> ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) ) ) ).
% block_complement_subset_points
thf(fact_351_block__complement__elem__iff,axiom,
! [Ps: set_a,Bl2: set_a] :
( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ Ps )
=> ~ ( member_a @ X2 @ Bl2 ) ) ) ) ) ).
% block_complement_elem_iff
thf(fact_352_block__comp__elem__alt__right,axiom,
! [Ps: set_a,Bl2: set_a] :
( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ Ps )
=> ~ ( member_a @ X3 @ Bl2 ) )
=> ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 ) ) ) ) ).
% block_comp_elem_alt_right
thf(fact_353_block__comp__elem__alt__left,axiom,
! [X: a,Bl2: set_a,Ps: set_a] :
( ( member_a @ X @ Bl2 )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 ) )
=> ~ ( member_a @ X @ Ps ) ) ) ).
% block_comp_elem_alt_left
thf(fact_354_add__block__rep__number__not__in,axiom,
! [X: a,B2: set_a] :
( ~ ( member_a @ X @ B2 )
=> ( ( design6637022207325878697mber_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B2 ) @ X )
= ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) ) ) ).
% add_block_rep_number_not_in
thf(fact_355_delete__point__blocks__sub,axiom,
! [B2: set_a,P3: a] :
( ( member_set_a @ B2 @ ( set_mset_set_a @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P3 ) ) )
=> ~ ! [Bl: set_a] :
~ ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
& ( ord_less_eq_set_a @ B2 @ Bl ) ) ) ).
% delete_point_blocks_sub
thf(fact_356_wf__list,axiom,
! [B2: set_a] :
( ( member_set_a @ B2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ord_less_eq_set_a @ B2 @ ( set_a2 @ v_s ) ) ) ).
% wf_list
thf(fact_357_block__complement__inv,axiom,
! [Bl2: set_a,Bl23: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 )
= Bl23 )
=> ( ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl23 )
= Bl2 ) ) ) ).
% block_complement_inv
thf(fact_358_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_359_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_360_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_361_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_362_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_363_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_364_add__del__block__inv,axiom,
! [Bl2: set_a] :
( ( ord_less_eq_set_a @ Bl2 @ ( set_a2 @ v_s ) )
=> ( ( design1146539425385464078lock_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) @ Bl2 )
= ( mset_set_a @ b_s ) ) ) ).
% add_del_block_inv
thf(fact_365_add__block__fin__cond,axiom,
! [B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( set_a2 @ v_s ) )
=> ( design9187838744727572296stem_a @ ( set_a2 @ v_s ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B2 ) ) ) ).
% add_block_fin_cond
thf(fact_366_add__block__wf__cond,axiom,
! [B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( set_a2 @ v_s ) )
=> ( design1863209521793301785stem_a @ ( set_a2 @ v_s ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B2 ) ) ) ).
% add_block_wf_cond
thf(fact_367_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_368_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_369_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_370_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_371_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_372_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_373_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_374_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_375_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_pos
thf(fact_376_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_pos
thf(fact_377_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_nonneg
thf(fact_378_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_nonneg
thf(fact_379_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_380_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_381_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_382_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_383_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_384_subset__code_I1_J,axiom,
! [Xs: list_mat_int,B: set_mat_int] :
( ( ord_le5299038897506728741at_int @ ( set_mat_int2 @ Xs ) @ B )
= ( ! [X2: mat_int] :
( ( member_mat_int @ X2 @ ( set_mat_int2 @ Xs ) )
=> ( member_mat_int @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_385_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_386_subset__code_I1_J,axiom,
! [Xs: list_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_387_subset__code_I1_J,axiom,
! [Xs: list_mat_real,B: set_mat_real] :
( ( ord_le6763538137481655845t_real @ ( set_mat_real2 @ Xs ) @ B )
= ( ! [X2: mat_real] :
( ( member_mat_real @ X2 @ ( set_mat_real2 @ Xs ) )
=> ( member_mat_real @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_388_subset__code_I1_J,axiom,
! [Xs: list_vec_int,B: set_vec_int] :
( ( ord_le5046699514128596377ec_int @ ( set_vec_int2 @ Xs ) @ B )
= ( ! [X2: vec_int] :
( ( member_vec_int @ X2 @ ( set_vec_int2 @ Xs ) )
=> ( member_vec_int @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_389_subset__code_I1_J,axiom,
! [Xs: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_390_subset__code_I1_J,axiom,
! [Xs: list_int,B: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B )
= ( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ( member_int @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_391_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_392_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_393_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_394_set__obtain__sublist,axiom,
! [S: set_vec_int,Ls: list_vec_int] :
( ( ord_le5046699514128596377ec_int @ S @ ( set_vec_int2 @ Ls ) )
=> ~ ! [Ss: list_vec_int] :
( ( distinct_vec_int @ Ss )
=> ( S
!= ( set_vec_int2 @ Ss ) ) ) ) ).
% set_obtain_sublist
thf(fact_395_set__obtain__sublist,axiom,
! [S: set_a,Ls: list_a] :
( ( ord_less_eq_set_a @ S @ ( set_a2 @ Ls ) )
=> ~ ! [Ss: list_a] :
( ( distinct_a @ Ss )
=> ( S
!= ( set_a2 @ Ss ) ) ) ) ).
% set_obtain_sublist
thf(fact_396_set__obtain__sublist,axiom,
! [S: set_int,Ls: list_int] :
( ( ord_less_eq_set_int @ S @ ( set_int2 @ Ls ) )
=> ~ ! [Ss: list_int] :
( ( distinct_int @ Ss )
=> ( S
!= ( set_int2 @ Ss ) ) ) ) ).
% set_obtain_sublist
thf(fact_397_set__obtain__sublist,axiom,
! [S: set_nat,Ls: list_nat] :
( ( ord_less_eq_set_nat @ S @ ( set_nat2 @ Ls ) )
=> ~ ! [Ss: list_nat] :
( ( distinct_nat @ Ss )
=> ( S
!= ( set_nat2 @ Ss ) ) ) ) ).
% set_obtain_sublist
thf(fact_398_ordered__incidence__system_Oblocks__list__empty__iff,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( B_s = nil_set_a )
= ( ( mset_set_a @ B_s )
= zero_z5079479921072680283_set_a ) ) ) ).
% ordered_incidence_system.blocks_list_empty_iff
thf(fact_399_ordered__incidence__system_Oblocks__list__empty__iff,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( B_s = nil_set_nat )
= ( ( mset_set_nat @ B_s )
= zero_z3157962936165190495et_nat ) ) ) ).
% ordered_incidence_system.blocks_list_empty_iff
thf(fact_400_ordered__incidence__system_Owf__list,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int,B2: set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s @ B_s )
=> ( ( member_set_vec_int @ B2 @ ( set_mset_set_vec_int @ ( mset_set_vec_int @ B_s ) ) )
=> ( ord_le5046699514128596377ec_int @ B2 @ ( set_vec_int2 @ V_s ) ) ) ) ).
% ordered_incidence_system.wf_list
thf(fact_401_ordered__incidence__system_Owf__list,axiom,
! [V_s: list_int,B_s: list_set_int,B2: set_int] :
( ( incide6996049454332333349em_int @ V_s @ B_s )
=> ( ( member_set_int @ B2 @ ( set_mset_set_int @ ( mset_set_int @ B_s ) ) )
=> ( ord_less_eq_set_int @ B2 @ ( set_int2 @ V_s ) ) ) ) ).
% ordered_incidence_system.wf_list
thf(fact_402_ordered__incidence__system_Owf__list,axiom,
! [V_s: list_a,B_s: list_set_a,B2: set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( member_set_a @ B2 @ ( set_mset_set_a @ ( mset_set_a @ B_s ) ) )
=> ( ord_less_eq_set_a @ B2 @ ( set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.wf_list
thf(fact_403_ordered__incidence__system_Owf__list,axiom,
! [V_s: list_nat,B_s: list_set_nat,B2: set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( member_set_nat @ B2 @ ( set_mset_set_nat @ ( mset_set_nat @ B_s ) ) )
=> ( ord_less_eq_set_nat @ B2 @ ( set_nat2 @ V_s ) ) ) ) ).
% ordered_incidence_system.wf_list
thf(fact_404_ordered__incidence__system_Ointro,axiom,
! [B_s: list_set_vec_int,V_s: list_vec_int] :
( ! [B5: set_vec_int] :
( ( member_set_vec_int @ B5 @ ( set_mset_set_vec_int @ ( mset_set_vec_int @ B_s ) ) )
=> ( ord_le5046699514128596377ec_int @ B5 @ ( set_vec_int2 @ V_s ) ) )
=> ( ( distinct_vec_int @ V_s )
=> ( incide9132343510602215008ec_int @ V_s @ B_s ) ) ) ).
% ordered_incidence_system.intro
thf(fact_405_ordered__incidence__system_Ointro,axiom,
! [B_s: list_set_int,V_s: list_int] :
( ! [B5: set_int] :
( ( member_set_int @ B5 @ ( set_mset_set_int @ ( mset_set_int @ B_s ) ) )
=> ( ord_less_eq_set_int @ B5 @ ( set_int2 @ V_s ) ) )
=> ( ( distinct_int @ V_s )
=> ( incide6996049454332333349em_int @ V_s @ B_s ) ) ) ).
% ordered_incidence_system.intro
thf(fact_406_ordered__incidence__system_Ointro,axiom,
! [B_s: list_set_a,V_s: list_a] :
( ! [B5: set_a] :
( ( member_set_a @ B5 @ ( set_mset_set_a @ ( mset_set_a @ B_s ) ) )
=> ( ord_less_eq_set_a @ B5 @ ( set_a2 @ V_s ) ) )
=> ( ( distinct_a @ V_s )
=> ( incide1624170830610365509stem_a @ V_s @ B_s ) ) ) ).
% ordered_incidence_system.intro
thf(fact_407_ordered__incidence__system_Ointro,axiom,
! [B_s: list_set_nat,V_s: list_nat] :
( ! [B5: set_nat] :
( ( member_set_nat @ B5 @ ( set_mset_set_nat @ ( mset_set_nat @ B_s ) ) )
=> ( ord_less_eq_set_nat @ B5 @ ( set_nat2 @ V_s ) ) )
=> ( ( distinct_nat @ V_s )
=> ( incide6998539924841383625em_nat @ V_s @ B_s ) ) ) ).
% ordered_incidence_system.intro
thf(fact_408_ordered__incidence__system__def,axiom,
( incide9132343510602215008ec_int
= ( ^ [V_s2: list_vec_int,B_s2: list_set_vec_int] :
( ! [B3: set_vec_int] :
( ( member_set_vec_int @ B3 @ ( set_mset_set_vec_int @ ( mset_set_vec_int @ B_s2 ) ) )
=> ( ord_le5046699514128596377ec_int @ B3 @ ( set_vec_int2 @ V_s2 ) ) )
& ( distinct_vec_int @ V_s2 ) ) ) ) ).
% ordered_incidence_system_def
thf(fact_409_ordered__incidence__system__def,axiom,
( incide6996049454332333349em_int
= ( ^ [V_s2: list_int,B_s2: list_set_int] :
( ! [B3: set_int] :
( ( member_set_int @ B3 @ ( set_mset_set_int @ ( mset_set_int @ B_s2 ) ) )
=> ( ord_less_eq_set_int @ B3 @ ( set_int2 @ V_s2 ) ) )
& ( distinct_int @ V_s2 ) ) ) ) ).
% ordered_incidence_system_def
thf(fact_410_ordered__incidence__system__def,axiom,
( incide1624170830610365509stem_a
= ( ^ [V_s2: list_a,B_s2: list_set_a] :
( ! [B3: set_a] :
( ( member_set_a @ B3 @ ( set_mset_set_a @ ( mset_set_a @ B_s2 ) ) )
=> ( ord_less_eq_set_a @ B3 @ ( set_a2 @ V_s2 ) ) )
& ( distinct_a @ V_s2 ) ) ) ) ).
% ordered_incidence_system_def
thf(fact_411_ordered__incidence__system__def,axiom,
( incide6998539924841383625em_nat
= ( ^ [V_s2: list_nat,B_s2: list_set_nat] :
( ! [B3: set_nat] :
( ( member_set_nat @ B3 @ ( set_mset_set_nat @ ( mset_set_nat @ B_s2 ) ) )
=> ( ord_less_eq_set_nat @ B3 @ ( set_nat2 @ V_s2 ) ) )
& ( distinct_nat @ V_s2 ) ) ) ) ).
% ordered_incidence_system_def
thf(fact_412_complement__proper__design,axiom,
( ! [Bl: set_a] :
( ( 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 ) ) ) ) )
=> ( design7287791228148780576sign_a @ ( set_a2 @ v_s ) @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ).
% complement_proper_design
thf(fact_413_block__comp__incomplete,axiom,
! [Bl2: set_a] :
( ( ( 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 ) ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 ) ) ) ) ).
% block_comp_incomplete
thf(fact_414_add__block__sub__sys,axiom,
! [B2: set_a] : ( sub_su7923802003039619913stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( sup_sup_set_a @ ( set_a2 @ v_s ) @ B2 ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B2 ) ) ).
% add_block_sub_sys
thf(fact_415_add__block__wf,axiom,
! [B2: set_a] : ( design1863209521793301785stem_a @ ( sup_sup_set_a @ ( set_a2 @ v_s ) @ B2 ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B2 ) ) ).
% add_block_wf
thf(fact_416_complete__block__all__subsets,axiom,
! [Bl2: set_a,Ps: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ( finite_card_a @ Bl2 )
= ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
=> ( ord_less_eq_set_a @ Ps @ Bl2 ) ) ) ) ).
% complete_block_all_subsets
thf(fact_417_delete__point__index__eq,axiom,
! [Ps: set_a,P3: a] :
( ( ord_less_eq_set_a @ Ps @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P3 ) )
=> ( ( design254580327166089565ndex_a @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P3 ) @ Ps )
= ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) ) ) ).
% delete_point_index_eq
thf(fact_418_incomplete__imp__incomp__block,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 ) ) ) ) ) ).
% incomplete_imp_incomp_block
thf(fact_419_v__non__zero,axiom,
ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) ).
% v_non_zero
thf(fact_420_sys__block__sizes__obtain__bl,axiom,
! [X: nat] :
( ( member_nat @ X @ ( 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 )
= X ) ) ) ).
% sys_block_sizes_obtain_bl
thf(fact_421_sys__block__sizes__in,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_nat @ ( finite_card_a @ Bl2 ) @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ) ) ).
% sys_block_sizes_in
thf(fact_422_incomplete__imp__proper__subset,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ord_less_set_a @ Bl2 @ ( set_a2 @ v_s ) ) ) ).
% incomplete_imp_proper_subset
thf(fact_423_add__block__index__not__in,axiom,
! [Ps: set_a,B2: set_a] :
( ~ ( ord_less_eq_set_a @ Ps @ B2 )
=> ( ( design254580327166089565ndex_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B2 ) @ Ps )
= ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) ) ) ).
% add_block_index_not_in
thf(fact_424_block__size__gt__0,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ Bl2 ) ) ) ).
% block_size_gt_0
thf(fact_425_block__size__lt__order,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ).
% block_size_lt_order
thf(fact_426_complete__block__size__eq__points,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ( finite_card_a @ Bl2 )
= ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( Bl2
= ( set_a2 @ v_s ) ) ) ) ).
% complete_block_size_eq_points
thf(fact_427_incomplete__block__proper__subset,axiom,
! [Bl2: set_a] :
( ( ( 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 ) ) ) )
=> ( ord_less_set_a @ Bl2 @ ( set_a2 @ v_s ) ) ) ).
% incomplete_block_proper_subset
thf(fact_428_incomplete__alt__in,axiom,
! [Bl2: set_a] :
( ( ( 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 ) ) ) )
=> ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% incomplete_alt_in
thf(fact_429_incomplete__alt__size,axiom,
! [Bl2: set_a] :
( ( ( 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 ) ) ) )
=> ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ).
% incomplete_alt_size
thf(fact_430_points__index__ps__nin,axiom,
! [Ps: set_a] :
( ~ ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
=> ( ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps )
= zero_zero_nat ) ) ).
% points_index_ps_nin
thf(fact_431_points__index__zero,axiom,
! [Ps: set_a] :
( ( ord_less_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ ( finite_card_a @ Ps ) )
=> ( ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps )
= zero_zero_nat ) ) ).
% points_index_zero
thf(fact_432_set__union,axiom,
! [Xs: list_vec_int,Ys: list_vec_int] :
( ( set_vec_int2 @ ( union_vec_int @ Xs @ Ys ) )
= ( sup_sup_set_vec_int @ ( set_vec_int2 @ Xs ) @ ( set_vec_int2 @ Ys ) ) ) ).
% set_union
thf(fact_433_set__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( set_a2 @ ( union_a @ Xs @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).
% set_union
thf(fact_434_set__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_435_incomplete__alt__imp,axiom,
! [Bl2: set_a] :
( ( 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 ) ) )
=> ( ( 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 ) ) ) ) ) ) ).
% incomplete_alt_imp
thf(fact_436_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_437_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_438_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_439_cols__take__cols__subset,axiom,
! [A: mat_int,Inds: list_nat] : ( ord_le5046699514128596377ec_int @ ( set_vec_int2 @ ( cols_int @ ( more_take_cols_int @ A @ Inds ) ) ) @ ( set_vec_int2 @ ( cols_int @ A ) ) ) ).
% cols_take_cols_subset
thf(fact_440_point__indices__elem__in,axiom,
! [Ps: set_a,T: nat] :
( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
=> ( ( ( finite_card_a @ Ps )
= T )
=> ( member_nat @ ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) @ ( design328527185268214962ices_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ T ) ) ) ) ).
% point_indices_elem_in
thf(fact_441_block__comp__incomplete__nempty,axiom,
! [Bl2: set_a] :
( ( ( 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 ) ) ) )
=> ( ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 )
!= bot_bot_set_a ) ) ).
% block_comp_incomplete_nempty
thf(fact_442_add__block__fin,axiom,
! [B2: set_a] :
( ( finite_finite_a @ B2 )
=> ( design9187838744727572296stem_a @ ( sup_sup_set_a @ ( set_a2 @ v_s ) @ B2 ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B2 ) ) ) ).
% add_block_fin
thf(fact_443_N__carrier__mat__01__lift,axiom,
member_mat_real @ ( matrix1811533390045330843t_real @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) @ ( carrier_mat_real @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% N_carrier_mat_01_lift
thf(fact_444_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_445_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_446_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_447__092_060open_062N_A_092_060in_062_Acarrier__mat_A_092_060v_062_A_Ilength_A_092_060B_062s_J_092_060close_062,axiom,
member_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( carrier_mat_int @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ ( size_size_list_set_a @ b_s ) ) ).
% \<open>N \<in> carrier_mat \<v> (length \<B>s)\<close>
thf(fact_448_design__points__nempty,axiom,
( ( set_a2 @ v_s )
!= bot_bot_set_a ) ).
% design_points_nempty
thf(fact_449_finite__sets,axiom,
finite_finite_a @ ( set_a2 @ v_s ) ).
% finite_sets
thf(fact_450_b__non__zero,axiom,
( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
!= zero_zero_nat ) ).
% b_non_zero
thf(fact_451_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_452_blocks__nempty__alt,axiom,
! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( X4 != bot_bot_set_a ) ) ).
% blocks_nempty_alt
thf(fact_453_blocks__nempty,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( Bl2 != bot_bot_set_a ) ) ).
% blocks_nempty
thf(fact_454_del__point__block__count,axiom,
! [P3: a] :
( ( size_s6566526139600085008_set_a @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P3 ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% del_point_block_count
thf(fact_455_finite__blocks,axiom,
! [B2: set_a] :
( ( member_set_a @ B2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( finite_finite_a @ B2 ) ) ).
% finite_blocks
thf(fact_456_b__positive,axiom,
ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).
% b_positive
thf(fact_457_points__list__empty__iff,axiom,
( ( v_s = nil_a )
= ( ( set_a2 @ v_s )
= bot_bot_set_a ) ) ).
% points_list_empty_iff
thf(fact_458_del__block__b_I2_J,axiom,
! [Bl2: set_a] :
( ~ ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% del_block_b(2)
thf(fact_459_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_460_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_461_List_Ofinite__set,axiom,
! [Xs: list_vec_int] : ( finite8375754479472695874ec_int @ ( set_vec_int2 @ Xs ) ) ).
% List.finite_set
thf(fact_462_List_Ofinite__set,axiom,
! [Xs: list_a] : ( finite_finite_a @ ( set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_463_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_464_List_Ofinite__set,axiom,
! [Xs: list_set_a] : ( finite_finite_set_a @ ( set_set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_465_List_Ofinite__set,axiom,
! [Xs: list_set_nat] : ( finite1152437895449049373et_nat @ ( set_set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_466_carrier__matD_I1_J,axiom,
! [A: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( dim_row_int @ A )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_467_carrier__matD_I1_J,axiom,
! [A: mat_real,Nr: nat,Nc: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ( dim_row_real @ A )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_468_index__mult__mat_I2_J,axiom,
! [A: mat_int,B: mat_int] :
( ( dim_row_int @ ( times_times_mat_int @ A @ B ) )
= ( dim_row_int @ A ) ) ).
% index_mult_mat(2)
thf(fact_469_index__mult__mat_I2_J,axiom,
! [A: mat_real,B: mat_real] :
( ( dim_row_real @ ( times_times_mat_real @ A @ B ) )
= ( dim_row_real @ A ) ) ).
% index_mult_mat(2)
thf(fact_470_index__map__mat_I2_J,axiom,
! [F: int > real,A: mat_int] :
( ( dim_row_real @ ( map_mat_int_real @ F @ A ) )
= ( dim_row_int @ A ) ) ).
% index_map_mat(2)
thf(fact_471_index__map__mat_I2_J,axiom,
! [F: real > real,A: mat_real] :
( ( dim_row_real @ ( map_mat_real_real @ F @ A ) )
= ( dim_row_real @ A ) ) ).
% index_map_mat(2)
thf(fact_472_index__map__mat_I2_J,axiom,
! [F: real > int,A: mat_real] :
( ( dim_row_int @ ( map_mat_real_int @ F @ A ) )
= ( dim_row_real @ A ) ) ).
% index_map_mat(2)
thf(fact_473_index__map__mat_I2_J,axiom,
! [F: int > int,A: mat_int] :
( ( dim_row_int @ ( map_mat_int_int @ F @ A ) )
= ( dim_row_int @ A ) ) ).
% index_map_mat(2)
thf(fact_474_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_475_pow__mat__dim_I1_J,axiom,
! [A: mat_int,K: nat] :
( ( dim_row_int @ ( pow_mat_int @ A @ K ) )
= ( dim_row_int @ A ) ) ).
% pow_mat_dim(1)
thf(fact_476_lift__01__mat__simp_I1_J,axiom,
! [M: mat_int] :
( ( dim_row_int @ ( matrix323868623736973467nt_int @ M ) )
= ( dim_row_int @ M ) ) ).
% lift_01_mat_simp(1)
thf(fact_477_dim__row__take__cols,axiom,
! [A: mat_int,Ls: list_nat] :
( ( dim_row_int @ ( more_take_cols_int @ A @ Ls ) )
= ( dim_row_int @ A ) ) ).
% dim_row_take_cols
thf(fact_478_set__empty2,axiom,
! [Xs: list_set_a] :
( ( bot_bot_set_set_a
= ( set_set_a2 @ Xs ) )
= ( Xs = nil_set_a ) ) ).
% set_empty2
thf(fact_479_set__empty2,axiom,
! [Xs: list_vec_int] :
( ( bot_bot_set_vec_int
= ( set_vec_int2 @ Xs ) )
= ( Xs = nil_vec_int ) ) ).
% set_empty2
thf(fact_480_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_481_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_482_set__empty2,axiom,
! [Xs: list_int] :
( ( bot_bot_set_int
= ( set_int2 @ Xs ) )
= ( Xs = nil_int ) ) ).
% set_empty2
thf(fact_483_set__empty,axiom,
! [Xs: list_set_a] :
( ( ( set_set_a2 @ Xs )
= bot_bot_set_set_a )
= ( Xs = nil_set_a ) ) ).
% set_empty
thf(fact_484_set__empty,axiom,
! [Xs: list_vec_int] :
( ( ( set_vec_int2 @ Xs )
= bot_bot_set_vec_int )
= ( Xs = nil_vec_int ) ) ).
% set_empty
thf(fact_485_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_486_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_487_set__empty,axiom,
! [Xs: list_int] :
( ( ( set_int2 @ Xs )
= bot_bot_set_int )
= ( Xs = nil_int ) ) ).
% set_empty
thf(fact_488_length__0__conv,axiom,
! [Xs: list_set_nat] :
( ( ( size_s3254054031482475050et_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_set_nat ) ) ).
% length_0_conv
thf(fact_489_length__0__conv,axiom,
! [Xs: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_set_a ) ) ).
% length_0_conv
thf(fact_490_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_491_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_492_add__block__sub__des,axiom,
! [B2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( B2 != 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 ) @ B2 ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B2 ) ) ) ) ).
% add_block_sub_des
thf(fact_493_pow__mat__dim__square_I1_J,axiom,
! [A: mat_int,N: nat,K: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( dim_row_int @ ( pow_mat_int @ A @ K ) )
= N ) ) ).
% pow_mat_dim_square(1)
thf(fact_494_pow__mat__dim__square_I1_J,axiom,
! [A: mat_real,N: nat,K: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( dim_row_real @ ( pow_mat_real @ A @ K ) )
= N ) ) ).
% pow_mat_dim_square(1)
thf(fact_495_length__greater__0__conv,axiom,
! [Xs: list_set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) )
= ( Xs != nil_set_nat ) ) ).
% length_greater_0_conv
thf(fact_496_length__greater__0__conv,axiom,
! [Xs: list_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_set_a @ Xs ) )
= ( Xs != nil_set_a ) ) ).
% length_greater_0_conv
thf(fact_497_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_498_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_499_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_500_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_501_finite__sysI,axiom,
( ( finite_finite_a @ ( set_a2 @ v_s ) )
=> ( design9187838744727572296stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ).
% finite_sysI
thf(fact_502_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_503_neq__if__length__neq,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( ( size_s3254054031482475050et_nat @ Xs )
!= ( size_s3254054031482475050et_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_504_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_505_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_506_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_507_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_set_nat] :
( ( size_s3254054031482475050et_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_508_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_set_a] :
( ( size_size_list_set_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_509_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_510_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_511_inc__mat__dim__row,axiom,
! [Vs2: list_set_nat,Bs2: list_set_set_nat] :
( ( dim_row_int @ ( incide8356258661762153887at_int @ Vs2 @ Bs2 ) )
= ( size_s3254054031482475050et_nat @ Vs2 ) ) ).
% inc_mat_dim_row
thf(fact_512_inc__mat__dim__row,axiom,
! [Vs2: list_set_a,Bs2: list_set_set_a] :
( ( dim_row_int @ ( incide2330559700202178271_a_int @ Vs2 @ Bs2 ) )
= ( size_size_list_set_a @ Vs2 ) ) ).
% inc_mat_dim_row
thf(fact_513_inc__mat__dim__row,axiom,
! [Vs2: list_a,Bs2: list_set_a] :
( ( dim_row_int @ ( incide7016682120514301311_a_int @ Vs2 @ Bs2 ) )
= ( size_size_list_a @ Vs2 ) ) ).
% inc_mat_dim_row
thf(fact_514_inc__mat__dim__row,axiom,
! [Vs2: list_nat,Bs2: list_set_nat] :
( ( dim_row_int @ ( incide1177982898701834729at_int @ Vs2 @ Bs2 ) )
= ( size_size_list_nat @ Vs2 ) ) ).
% inc_mat_dim_row
thf(fact_515_ordered__incidence__system_Oblocks__list__length,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( size_size_list_set_a @ B_s )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ B_s ) ) ) ) ).
% ordered_incidence_system.blocks_list_length
thf(fact_516_ordered__incidence__system_Oblocks__list__length,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( size_s3254054031482475050et_nat @ B_s )
= ( size_s7462436076474991978et_nat @ ( mset_set_nat @ B_s ) ) ) ) ).
% ordered_incidence_system.blocks_list_length
thf(fact_517_length__induct,axiom,
! [P: list_set_nat > $o,Xs: list_set_nat] :
( ! [Xs2: list_set_nat] :
( ! [Ys2: list_set_nat] :
( ( ord_less_nat @ ( size_s3254054031482475050et_nat @ Ys2 ) @ ( size_s3254054031482475050et_nat @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_518_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_519_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_520_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_521_finite__list,axiom,
! [A: set_vec_int] :
( ( finite8375754479472695874ec_int @ A )
=> ? [Xs2: list_vec_int] :
( ( set_vec_int2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_522_finite__list,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ? [Xs2: list_a] :
( ( set_a2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_523_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_524_finite__list,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ? [Xs2: list_set_a] :
( ( set_set_a2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_525_finite__list,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ? [Xs2: list_set_nat] :
( ( set_set_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_526_list_Osize_I3_J,axiom,
( ( size_s3254054031482475050et_nat @ nil_set_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_527_list_Osize_I3_J,axiom,
( ( size_size_list_set_a @ nil_set_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_528_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_529_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_530_card__length,axiom,
! [Xs: list_vec_int] : ( ord_less_eq_nat @ ( finite_card_vec_int @ ( set_vec_int2 @ Xs ) ) @ ( size_s6547088122282965643ec_int @ Xs ) ) ).
% card_length
thf(fact_531_card__length,axiom,
! [Xs: list_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( set_set_nat2 @ Xs ) ) @ ( size_s3254054031482475050et_nat @ Xs ) ) ).
% card_length
thf(fact_532_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_533_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_534_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_535_finite__distinct__list,axiom,
! [A: set_vec_int] :
( ( finite8375754479472695874ec_int @ A )
=> ? [Xs2: list_vec_int] :
( ( ( set_vec_int2 @ Xs2 )
= A )
& ( distinct_vec_int @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_536_finite__distinct__list,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ? [Xs2: list_a] :
( ( ( set_a2 @ Xs2 )
= A )
& ( distinct_a @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_537_finite__distinct__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( ( set_nat2 @ Xs2 )
= A )
& ( distinct_nat @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_538_finite__distinct__list,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ? [Xs2: list_set_a] :
( ( ( set_set_a2 @ Xs2 )
= A )
& ( distinct_set_a @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_539_finite__distinct__list,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ? [Xs2: list_set_nat] :
( ( ( set_set_nat2 @ Xs2 )
= A )
& ( distinct_set_nat @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_540_empty__set,axiom,
( bot_bot_set_set_a
= ( set_set_a2 @ nil_set_a ) ) ).
% empty_set
thf(fact_541_empty__set,axiom,
( bot_bot_set_vec_int
= ( set_vec_int2 @ nil_vec_int ) ) ).
% empty_set
thf(fact_542_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_543_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_544_empty__set,axiom,
( bot_bot_set_int
= ( set_int2 @ nil_int ) ) ).
% empty_set
thf(fact_545_similar__mat__witD_I7_J,axiom,
! [N: nat,A: mat_int,B: mat_int,P: mat_int,Q: mat_int] :
( ( N
= ( dim_row_int @ A ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( member_mat_int @ Q @ ( carrier_mat_int @ N @ N ) ) ) ) ).
% similar_mat_witD(7)
thf(fact_546_similar__mat__witD_I7_J,axiom,
! [N: nat,A: mat_real,B: mat_real,P: mat_real,Q: mat_real] :
( ( N
= ( dim_row_real @ A ) )
=> ( ( similar_mat_wit_real @ A @ B @ P @ Q )
=> ( member_mat_real @ Q @ ( carrier_mat_real @ N @ N ) ) ) ) ).
% similar_mat_witD(7)
thf(fact_547_similar__mat__witD_I6_J,axiom,
! [N: nat,A: mat_int,B: mat_int,P: mat_int,Q: mat_int] :
( ( N
= ( dim_row_int @ A ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( member_mat_int @ P @ ( carrier_mat_int @ N @ N ) ) ) ) ).
% similar_mat_witD(6)
thf(fact_548_similar__mat__witD_I6_J,axiom,
! [N: nat,A: mat_real,B: mat_real,P: mat_real,Q: mat_real] :
( ( N
= ( dim_row_real @ A ) )
=> ( ( similar_mat_wit_real @ A @ B @ P @ Q )
=> ( member_mat_real @ P @ ( carrier_mat_real @ N @ N ) ) ) ) ).
% similar_mat_witD(6)
thf(fact_549_similar__mat__witD_I5_J,axiom,
! [N: nat,A: mat_int,B: mat_int,P: mat_int,Q: mat_int] :
( ( N
= ( dim_row_int @ A ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( member_mat_int @ B @ ( carrier_mat_int @ N @ N ) ) ) ) ).
% similar_mat_witD(5)
thf(fact_550_similar__mat__witD_I5_J,axiom,
! [N: nat,A: mat_real,B: mat_real,P: mat_real,Q: mat_real] :
( ( N
= ( dim_row_real @ A ) )
=> ( ( similar_mat_wit_real @ A @ B @ P @ Q )
=> ( member_mat_real @ B @ ( carrier_mat_real @ N @ N ) ) ) ) ).
% similar_mat_witD(5)
thf(fact_551_similar__mat__witD_I4_J,axiom,
! [N: nat,A: mat_int,B: mat_int,P: mat_int,Q: mat_int] :
( ( N
= ( dim_row_int @ A ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) ) ) ) ).
% similar_mat_witD(4)
thf(fact_552_similar__mat__witD_I4_J,axiom,
! [N: nat,A: mat_real,B: mat_real,P: mat_real,Q: mat_real] :
( ( N
= ( dim_row_real @ A ) )
=> ( ( similar_mat_wit_real @ A @ B @ P @ Q )
=> ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) ) ) ) ).
% similar_mat_witD(4)
thf(fact_553_similar__mat__witD_I3_J,axiom,
! [N: nat,A: mat_int,B: mat_int,P: mat_int,Q: mat_int] :
( ( N
= ( dim_row_int @ A ) )
=> ( ( similar_mat_wit_int @ A @ B @ P @ Q )
=> ( A
= ( times_times_mat_int @ ( times_times_mat_int @ P @ B ) @ Q ) ) ) ) ).
% similar_mat_witD(3)
thf(fact_554_similar__mat__witD_I3_J,axiom,
! [N: nat,A: mat_real,B: mat_real,P: mat_real,Q: mat_real] :
( ( N
= ( dim_row_real @ A ) )
=> ( ( similar_mat_wit_real @ A @ B @ P @ Q )
=> ( A
= ( times_times_mat_real @ ( times_times_mat_real @ P @ B ) @ Q ) ) ) ) ).
% similar_mat_witD(3)
thf(fact_555_length__pos__if__in__set,axiom,
! [X: mat_int,Xs: list_mat_int] :
( ( member_mat_int @ X @ ( set_mat_int2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6799427505661098007at_int @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_556_length__pos__if__in__set,axiom,
! [X: mat_real,Xs: list_mat_real] :
( ( member_mat_real @ X @ ( set_mat_real2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s7383440822540753431t_real @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_557_length__pos__if__in__set,axiom,
! [X: vec_int,Xs: list_vec_int] :
( ( member_vec_int @ X @ ( set_vec_int2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6547088122282965643ec_int @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_558_length__pos__if__in__set,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_559_length__pos__if__in__set,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_set_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_560_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_561_length__pos__if__in__set,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_562_distinct__card,axiom,
! [Xs: list_vec_int] :
( ( distinct_vec_int @ Xs )
=> ( ( finite_card_vec_int @ ( set_vec_int2 @ Xs ) )
= ( size_s6547088122282965643ec_int @ Xs ) ) ) ).
% distinct_card
thf(fact_563_distinct__card,axiom,
! [Xs: list_set_nat] :
( ( distinct_set_nat @ Xs )
=> ( ( finite_card_set_nat @ ( set_set_nat2 @ Xs ) )
= ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_564_distinct__card,axiom,
! [Xs: list_set_a] :
( ( distinct_set_a @ Xs )
=> ( ( finite_card_set_a @ ( set_set_a2 @ Xs ) )
= ( size_size_list_set_a @ Xs ) ) ) ).
% distinct_card
thf(fact_565_distinct__card,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_566_distinct__card,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( ( finite_card_a @ ( set_a2 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ) ).
% distinct_card
thf(fact_567_card__distinct,axiom,
! [Xs: list_vec_int] :
( ( ( finite_card_vec_int @ ( set_vec_int2 @ Xs ) )
= ( size_s6547088122282965643ec_int @ Xs ) )
=> ( distinct_vec_int @ Xs ) ) ).
% card_distinct
thf(fact_568_card__distinct,axiom,
! [Xs: list_set_nat] :
( ( ( finite_card_set_nat @ ( set_set_nat2 @ Xs ) )
= ( size_s3254054031482475050et_nat @ Xs ) )
=> ( distinct_set_nat @ Xs ) ) ).
% card_distinct
thf(fact_569_card__distinct,axiom,
! [Xs: list_set_a] :
( ( ( finite_card_set_a @ ( set_set_a2 @ Xs ) )
= ( size_size_list_set_a @ Xs ) )
=> ( distinct_set_a @ Xs ) ) ).
% card_distinct
thf(fact_570_card__distinct,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) )
=> ( distinct_nat @ Xs ) ) ).
% card_distinct
thf(fact_571_card__distinct,axiom,
! [Xs: list_a] :
( ( ( finite_card_a @ ( set_a2 @ Xs ) )
= ( size_size_list_a @ Xs ) )
=> ( distinct_a @ Xs ) ) ).
% card_distinct
thf(fact_572_ordered__incidence__system_Opoints__list__length,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s @ B_s )
=> ( ( size_s6547088122282965643ec_int @ V_s )
= ( finite_card_vec_int @ ( set_vec_int2 @ V_s ) ) ) ) ).
% ordered_incidence_system.points_list_length
thf(fact_573_ordered__incidence__system_Opoints__list__length,axiom,
! [V_s: list_set_nat,B_s: list_set_set_nat] :
( ( incide937065307065650815et_nat @ V_s @ B_s )
=> ( ( size_s3254054031482475050et_nat @ V_s )
= ( finite_card_set_nat @ ( set_set_nat2 @ V_s ) ) ) ) ).
% ordered_incidence_system.points_list_length
thf(fact_574_ordered__incidence__system_Opoints__list__length,axiom,
! [V_s: list_set_a,B_s: list_set_set_a] :
( ( incide2166342132139297189_set_a @ V_s @ B_s )
=> ( ( size_size_list_set_a @ V_s )
= ( finite_card_set_a @ ( set_set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.points_list_length
thf(fact_575_ordered__incidence__system_Opoints__list__length,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( size_size_list_a @ V_s )
= ( finite_card_a @ ( set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.points_list_length
thf(fact_576_ordered__incidence__system_Opoints__list__length,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( size_size_list_nat @ V_s )
= ( finite_card_nat @ ( set_nat2 @ V_s ) ) ) ) ).
% ordered_incidence_system.points_list_length
thf(fact_577_ordered__incidence__system_Opoints__list__empty__iff,axiom,
! [V_s: list_set_a,B_s: list_set_set_a] :
( ( incide2166342132139297189_set_a @ V_s @ B_s )
=> ( ( V_s = nil_set_a )
= ( ( set_set_a2 @ V_s )
= bot_bot_set_set_a ) ) ) ).
% ordered_incidence_system.points_list_empty_iff
thf(fact_578_ordered__incidence__system_Opoints__list__empty__iff,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s @ B_s )
=> ( ( V_s = nil_vec_int )
= ( ( set_vec_int2 @ V_s )
= bot_bot_set_vec_int ) ) ) ).
% ordered_incidence_system.points_list_empty_iff
thf(fact_579_ordered__incidence__system_Opoints__list__empty__iff,axiom,
! [V_s: list_int,B_s: list_set_int] :
( ( incide6996049454332333349em_int @ V_s @ B_s )
=> ( ( V_s = nil_int )
= ( ( set_int2 @ V_s )
= bot_bot_set_int ) ) ) ).
% ordered_incidence_system.points_list_empty_iff
thf(fact_580_ordered__incidence__system_Opoints__list__empty__iff,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( V_s = nil_a )
= ( ( set_a2 @ V_s )
= bot_bot_set_a ) ) ) ).
% ordered_incidence_system.points_list_empty_iff
thf(fact_581_ordered__incidence__system_Opoints__list__empty__iff,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( V_s = nil_nat )
= ( ( set_nat2 @ V_s )
= bot_bot_set_nat ) ) ) ).
% ordered_incidence_system.points_list_empty_iff
thf(fact_582_ordered__incidence__system_Odim__row__is__v,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s @ B_s )
=> ( ( dim_row_int @ ( incide2612794423365091098nt_int @ V_s @ B_s ) )
= ( finite_card_vec_int @ ( set_vec_int2 @ V_s ) ) ) ) ).
% ordered_incidence_system.dim_row_is_v
thf(fact_583_ordered__incidence__system_Odim__row__is__v,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dim_row_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) )
= ( finite_card_a @ ( set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.dim_row_is_v
thf(fact_584_ordered__incidence__system_Odim__row__is__v,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dim_row_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) )
= ( finite_card_nat @ ( set_nat2 @ V_s ) ) ) ) ).
% ordered_incidence_system.dim_row_is_v
thf(fact_585_ordered__incidence__system_Otranspose__N__mult__dim_I1_J,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s @ B_s )
=> ( ( dim_row_int @ ( times_times_mat_int @ ( incide2612794423365091098nt_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide2612794423365091098nt_int @ V_s @ B_s ) ) ) )
= ( finite_card_vec_int @ ( set_vec_int2 @ V_s ) ) ) ) ).
% ordered_incidence_system.transpose_N_mult_dim(1)
thf(fact_586_ordered__incidence__system_Otranspose__N__mult__dim_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dim_row_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) )
= ( finite_card_a @ ( set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.transpose_N_mult_dim(1)
thf(fact_587_ordered__incidence__system_Otranspose__N__mult__dim_I1_J,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dim_row_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) )
= ( finite_card_nat @ ( set_nat2 @ V_s ) ) ) ) ).
% ordered_incidence_system.transpose_N_mult_dim(1)
thf(fact_588_ordered__incidence__system_ON__carrier__mat,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s @ B_s )
=> ( member_mat_int @ ( incide2612794423365091098nt_int @ V_s @ B_s ) @ ( carrier_mat_int @ ( finite_card_vec_int @ ( set_vec_int2 @ V_s ) ) @ ( size_s3706287384392134315ec_int @ ( mset_set_vec_int @ B_s ) ) ) ) ) ).
% ordered_incidence_system.N_carrier_mat
thf(fact_589_ordered__incidence__system_ON__carrier__mat,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( 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 ) ) ) ) ) ).
% ordered_incidence_system.N_carrier_mat
thf(fact_590_ordered__incidence__system_ON__carrier__mat,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( member_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( carrier_mat_int @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) @ ( size_s7462436076474991978et_nat @ ( mset_set_nat @ B_s ) ) ) ) ) ).
% ordered_incidence_system.N_carrier_mat
thf(fact_591_ordered__incidence__system_ON__carrier__mat__01__lift,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s @ B_s )
=> ( member_mat_real @ ( matrix1811533390045330843t_real @ ( incide2612794423365091098nt_int @ V_s @ B_s ) ) @ ( carrier_mat_real @ ( finite_card_vec_int @ ( set_vec_int2 @ V_s ) ) @ ( size_s3706287384392134315ec_int @ ( mset_set_vec_int @ B_s ) ) ) ) ) ).
% ordered_incidence_system.N_carrier_mat_01_lift
thf(fact_592_ordered__incidence__system_ON__carrier__mat__01__lift,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s @ B_s )
=> ( member_mat_int @ ( matrix323868623736973467nt_int @ ( incide2612794423365091098nt_int @ V_s @ B_s ) ) @ ( carrier_mat_int @ ( finite_card_vec_int @ ( set_vec_int2 @ V_s ) ) @ ( size_s3706287384392134315ec_int @ ( mset_set_vec_int @ B_s ) ) ) ) ) ).
% ordered_incidence_system.N_carrier_mat_01_lift
thf(fact_593_ordered__incidence__system_ON__carrier__mat__01__lift,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( member_mat_real @ ( matrix1811533390045330843t_real @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) @ ( carrier_mat_real @ ( finite_card_a @ ( set_a2 @ V_s ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ B_s ) ) ) ) ) ).
% ordered_incidence_system.N_carrier_mat_01_lift
thf(fact_594_ordered__incidence__system_ON__carrier__mat__01__lift,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( 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 ) ) ) ) ) ).
% ordered_incidence_system.N_carrier_mat_01_lift
thf(fact_595_ordered__incidence__system_ON__carrier__mat__01__lift,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( member_mat_real @ ( matrix1811533390045330843t_real @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) @ ( carrier_mat_real @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) @ ( size_s7462436076474991978et_nat @ ( mset_set_nat @ B_s ) ) ) ) ) ).
% ordered_incidence_system.N_carrier_mat_01_lift
thf(fact_596_ordered__incidence__system_ON__carrier__mat__01__lift,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( member_mat_int @ ( matrix323868623736973467nt_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) @ ( carrier_mat_int @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) @ ( size_s7462436076474991978et_nat @ ( mset_set_nat @ B_s ) ) ) ) ) ).
% ordered_incidence_system.N_carrier_mat_01_lift
thf(fact_597_del__block__proper,axiom,
! [Bl2: 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 ) @ Bl2 ) ) ) ).
% del_block_proper
thf(fact_598_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_599_add__block__design,axiom,
! [Bl2: set_a] :
( ( finite_finite_a @ Bl2 )
=> ( ( Bl2 != bot_bot_set_a )
=> ( design_design_a @ ( sup_sup_set_a @ ( set_a2 @ v_s ) @ Bl2 ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) ) ) ) ).
% add_block_design
thf(fact_600_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_601_wf__design__iff,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( 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 @ Bl2 @ ( set_a2 @ v_s ) )
& ( finite_finite_a @ ( set_a2 @ v_s ) )
& ( Bl2 != bot_bot_set_a ) ) ) ) ).
% wf_design_iff
thf(fact_602_complement__design,axiom,
( ! [Bl: set_a] :
( ( 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 ) ) ) ) )
=> ( design_design_a @ ( set_a2 @ v_s ) @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ).
% complement_design
thf(fact_603_points__list__length,axiom,
( ( size_size_list_a @ v_s )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% points_list_length
thf(fact_604_finite__block__sizes,axiom,
finite_finite_nat @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ).
% finite_block_sizes
thf(fact_605_finite__design__support,axiom,
finite_finite_set_a @ ( design5397942185814921632port_a @ ( mset_set_a @ b_s ) ) ).
% finite_design_support
thf(fact_606_t__design__min__v,axiom,
ord_less_nat @ one_one_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) ).
% t_design_min_v
thf(fact_607_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_608_wf__design,axiom,
design_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).
% wf_design
thf(fact_609_replication__numbers__finite,axiom,
finite_finite_nat @ ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% replication_numbers_finite
thf(fact_610_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_611_delete__block__design,axiom,
! [Bl2: set_a] : ( design_design_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) ) ).
% delete_block_design
thf(fact_612_add__point__design,axiom,
! [P3: a] : ( design_design_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( mset_set_a @ b_s ) ) ).
% add_point_design
thf(fact_613_ideal_Oscale__one,axiom,
! [X: int] :
( ( times_times_int @ one_one_int @ X )
= X ) ).
% ideal.scale_one
thf(fact_614_ideal_Oscale__one,axiom,
! [X: real] :
( ( times_times_real @ one_one_real @ X )
= X ) ).
% ideal.scale_one
thf(fact_615_mult_Oright__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.right_neutral
thf(fact_616_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_617_mult_Oright__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.right_neutral
thf(fact_618_mult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% mult_1
thf(fact_619_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_620_mult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% mult_1
thf(fact_621_of__int__hom_Ohom__one,axiom,
( ( ring_1_of_int_real @ one_one_int )
= one_one_real ) ).
% of_int_hom.hom_one
thf(fact_622_of__int__hom_Ohom__one,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_hom.hom_one
thf(fact_623_of__int__hom_Ohom__1__iff,axiom,
! [X: int] :
( ( ( ring_1_of_int_real @ X )
= one_one_real )
= ( X = one_one_int ) ) ).
% of_int_hom.hom_1_iff
thf(fact_624_of__int__hom_Ohom__1__iff,axiom,
! [X: int] :
( ( ( ring_1_of_int_int @ X )
= one_one_int )
= ( X = one_one_int ) ) ).
% of_int_hom.hom_1_iff
thf(fact_625_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= one_one_real )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_626_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= one_one_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_627_carrier__matD_I2_J,axiom,
! [A: mat_int,Nr: nat,Nc: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( dim_col_int @ A )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_628_carrier__matD_I2_J,axiom,
! [A: mat_real,Nr: nat,Nc: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ( dim_col_real @ A )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_629_index__mult__mat_I3_J,axiom,
! [A: mat_int,B: mat_int] :
( ( dim_col_int @ ( times_times_mat_int @ A @ B ) )
= ( dim_col_int @ B ) ) ).
% index_mult_mat(3)
thf(fact_630_index__mult__mat_I3_J,axiom,
! [A: mat_real,B: mat_real] :
( ( dim_col_real @ ( times_times_mat_real @ A @ B ) )
= ( dim_col_real @ B ) ) ).
% index_mult_mat(3)
thf(fact_631_real__one,axiom,
( ( ring_H3062062880446257660of_int @ one_one_int )
= one_one_real ) ).
% real_one
thf(fact_632_cols__length,axiom,
! [A: mat_int] :
( ( size_s6547088122282965643ec_int @ ( cols_int @ A ) )
= ( dim_col_int @ A ) ) ).
% cols_length
thf(fact_633_index__map__mat_I3_J,axiom,
! [F: int > real,A: mat_int] :
( ( dim_col_real @ ( map_mat_int_real @ F @ A ) )
= ( dim_col_int @ A ) ) ).
% index_map_mat(3)
thf(fact_634_index__map__mat_I3_J,axiom,
! [F: real > real,A: mat_real] :
( ( dim_col_real @ ( map_mat_real_real @ F @ A ) )
= ( dim_col_real @ A ) ) ).
% index_map_mat(3)
thf(fact_635_index__map__mat_I3_J,axiom,
! [F: real > int,A: mat_real] :
( ( dim_col_int @ ( map_mat_real_int @ F @ A ) )
= ( dim_col_real @ A ) ) ).
% index_map_mat(3)
thf(fact_636_index__map__mat_I3_J,axiom,
! [F: int > int,A: mat_int] :
( ( dim_col_int @ ( map_mat_int_int @ F @ A ) )
= ( dim_col_int @ A ) ) ).
% index_map_mat(3)
thf(fact_637_strong__del__point__design,axiom,
! [P3: a] : ( design_design_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P3 ) ) ).
% strong_del_point_design
thf(fact_638_length__rows,axiom,
! [A: mat_int] :
( ( size_s6547088122282965643ec_int @ ( rows_int @ A ) )
= ( dim_row_int @ A ) ) ).
% length_rows
thf(fact_639_lift__01__mat__simp_I2_J,axiom,
! [M: mat_int] :
( ( dim_col_int @ ( matrix323868623736973467nt_int @ M ) )
= ( dim_col_int @ M ) ) ).
% lift_01_mat_simp(2)
thf(fact_640_add__block__design__cond,axiom,
! [Bl2: set_a] :
( ( ord_less_eq_set_a @ Bl2 @ ( set_a2 @ v_s ) )
=> ( ( Bl2 != bot_bot_set_a )
=> ( design_design_a @ ( set_a2 @ v_s ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) ) ) ) ).
% add_block_design_cond
thf(fact_641_carrier__matI,axiom,
! [A: mat_int,Nr: nat,Nc: nat] :
( ( ( dim_row_int @ A )
= Nr )
=> ( ( ( dim_col_int @ A )
= Nc )
=> ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_642_carrier__matI,axiom,
! [A: mat_real,Nr: nat,Nc: nat] :
( ( ( dim_row_real @ A )
= Nr )
=> ( ( ( dim_col_real @ A )
= Nc )
=> ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_643_index__transpose__mat_I3_J,axiom,
! [A: mat_int] :
( ( dim_col_int @ ( transpose_mat_int @ A ) )
= ( dim_row_int @ A ) ) ).
% index_transpose_mat(3)
thf(fact_644_index__transpose__mat_I3_J,axiom,
! [A: mat_real] :
( ( dim_col_real @ ( transpose_mat_real @ A ) )
= ( dim_row_real @ A ) ) ).
% index_transpose_mat(3)
thf(fact_645_index__transpose__mat_I2_J,axiom,
! [A: mat_int] :
( ( dim_row_int @ ( transpose_mat_int @ A ) )
= ( dim_col_int @ A ) ) ).
% index_transpose_mat(2)
thf(fact_646_index__transpose__mat_I2_J,axiom,
! [A: mat_real] :
( ( dim_row_real @ ( transpose_mat_real @ A ) )
= ( dim_col_real @ A ) ) ).
% index_transpose_mat(2)
thf(fact_647_pow__mat__dim__square_I2_J,axiom,
! [A: mat_int,N: nat,K: nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ N @ N ) )
=> ( ( dim_col_int @ ( pow_mat_int @ A @ K ) )
= N ) ) ).
% pow_mat_dim_square(2)
thf(fact_648_pow__mat__dim__square_I2_J,axiom,
! [A: mat_real,N: nat,K: nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( dim_col_real @ ( pow_mat_real @ A @ K ) )
= N ) ) ).
% pow_mat_dim_square(2)
thf(fact_649_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_650_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_651_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_652_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_653_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_654_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_655_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_656_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_657_pow__mat__dim_I2_J,axiom,
! [K: nat,A: mat_int] :
( ( ( K = zero_zero_nat )
=> ( ( dim_col_int @ ( pow_mat_int @ A @ K ) )
= ( dim_row_int @ A ) ) )
& ( ( K != zero_zero_nat )
=> ( ( dim_col_int @ ( pow_mat_int @ A @ K ) )
= ( dim_col_int @ A ) ) ) ) ).
% pow_mat_dim(2)
thf(fact_658_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_659_sub__designII,axiom,
! [U: set_a,A4: multiset_set_a] :
( ( design_design_a @ U @ A4 )
=> ( ( sub_su7923802003039619913stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ U @ A4 )
=> ( sub_sub_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ U @ A4 ) ) ) ).
% sub_designII
thf(fact_660_sub__designI,axiom,
! [U: set_a,A4: multiset_set_a] :
( ( design_design_a @ U @ A4 )
=> ( ( sub_su7923802003039619913stem_a @ U @ A4 @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) )
=> ( sub_sub_design_a @ U @ A4 @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ).
% sub_designI
thf(fact_661_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_662_of__int__hom_Ohom__1,axiom,
! [X: int] :
( ( ( ring_1_of_int_real @ X )
= one_one_real )
=> ( X = one_one_int ) ) ).
% of_int_hom.hom_1
thf(fact_663_of__int__hom_Ohom__1,axiom,
! [X: int] :
( ( ( ring_1_of_int_int @ X )
= one_one_int )
=> ( X = one_one_int ) ) ).
% of_int_hom.hom_1
thf(fact_664_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_665_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_666_dim__col__take__cols,axiom,
! [Inds: list_nat,A: mat_int] :
( ! [J2: nat] :
( ( member_nat @ J2 @ ( set_nat2 @ Inds ) )
=> ( ord_less_nat @ J2 @ ( dim_col_int @ A ) ) )
=> ( ( dim_col_int @ ( more_take_cols_int @ A @ Inds ) )
= ( size_size_list_nat @ Inds ) ) ) ).
% dim_col_take_cols
thf(fact_667_comm__monoid__mult__class_Omult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_668_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_669_comm__monoid__mult__class_Omult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_670_mult_Ocomm__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.comm_neutral
thf(fact_671_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_672_mult_Ocomm__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.comm_neutral
thf(fact_673_rows__inj,axiom,
! [A: mat_int,B: mat_int] :
( ( ( dim_col_int @ A )
= ( dim_col_int @ B ) )
=> ( ( ( rows_int @ A )
= ( rows_int @ B ) )
=> ( A = B ) ) ) ).
% rows_inj
thf(fact_674_carrier__mat__triv,axiom,
! [M2: mat_int] : ( member_mat_int @ M2 @ ( carrier_mat_int @ ( dim_row_int @ M2 ) @ ( dim_col_int @ M2 ) ) ) ).
% carrier_mat_triv
thf(fact_675_carrier__mat__triv,axiom,
! [M2: mat_real] : ( member_mat_real @ M2 @ ( carrier_mat_real @ ( dim_row_real @ M2 ) @ ( dim_col_real @ M2 ) ) ) ).
% carrier_mat_triv
thf(fact_676_inc__mat__dim__col,axiom,
! [Vs2: list_a,Bs2: list_set_a] :
( ( dim_col_int @ ( incide7016682120514301311_a_int @ Vs2 @ Bs2 ) )
= ( size_size_list_set_a @ Bs2 ) ) ).
% inc_mat_dim_col
thf(fact_677_inc__mat__dim__col,axiom,
! [Vs2: list_nat,Bs2: list_set_nat] :
( ( dim_col_int @ ( incide1177982898701834729at_int @ Vs2 @ Bs2 ) )
= ( size_s3254054031482475050et_nat @ Bs2 ) ) ).
% inc_mat_dim_col
thf(fact_678_finite__maxlen,axiom,
! [M: set_list_set_nat] :
( ( finite1091814263879798189et_nat @ M )
=> ? [N3: nat] :
! [X4: list_set_nat] :
( ( member_list_set_nat @ X4 @ M )
=> ( ord_less_nat @ ( size_s3254054031482475050et_nat @ X4 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_679_finite__maxlen,axiom,
! [M: set_list_set_a] :
( ( finite1971793804006318733_set_a @ M )
=> ? [N3: nat] :
! [X4: list_set_a] :
( ( member_list_set_a @ X4 @ M )
=> ( ord_less_nat @ ( size_size_list_set_a @ X4 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_680_finite__maxlen,axiom,
! [M: set_list_nat] :
( ( finite8100373058378681591st_nat @ M )
=> ? [N3: nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ M )
=> ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_681_finite__maxlen,axiom,
! [M: set_list_a] :
( ( finite_finite_list_a @ M )
=> ? [N3: nat] :
! [X4: list_a] :
( ( member_list_a @ X4 @ M )
=> ( ord_less_nat @ ( size_size_list_a @ X4 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_682_ordered__simple__design_Osimp__blocks__length__card,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide8427317466731264060gn_nat @ V_s @ B_s )
=> ( ( size_s3254054031482475050et_nat @ B_s )
= ( finite_card_set_nat @ ( set_set_nat2 @ B_s ) ) ) ) ).
% ordered_simple_design.simp_blocks_length_card
thf(fact_683_ordered__simple__design_Osimp__blocks__length__card,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide371748008924627346sign_a @ V_s @ B_s )
=> ( ( size_size_list_set_a @ B_s )
= ( finite_card_set_a @ ( set_set_a2 @ B_s ) ) ) ) ).
% ordered_simple_design.simp_blocks_length_card
thf(fact_684_ordered__design_Oaxioms_I2_J,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide7599675121277616863ec_int @ V_s @ B_s )
=> ( design3193161975243185419ec_int @ ( set_vec_int2 @ V_s ) @ ( mset_set_vec_int @ B_s ) ) ) ).
% ordered_design.axioms(2)
thf(fact_685_ordered__design_Oaxioms_I2_J,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide8999572217031194378gn_nat @ V_s @ B_s )
=> ( design_design_nat @ ( set_nat2 @ V_s ) @ ( mset_set_nat @ B_s ) ) ) ).
% ordered_design.axioms(2)
thf(fact_686_ordered__design_Oaxioms_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide2848671379600480836sign_a @ V_s @ B_s )
=> ( design_design_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) ) ) ).
% ordered_design.axioms(2)
thf(fact_687_lift__01__mat__carrier,axiom,
! [M: mat_int] : ( member_mat_real @ ( matrix1811533390045330843t_real @ M ) @ ( carrier_mat_real @ ( dim_row_int @ M ) @ ( dim_col_int @ M ) ) ) ).
% lift_01_mat_carrier
thf(fact_688_lift__01__mat__carrier,axiom,
! [M: mat_int] : ( member_mat_int @ ( matrix323868623736973467nt_int @ M ) @ ( carrier_mat_int @ ( dim_row_int @ M ) @ ( dim_col_int @ M ) ) ) ).
% lift_01_mat_carrier
thf(fact_689_take__cols__carrier__mat__strict,axiom,
! [A: mat_int,Nr: nat,Nc: nat,Inds: list_nat] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( set_nat2 @ Inds ) )
=> ( ord_less_nat @ I2 @ Nc ) )
=> ( member_mat_int @ ( more_take_cols_int @ A @ Inds ) @ ( carrier_mat_int @ Nr @ ( size_size_list_nat @ Inds ) ) ) ) ) ).
% take_cols_carrier_mat_strict
thf(fact_690_take__cols__carrier__mat__strict,axiom,
! [A: mat_real,Nr: nat,Nc: nat,Inds: list_nat] :
( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( set_nat2 @ Inds ) )
=> ( ord_less_nat @ I2 @ Nc ) )
=> ( member_mat_real @ ( more_take_cols_real @ A @ Inds ) @ ( carrier_mat_real @ Nr @ ( size_size_list_nat @ Inds ) ) ) ) ) ).
% take_cols_carrier_mat_strict
thf(fact_691_ordered__design_Ointro,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s @ B_s )
=> ( ( design3193161975243185419ec_int @ ( set_vec_int2 @ V_s ) @ ( mset_set_vec_int @ B_s ) )
=> ( incide7599675121277616863ec_int @ V_s @ B_s ) ) ) ).
% ordered_design.intro
thf(fact_692_ordered__design_Ointro,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( design_design_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) )
=> ( incide2848671379600480836sign_a @ V_s @ B_s ) ) ) ).
% ordered_design.intro
thf(fact_693_ordered__design_Ointro,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( design_design_nat @ ( set_nat2 @ V_s ) @ ( mset_set_nat @ B_s ) )
=> ( incide8999572217031194378gn_nat @ V_s @ B_s ) ) ) ).
% ordered_design.intro
thf(fact_694_ordered__design__def,axiom,
( incide7599675121277616863ec_int
= ( ^ [V_s2: list_vec_int,B_s2: list_set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s2 @ B_s2 )
& ( design3193161975243185419ec_int @ ( set_vec_int2 @ V_s2 ) @ ( mset_set_vec_int @ B_s2 ) ) ) ) ) ).
% ordered_design_def
thf(fact_695_ordered__design__def,axiom,
( incide2848671379600480836sign_a
= ( ^ [V_s2: list_a,B_s2: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s2 @ B_s2 )
& ( design_design_a @ ( set_a2 @ V_s2 ) @ ( mset_set_a @ B_s2 ) ) ) ) ) ).
% ordered_design_def
thf(fact_696_ordered__design__def,axiom,
( incide8999572217031194378gn_nat
= ( ^ [V_s2: list_nat,B_s2: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s2 @ B_s2 )
& ( design_design_nat @ ( set_nat2 @ V_s2 ) @ ( mset_set_nat @ B_s2 ) ) ) ) ) ).
% ordered_design_def
thf(fact_697_ordered__incidence__system_Odim__col__is__b,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dim_col_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ B_s ) ) ) ) ).
% ordered_incidence_system.dim_col_is_b
thf(fact_698_ordered__incidence__system_Odim__col__is__b,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dim_col_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) )
= ( size_s7462436076474991978et_nat @ ( mset_set_nat @ B_s ) ) ) ) ).
% ordered_incidence_system.dim_col_is_b
thf(fact_699_proper__inc__mat__def,axiom,
( incide294466202882093137at_int
= ( ^ [M3: mat_int] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_row_int @ M3 ) )
& ( ord_less_nat @ zero_zero_nat @ ( dim_col_int @ M3 ) ) ) ) ) ).
% proper_inc_mat_def
thf(fact_700_ordered__incidence__system_Otranspose__N__mult__dim_I2_J,axiom,
! [V_s: list_vec_int,B_s: list_set_vec_int] :
( ( incide9132343510602215008ec_int @ V_s @ B_s )
=> ( ( dim_col_int @ ( times_times_mat_int @ ( incide2612794423365091098nt_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide2612794423365091098nt_int @ V_s @ B_s ) ) ) )
= ( finite_card_vec_int @ ( set_vec_int2 @ V_s ) ) ) ) ).
% ordered_incidence_system.transpose_N_mult_dim(2)
thf(fact_701_ordered__incidence__system_Otranspose__N__mult__dim_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( dim_col_int @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) ) )
= ( finite_card_a @ ( set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.transpose_N_mult_dim(2)
thf(fact_702_ordered__incidence__system_Otranspose__N__mult__dim_I2_J,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( dim_col_int @ ( times_times_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ ( transpose_mat_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) ) )
= ( finite_card_nat @ ( set_nat2 @ V_s ) ) ) ) ).
% ordered_incidence_system.transpose_N_mult_dim(2)
thf(fact_703_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_704_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_705_add__point__sys__rep__numbers,axiom,
! [P3: a] :
( ( design8835372594653258411bers_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P3 ) @ ( mset_set_a @ b_s ) )
= ( sup_sup_set_nat @ ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) @ ( insert_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ P3 ) @ bot_bot_set_nat ) ) ) ).
% add_point_sys_rep_numbers
thf(fact_706_mult__cancel__left1,axiom,
! [C2: int,B2: int] :
( ( C2
= ( times_times_int @ C2 @ B2 ) )
= ( ( C2 = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_707_mult__cancel__left1,axiom,
! [C2: real,B2: real] :
( ( C2
= ( times_times_real @ C2 @ B2 ) )
= ( ( C2 = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_708_mult__cancel__left2,axiom,
! [C2: int,A2: int] :
( ( ( times_times_int @ C2 @ A2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_709_mult__cancel__left2,axiom,
! [C2: real,A2: real] :
( ( ( times_times_real @ C2 @ A2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_710_mult__cancel__right1,axiom,
! [C2: int,B2: int] :
( ( C2
= ( times_times_int @ B2 @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_711_mult__cancel__right1,axiom,
! [C2: real,B2: real] :
( ( C2
= ( times_times_real @ B2 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_712_mult__cancel__right,axiom,
! [A2: int,C2: int,B2: int] :
( ( ( times_times_int @ A2 @ C2 )
= ( times_times_int @ B2 @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_713_mult__cancel__right,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ C2 )
= ( times_times_nat @ B2 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_714_mult__cancel__right,axiom,
! [A2: real,C2: real,B2: real] :
( ( ( times_times_real @ A2 @ C2 )
= ( times_times_real @ B2 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_715_mult__cancel__left,axiom,
! [C2: int,A2: int,B2: int] :
( ( ( times_times_int @ C2 @ A2 )
= ( times_times_int @ C2 @ B2 ) )
= ( ( C2 = zero_zero_int )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_716_mult__cancel__left,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ( times_times_nat @ C2 @ A2 )
= ( times_times_nat @ C2 @ B2 ) )
= ( ( C2 = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_717_mult__cancel__left,axiom,
! [C2: real,A2: real,B2: real] :
( ( ( times_times_real @ C2 @ A2 )
= ( times_times_real @ C2 @ B2 ) )
= ( ( C2 = zero_zero_real )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_718_mult__eq__0__iff,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_719_mult__eq__0__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_720_mult__eq__0__iff,axiom,
! [A2: real,B2: real] :
( ( ( times_times_real @ A2 @ B2 )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_721_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_722_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_723_mult__zero__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_724_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_725_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_726_mult__zero__left,axiom,
! [A2: real] :
( ( times_times_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_727_mult__cancel__right2,axiom,
! [A2: int,C2: int] :
( ( ( times_times_int @ A2 @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_728_mult__cancel__right2,axiom,
! [A2: real,C2: real] :
( ( ( times_times_real @ A2 @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_729_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_730_pos__zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_731_mult__not__zero,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
!= zero_zero_int )
=> ( ( A2 != zero_zero_int )
& ( B2 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_732_mult__not__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_733_mult__not__zero,axiom,
! [A2: real,B2: real] :
( ( ( times_times_real @ A2 @ B2 )
!= zero_zero_real )
=> ( ( A2 != zero_zero_real )
& ( B2 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_734_divisors__zero,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
= zero_zero_int )
=> ( ( A2 = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_735_divisors__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_736_divisors__zero,axiom,
! [A2: real,B2: real] :
( ( ( times_times_real @ A2 @ B2 )
= zero_zero_real )
=> ( ( A2 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_737_no__zero__divisors,axiom,
! [A2: int,B2: int] :
( ( A2 != zero_zero_int )
=> ( ( B2 != zero_zero_int )
=> ( ( times_times_int @ A2 @ B2 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_738_no__zero__divisors,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_739_no__zero__divisors,axiom,
! [A2: real,B2: real] :
( ( A2 != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( times_times_real @ A2 @ B2 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_740_mult__left__cancel,axiom,
! [C2: int,A2: int,B2: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A2 )
= ( times_times_int @ C2 @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_741_mult__left__cancel,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A2 )
= ( times_times_nat @ C2 @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_742_mult__left__cancel,axiom,
! [C2: real,A2: real,B2: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A2 )
= ( times_times_real @ C2 @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_743_mult__right__cancel,axiom,
! [C2: int,A2: int,B2: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A2 @ C2 )
= ( times_times_int @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_744_mult__right__cancel,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C2 )
= ( times_times_nat @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_745_mult__right__cancel,axiom,
! [C2: real,A2: real,B2: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ C2 )
= ( times_times_real @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_746_ordered__design_Oall__cols__non__empty,axiom,
! [V_s: list_nat,B_s: list_set_nat,J: nat] :
( ( incide8999572217031194378gn_nat @ V_s @ B_s )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) ) )
=> ( incide6851923868969248411ol_int @ ( incide1177982898701834729at_int @ V_s @ B_s ) @ J ) ) ) ).
% ordered_design.all_cols_non_empty
thf(fact_747_ordered__design_Oall__cols__non__empty,axiom,
! [V_s: list_a,B_s: list_set_a,J: nat] :
( ( incide2848671379600480836sign_a @ V_s @ B_s )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) ) )
=> ( incide6851923868969248411ol_int @ ( incide7016682120514301311_a_int @ V_s @ B_s ) @ J ) ) ) ).
% ordered_design.all_cols_non_empty
thf(fact_748_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_749_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_750_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_751_zero__le__mult__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_752_zero__le__mult__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_753_mult__nonneg__nonpos2,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B2 @ A2 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_754_mult__nonneg__nonpos2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_755_mult__nonneg__nonpos2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_756_mult__nonpos__nonneg,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_757_mult__nonpos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_758_mult__nonpos__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_759_mult__nonneg__nonpos,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_760_mult__nonneg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_761_mult__nonneg__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_762_mult__nonneg__nonneg,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_763_mult__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_764_mult__nonneg__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_765_split__mult__neg__le,axiom,
! [A2: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_766_split__mult__neg__le,axiom,
! [A2: nat,B2: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_767_split__mult__neg__le,axiom,
! [A2: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_768_mult__le__0__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_769_mult__le__0__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_770_mult__right__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_771_mult__right__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_772_mult__right__mono,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_773_mult__right__mono__neg,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_774_mult__right__mono__neg,axiom,
! [B2: int,A2: int,C2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_775_mult__left__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_776_mult__left__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_777_mult__left__mono,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_778_mult__nonpos__nonpos,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_779_mult__nonpos__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_780_mult__left__mono__neg,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_781_mult__left__mono__neg,axiom,
! [B2: int,A2: int,C2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_782_split__mult__pos__le,axiom,
! [A2: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_783_split__mult__pos__le,axiom,
! [A2: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_784_zero__le__square,axiom,
! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_785_zero__le__square,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_786_mult__mono_H,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_787_mult__mono_H,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_788_mult__mono_H,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_789_mult__mono,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_790_mult__mono,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_791_mult__mono,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_792_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_793_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_794_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_795_mult__less__cancel__right__disj,axiom,
! [A2: real,C2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A2 @ B2 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_796_mult__less__cancel__right__disj,axiom,
! [A2: int,C2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A2 @ B2 ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_797_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_798_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_799_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_800_mult__strict__right__mono__neg,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_801_mult__strict__right__mono__neg,axiom,
! [B2: int,A2: int,C2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_802_mult__less__cancel__left__disj,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A2 @ B2 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_803_mult__less__cancel__left__disj,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A2 @ B2 ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_804_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_805_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_806_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_807_mult__strict__left__mono__neg,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_808_mult__strict__left__mono__neg,axiom,
! [B2: int,A2: int,C2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_809_mult__less__cancel__left__pos,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_real @ A2 @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_810_mult__less__cancel__left__pos,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_811_mult__less__cancel__left__neg,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_real @ B2 @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_812_mult__less__cancel__left__neg,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
= ( ord_less_int @ B2 @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_813_zero__less__mult__pos2,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_814_zero__less__mult__pos2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_815_zero__less__mult__pos2,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_816_zero__less__mult__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_817_zero__less__mult__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_818_zero__less__mult__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_819_zero__less__mult__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_820_zero__less__mult__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_821_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B2 @ A2 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_822_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_823_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_824_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_825_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_826_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_827_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_828_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_829_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_830_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_831_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_832_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_833_mult__less__0__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_834_mult__less__0__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_835_not__square__less__zero,axiom,
! [A2: real] :
~ ( ord_less_real @ ( times_times_real @ A2 @ A2 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_836_not__square__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_837_mult__neg__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_838_mult__neg__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_839_less__1__mult,axiom,
! [M2: real,N: real] :
( ( ord_less_real @ one_one_real @ M2 )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_840_less__1__mult,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_841_less__1__mult,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ one_one_int @ M2 )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_842_mult__le__cancel__left,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A2 @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_843_mult__le__cancel__left,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A2 @ B2 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_844_mult__le__cancel__right,axiom,
! [A2: real,C2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A2 @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_845_mult__le__cancel__right,axiom,
! [A2: int,C2: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A2 @ B2 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_846_mult__left__less__imp__less,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_847_mult__left__less__imp__less,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_848_mult__left__less__imp__less,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_849_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_850_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_851_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_852_mult__less__cancel__left,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A2 @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_853_mult__less__cancel__left,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A2 @ B2 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_854_mult__right__less__imp__less,axiom,
! [A2: real,C2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_855_mult__right__less__imp__less,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_856_mult__right__less__imp__less,axiom,
! [A2: int,C2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_857_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_858_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_859_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_860_mult__less__cancel__right,axiom,
! [A2: real,C2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A2 @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_861_mult__less__cancel__right,axiom,
! [A2: int,C2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A2 @ B2 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_862_mult__le__cancel__left__neg,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_eq_real @ B2 @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_863_mult__le__cancel__left__neg,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
= ( ord_less_eq_int @ B2 @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_864_mult__le__cancel__left__pos,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_eq_real @ A2 @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_865_mult__le__cancel__left__pos,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_866_mult__left__le__imp__le,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_867_mult__left__le__imp__le,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_868_mult__left__le__imp__le,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_869_mult__right__le__imp__le,axiom,
! [A2: real,C2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_870_mult__right__le__imp__le,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_871_mult__right__le__imp__le,axiom,
! [A2: int,C2: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_872_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_873_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_874_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_875_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_876_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_877_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_878_mult__left__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_879_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_880_mult__right__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_881_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_882_mult__le__one,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ B2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_883_mult__le__one,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_884_mult__le__one,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_885_mult__left__le,axiom,
! [C2: real,A2: real] :
( ( ord_less_eq_real @ C2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_886_mult__left__le,axiom,
! [C2: nat,A2: nat] :
( ( ord_less_eq_nat @ C2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_887_mult__left__le,axiom,
! [C2: int,A2: int] :
( ( ord_less_eq_int @ C2 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_888_mult__le__cancel__left1,axiom,
! [C2: real,B2: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_889_mult__le__cancel__left1,axiom,
! [C2: int,B2: int] :
( ( ord_less_eq_int @ C2 @ ( times_times_int @ C2 @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ one_one_int @ B2 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_890_mult__le__cancel__left2,axiom,
! [C2: real,A2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A2 @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A2 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_891_mult__le__cancel__left2,axiom,
! [C2: int,A2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A2 @ one_one_int ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_892_mult__le__cancel__right1,axiom,
! [C2: real,B2: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_893_mult__le__cancel__right1,axiom,
! [C2: int,B2: int] :
( ( ord_less_eq_int @ C2 @ ( times_times_int @ B2 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ one_one_int @ B2 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_894_mult__le__cancel__right2,axiom,
! [A2: real,C2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A2 @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A2 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_895_mult__le__cancel__right2,axiom,
! [A2: int,C2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A2 @ one_one_int ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_896_mult__less__cancel__left1,axiom,
! [C2: real,B2: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_897_mult__less__cancel__left1,axiom,
! [C2: int,B2: int] :
( ( ord_less_int @ C2 @ ( times_times_int @ C2 @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ one_one_int @ B2 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_898_mult__less__cancel__left2,axiom,
! [C2: real,A2: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A2 @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_899_mult__less__cancel__left2,axiom,
! [C2: int,A2: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A2 @ one_one_int ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_900_mult__less__cancel__right1,axiom,
! [C2: real,B2: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_901_mult__less__cancel__right1,axiom,
! [C2: int,B2: int] :
( ( ord_less_int @ C2 @ ( times_times_int @ B2 @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ one_one_int @ B2 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_902_mult__less__cancel__right2,axiom,
! [A2: real,C2: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A2 @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_903_mult__less__cancel__right2,axiom,
! [A2: int,C2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A2 @ one_one_int ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_904_preserve__non__empty,axiom,
! [F: int > real,J: nat] :
( ( matrix8987650032624085902t_real @ F )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( incide6851923868969248411ol_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ J )
= ( incide8049862060206209947l_real @ ( map_mat_int_real @ F @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) @ J ) ) ) ) ).
% preserve_non_empty
thf(fact_905_preserve__non__empty,axiom,
! [F: int > int,J: nat] :
( ( matrix4192408864422109198nt_int @ F )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( incide6851923868969248411ol_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ J )
= ( incide6851923868969248411ol_int @ ( map_mat_int_int @ F @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) @ J ) ) ) ) ).
% preserve_non_empty
thf(fact_906_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_907_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_908_non__empty__col__map__conv,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 )
= ( ( nth_set_a @ b_s @ J )
!= bot_bot_set_a ) ) ) ).
% non_empty_col_map_conv
thf(fact_909_dual__blocks__nempty,axiom,
! [Bl2: set_nat] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ v_s ) )
=> ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X3 ) ) )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( Bl2 != bot_bot_set_nat ) ) ) ).
% dual_blocks_nempty
thf(fact_910_add__block__index__in,axiom,
! [Ps: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ Ps @ B2 )
=> ( ( design254580327166089565ndex_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B2 ) @ Ps )
= ( plus_plus_nat @ ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) @ one_one_nat ) ) ) ).
% add_block_index_in
thf(fact_911_add__point__def,axiom,
! [P3: a] :
( ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P3 )
= ( insert_a @ P3 @ ( set_a2 @ v_s ) ) ) ).
% add_point_def
thf(fact_912_add__left__cancel,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C2 ) )
= ( B2 = C2 ) ) ).
% add_left_cancel
thf(fact_913_add__left__cancel,axiom,
! [A2: int,B2: int,C2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C2 ) )
= ( B2 = C2 ) ) ).
% add_left_cancel
thf(fact_914_add__right__cancel,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C2 @ A2 ) )
= ( B2 = C2 ) ) ).
% add_right_cancel
thf(fact_915_add__right__cancel,axiom,
! [B2: int,A2: int,C2: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C2 @ A2 ) )
= ( B2 = C2 ) ) ).
% add_right_cancel
thf(fact_916_dual__sys__b__non__zero,axiom,
( ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
!= zero_zero_nat ) ).
% dual_sys_b_non_zero
thf(fact_917_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_918_dual__blocks__b,axiom,
( ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% dual_blocks_b
thf(fact_919_dual__sys_Oblock__set__nempty__imp__block__ex,axiom,
( ( ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s )
!= zero_z3157962936165190495et_nat )
=> ? [Bl: set_nat] : ( member_set_nat @ Bl @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).
% dual_sys.block_set_nempty_imp_block_ex
thf(fact_920_valid__blocks__index__obtains,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ~ ! [J2: nat] :
~ ( ( ( nth_set_a @ b_s @ J2 )
= Bl2 )
& ( ord_less_nat @ J2 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ).
% valid_blocks_index_obtains
thf(fact_921_valid__blocks__index__cons,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ? [J2: nat] :
( ( ( nth_set_a @ b_s @ J2 )
= Bl2 )
& ( ord_less_nat @ J2 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ).
% valid_blocks_index_cons
thf(fact_922_valid__blocks__index,axiom,
! [J: nat] :
( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_set_a @ ( nth_set_a @ b_s @ J ) @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% valid_blocks_index
thf(fact_923_add__block__rep__number__in,axiom,
! [X: a,B2: set_a] :
( ( member_a @ X @ B2 )
=> ( ( design6637022207325878697mber_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B2 ) @ X )
= ( plus_plus_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) @ one_one_nat ) ) ) ).
% add_block_rep_number_in
thf(fact_924_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_925_dual__sys_Ofinite__blocks,axiom,
! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( finite_finite_nat @ B2 ) ) ).
% dual_sys.finite_blocks
thf(fact_926_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_927_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_928_add_Oright__neutral,axiom,
! [A2: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ A2 @ zero_z5079479921072680283_set_a )
= A2 ) ).
% add.right_neutral
thf(fact_929_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_930_add_Oright__neutral,axiom,
! [A2: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ A2 @ zero_z3157962936165190495et_nat )
= A2 ) ).
% add.right_neutral
thf(fact_931_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_932_add__cancel__left__left,axiom,
! [B2: nat,A2: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_933_add__cancel__left__left,axiom,
! [B2: multiset_set_a,A2: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ B2 @ A2 )
= A2 )
= ( B2 = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_left_left
thf(fact_934_add__cancel__left__left,axiom,
! [B2: int,A2: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_935_add__cancel__left__left,axiom,
! [B2: multiset_set_nat,A2: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ B2 @ A2 )
= A2 )
= ( B2 = zero_z3157962936165190495et_nat ) ) ).
% add_cancel_left_left
thf(fact_936_add__cancel__left__right,axiom,
! [A2: nat,B2: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_937_add__cancel__left__right,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ A2 @ B2 )
= A2 )
= ( B2 = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_left_right
thf(fact_938_add__cancel__left__right,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_939_add__cancel__left__right,axiom,
! [A2: multiset_set_nat,B2: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ A2 @ B2 )
= A2 )
= ( B2 = zero_z3157962936165190495et_nat ) ) ).
% add_cancel_left_right
thf(fact_940_add__cancel__right__left,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_941_add__cancel__right__left,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( A2
= ( plus_p2331992037799027419_set_a @ B2 @ A2 ) )
= ( B2 = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_right_left
thf(fact_942_add__cancel__right__left,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ B2 @ A2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_943_add__cancel__right__left,axiom,
! [A2: multiset_set_nat,B2: multiset_set_nat] :
( ( A2
= ( plus_p8712254050562127327et_nat @ B2 @ A2 ) )
= ( B2 = zero_z3157962936165190495et_nat ) ) ).
% add_cancel_right_left
thf(fact_944_add__cancel__right__right,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_945_add__cancel__right__right,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( A2
= ( plus_p2331992037799027419_set_a @ A2 @ B2 ) )
= ( B2 = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_right_right
thf(fact_946_add__cancel__right__right,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ A2 @ B2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_947_add__cancel__right__right,axiom,
! [A2: multiset_set_nat,B2: multiset_set_nat] :
( ( A2
= ( plus_p8712254050562127327et_nat @ A2 @ B2 ) )
= ( B2 = zero_z3157962936165190495et_nat ) ) ).
% add_cancel_right_right
thf(fact_948_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_949_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_950_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_951_add__0,axiom,
! [A2: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ A2 )
= A2 ) ).
% add_0
thf(fact_952_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_953_add__0,axiom,
! [A2: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ zero_z3157962936165190495et_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_954_add__le__cancel__left,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_955_add__le__cancel__left,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_956_add__le__cancel__right,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_957_add__le__cancel__right,axiom,
! [A2: int,C2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_958_add__less__cancel__left,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_959_add__less__cancel__left,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_960_add__less__cancel__right,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_961_add__less__cancel__right,axiom,
! [A2: int,C2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_962_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_add
thf(fact_963_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_add
thf(fact_964_of__int__hom_Ohom__add,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) ) ) ).
% of_int_hom.hom_add
thf(fact_965_of__int__hom_Ohom__add,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) ) ) ).
% of_int_hom.hom_add
thf(fact_966_dual__blocks__size__is__rep__obtain,axiom,
! [Bl2: set_nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ v_s ) )
=> ( ( finite_card_nat @ Bl2 )
!= ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X3 ) ) ) ) ).
% dual_blocks_size_is_rep_obtain
thf(fact_967_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_968_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_969_le__add__same__cancel2,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ B2 @ A2 ) )
= ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_970_le__add__same__cancel2,axiom,
! [A2: multiset_set_nat,B2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ A2 @ ( plus_p8712254050562127327et_nat @ B2 @ A2 ) )
= ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_971_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_972_le__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_973_le__add__same__cancel1,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) )
= ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_974_le__add__same__cancel1,axiom,
! [A2: multiset_set_nat,B2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ A2 @ ( plus_p8712254050562127327et_nat @ A2 @ B2 ) )
= ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_975_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_976_le__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_977_add__le__same__cancel2,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) @ B2 )
= ( ord_le7905258569527593284_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ).
% add_le_same_cancel2
thf(fact_978_add__le__same__cancel2,axiom,
! [A2: multiset_set_nat,B2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ B2 ) @ B2 )
= ( ord_le4034546139768944438et_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ).
% add_le_same_cancel2
thf(fact_979_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_980_add__le__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_981_add__le__same__cancel1,axiom,
! [B2: multiset_set_a,A2: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ B2 @ A2 ) @ B2 )
= ( ord_le7905258569527593284_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ).
% add_le_same_cancel1
thf(fact_982_add__le__same__cancel1,axiom,
! [B2: multiset_set_nat,A2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ B2 @ A2 ) @ B2 )
= ( ord_le4034546139768944438et_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ).
% add_le_same_cancel1
thf(fact_983_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_984_add__le__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_985_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_986_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_987_less__add__same__cancel2,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ B2 @ A2 ) )
= ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_988_less__add__same__cancel2,axiom,
! [A2: multiset_set_nat,B2: multiset_set_nat] :
( ( ord_le6127582356673144618et_nat @ A2 @ ( plus_p8712254050562127327et_nat @ B2 @ A2 ) )
= ( ord_le6127582356673144618et_nat @ zero_z3157962936165190495et_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_989_less__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_990_less__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_991_less__add__same__cancel1,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) )
= ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_992_less__add__same__cancel1,axiom,
! [A2: multiset_set_nat,B2: multiset_set_nat] :
( ( ord_le6127582356673144618et_nat @ A2 @ ( plus_p8712254050562127327et_nat @ A2 @ B2 ) )
= ( ord_le6127582356673144618et_nat @ zero_z3157962936165190495et_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_993_less__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_994_less__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_995_add__less__same__cancel2,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) @ B2 )
= ( ord_le5765082015083327056_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ).
% add_less_same_cancel2
thf(fact_996_add__less__same__cancel2,axiom,
! [A2: multiset_set_nat,B2: multiset_set_nat] :
( ( ord_le6127582356673144618et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ B2 ) @ B2 )
= ( ord_le6127582356673144618et_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ).
% add_less_same_cancel2
thf(fact_997_add__less__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_998_add__less__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_999_add__less__same__cancel1,axiom,
! [B2: multiset_set_a,A2: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ B2 @ A2 ) @ B2 )
= ( ord_le5765082015083327056_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ).
% add_less_same_cancel1
thf(fact_1000_add__less__same__cancel1,axiom,
! [B2: multiset_set_nat,A2: multiset_set_nat] :
( ( ord_le6127582356673144618et_nat @ ( plus_p8712254050562127327et_nat @ B2 @ A2 ) @ B2 )
= ( ord_le6127582356673144618et_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ).
% add_less_same_cancel1
thf(fact_1001_add__less__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1002_add__less__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1003_dual__sys_Ofinite__design__support,axiom,
finite1152437895449049373et_nat @ ( design4862117536649126062rt_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).
% dual_sys.finite_design_support
thf(fact_1004_dual__sys_Odesign__support__def,axiom,
( ( design4862117536649126062rt_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
= ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).
% dual_sys.design_support_def
thf(fact_1005_ring__class_Oring__distribs_I2_J,axiom,
! [A2: int,B2: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1006_ring__class_Oring__distribs_I2_J,axiom,
! [A2: real,B2: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1007_ring__class_Oring__distribs_I1_J,axiom,
! [A2: int,B2: int,C2: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1008_ring__class_Oring__distribs_I1_J,axiom,
! [A2: real,B2: real,C2: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ B2 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1009_comm__semiring__class_Odistrib,axiom,
! [A2: int,B2: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1010_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1011_comm__semiring__class_Odistrib,axiom,
! [A2: real,B2: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1012_distrib__left,axiom,
! [A2: int,B2: int,C2: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ).
% distrib_left
thf(fact_1013_distrib__left,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C2 ) ) ) ).
% distrib_left
thf(fact_1014_distrib__left,axiom,
! [A2: real,B2: real,C2: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ B2 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) ) ).
% distrib_left
thf(fact_1015_distrib__right,axiom,
! [A2: int,B2: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ).
% distrib_right
thf(fact_1016_distrib__right,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% distrib_right
thf(fact_1017_distrib__right,axiom,
! [A2: real,B2: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ).
% distrib_right
thf(fact_1018_combine__common__factor,axiom,
! [A2: int,E: int,B2: int,C2: int] :
( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_1019_combine__common__factor,axiom,
! [A2: nat,E: nat,B2: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_1020_combine__common__factor,axiom,
! [A2: real,E: real,B2: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_1021_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_1022_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_1023_add_Ocomm__neutral,axiom,
! [A2: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ A2 @ zero_z5079479921072680283_set_a )
= A2 ) ).
% add.comm_neutral
thf(fact_1024_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_1025_add_Ocomm__neutral,axiom,
! [A2: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ A2 @ zero_z3157962936165190495et_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_1026_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_1027_comm__monoid__add__class_Oadd__0,axiom,
! [A2: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_1028_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_1029_comm__monoid__add__class_Oadd__0,axiom,
! [A2: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ zero_z3157962936165190495et_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_1030_add__le__imp__le__right,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1031_add__le__imp__le__right,axiom,
! [A2: int,C2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1032_add__le__imp__le__left,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1033_add__le__imp__le__left,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1034_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_1035_add__right__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add_right_mono
thf(fact_1036_add__right__mono,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) ) ) ).
% add_right_mono
thf(fact_1037_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C4: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C4 ) ) ) ).
% less_eqE
thf(fact_1038_add__left__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) ) ) ).
% add_left_mono
thf(fact_1039_add__left__mono,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) ) ) ).
% add_left_mono
thf(fact_1040_add__mono,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_1041_add__mono,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_1042_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_1043_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1044_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_1045_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1046_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_1047_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1048_add__less__imp__less__right,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_1049_add__less__imp__less__right,axiom,
! [A2: int,C2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_1050_add__less__imp__less__left,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_1051_add__less__imp__less__left,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_1052_add__strict__right__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1053_add__strict__right__mono,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1054_add__strict__left__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_1055_add__strict__left__mono,axiom,
! [A2: int,B2: int,C2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_1056_add__strict__mono,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1057_add__strict__mono,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1058_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_1059_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1060_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_1061_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1062_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_1063_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1064_mult__hom_Ohom__add,axiom,
! [C2: int,X: int,Y: int] :
( ( times_times_int @ C2 @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( times_times_int @ C2 @ X ) @ ( times_times_int @ C2 @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_1065_mult__hom_Ohom__add,axiom,
! [C2: nat,X: nat,Y: nat] :
( ( times_times_nat @ C2 @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ C2 @ X ) @ ( times_times_nat @ C2 @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_1066_mult__hom_Ohom__add,axiom,
! [C2: real,X: real,Y: real] :
( ( times_times_real @ C2 @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ C2 @ X ) @ ( times_times_real @ C2 @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_1067_ideal_Oscale__right__distrib,axiom,
! [A2: int,X: int,Y: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ X ) @ ( times_times_int @ A2 @ Y ) ) ) ).
% ideal.scale_right_distrib
thf(fact_1068_ideal_Oscale__right__distrib,axiom,
! [A2: real,X: real,Y: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ X ) @ ( times_times_real @ A2 @ Y ) ) ) ).
% ideal.scale_right_distrib
thf(fact_1069_ideal_Oscale__left__distrib,axiom,
! [A2: int,B2: int,X: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ X )
= ( plus_plus_int @ ( times_times_int @ A2 @ X ) @ ( times_times_int @ B2 @ X ) ) ) ).
% ideal.scale_left_distrib
thf(fact_1070_ideal_Oscale__left__distrib,axiom,
! [A2: real,B2: real,X: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ X )
= ( plus_plus_real @ ( times_times_real @ A2 @ X ) @ ( times_times_real @ B2 @ X ) ) ) ).
% ideal.scale_left_distrib
thf(fact_1071_More__Matrix_Oarith,axiom,
! [D: nat,A2: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ D @ A2 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ D ) @ C2 ) @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% More_Matrix.arith
thf(fact_1072_sum__mat__add,axiom,
! [A: mat_real,Nr: nat,Nc: nat,B: mat_real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ( member_mat_real @ B @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ( comm_m8678487124704766304t_real @ ( plus_plus_mat_real @ A @ B ) )
= ( plus_plus_real @ ( comm_m8678487124704766304t_real @ A ) @ ( comm_m8678487124704766304t_real @ B ) ) ) ) ) ).
% sum_mat_add
thf(fact_1073_sum__mat__add,axiom,
! [A: mat_nat,Nr: nat,Nc: nat,B: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ B @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( comm_m4056229327131402372at_nat @ ( plus_plus_mat_nat @ A @ B ) )
= ( plus_plus_nat @ ( comm_m4056229327131402372at_nat @ A ) @ ( comm_m4056229327131402372at_nat @ B ) ) ) ) ) ).
% sum_mat_add
thf(fact_1074_sum__mat__add,axiom,
! [A: mat_int,Nr: nat,Nc: nat,B: mat_int] :
( ( member_mat_int @ A @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( comm_m4053738856622352096at_int @ ( plus_plus_mat_int @ A @ B ) )
= ( plus_plus_int @ ( comm_m4053738856622352096at_int @ A ) @ ( comm_m4053738856622352096at_int @ B ) ) ) ) ) ).
% sum_mat_add
thf(fact_1075_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1076_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B2: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1077_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_1078_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1079_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_1080_group__cancel_Oadd1,axiom,
! [A: int,K: int,A2: int,B2: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_1081_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_1082_group__cancel_Oadd2,axiom,
! [B: int,K: int,B2: int,A2: int] :
( ( B
= ( plus_plus_int @ K @ B2 ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_1083_add_Oassoc,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add.assoc
thf(fact_1084_add_Oassoc,axiom,
! [A2: int,B2: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) ) ) ).
% add.assoc
thf(fact_1085_add_Oleft__cancel,axiom,
! [A2: int,B2: int,C2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C2 ) )
= ( B2 = C2 ) ) ).
% add.left_cancel
thf(fact_1086_add_Oright__cancel,axiom,
! [B2: int,A2: int,C2: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C2 @ A2 ) )
= ( B2 = C2 ) ) ).
% add.right_cancel
thf(fact_1087_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_1088_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_1089_class__semiring_Oadd_Ofactors__equal,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( A2 = B2 )
=> ( ( C2 = D )
=> ( ( plus_plus_nat @ A2 @ C2 )
= ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_1090_class__semiring_Oadd_Ofactors__equal,axiom,
! [A2: int,B2: int,C2: int,D: int] :
( ( A2 = B2 )
=> ( ( C2 = D )
=> ( ( plus_plus_int @ A2 @ C2 )
= ( plus_plus_int @ B2 @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_1091_add_Oleft__commute,axiom,
! [B2: int,A2: int,C2: int] :
( ( plus_plus_int @ B2 @ ( plus_plus_int @ A2 @ C2 ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) ) ) ).
% add.left_commute
thf(fact_1092_str__del__block__del__point,axiom,
! [X: a] :
( ~ ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design4241783006516448631lock_a @ ( mset_set_a @ b_s ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ X ) ) ) ).
% str_del_block_del_point
thf(fact_1093_dual__sys_Oadd__point__existing__blocks,axiom,
! [Bs4: set_set_nat,P3: nat] :
( ! [Bl: set_nat] :
( ( member_set_nat @ Bl @ Bs4 )
=> ( member_nat @ P3 @ Bl ) )
=> ( ( design5698312687278145166ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 @ Bs4 )
= ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).
% dual_sys.add_point_existing_blocks
thf(fact_1094_mat__block__size__N__col,axiom,
! [J: nat] :
( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( incide7086334917823153261ze_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ J )
= ( finite_card_a @ ( nth_set_a @ b_s @ J ) ) ) ) ).
% mat_block_size_N_col
thf(fact_1095_mat__block__size__conv,axiom,
! [J: nat] :
( ( ord_less_nat @ J @ ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( finite_card_a @ ( nth_set_a @ b_s @ J ) )
= ( incide7086334917823153261ze_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ J ) ) ) ).
% mat_block_size_conv
thf(fact_1096_valid__points__index,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( member_a @ ( nth_a @ v_s @ I ) @ ( set_a2 @ v_s ) ) ) ).
% valid_points_index
thf(fact_1097_valid__points__index__cons,axiom,
! [X: a] :
( ( member_a @ X @ ( set_a2 @ v_s ) )
=> ? [I2: nat] :
( ( ( nth_a @ v_s @ I2 )
= X )
& ( ord_less_nat @ I2 @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ) ).
% valid_points_index_cons
thf(fact_1098_valid__points__index__obtains,axiom,
! [X: a] :
( ( member_a @ X @ ( set_a2 @ v_s ) )
=> ~ ! [I2: nat] :
~ ( ( ( nth_a @ v_s @ I2 )
= X )
& ( ord_less_nat @ I2 @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ) ).
% valid_points_index_obtains
thf(fact_1099_block__points__valid__point__index,axiom,
! [Bl2: set_a,X: a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_a @ X @ Bl2 )
=> ~ ! [I2: nat] :
~ ( ( ord_less_nat @ I2 @ ( size_size_list_a @ v_s ) )
& ( ( nth_a @ v_s @ I2 )
= X ) ) ) ) ).
% block_points_valid_point_index
thf(fact_1100_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1101_incidence__cond__indexed,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design3210447939978979927dent_a @ ( mset_set_a @ b_s ) @ ( nth_a @ v_s @ I ) @ ( nth_set_a @ b_s @ J ) )
= ( member_a @ ( nth_a @ v_s @ I ) @ ( nth_set_a @ b_s @ J ) ) ) ) ) ).
% incidence_cond_indexed
thf(fact_1102_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1103_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1104_int__distrib_I1_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1105_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(2)
thf(fact_1106_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1107_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1108_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1109_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1110_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1111_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1112_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1113_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1114_incidence__mat__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 ) ) )
=> ( ( incide1709061621920695677ex_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( insert_nat @ I1 @ ( insert_nat @ I22 @ bot_bot_set_nat ) ) )
= ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ ( insert_a @ ( nth_a @ v_s @ I1 ) @ ( insert_a @ ( nth_a @ v_s @ I22 ) @ bot_bot_set_a ) ) ) ) ) ) ).
% incidence_mat_two_index
thf(fact_1115_mat__rep__num__N__row,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( incide7000514267430604580um_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ I )
= ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ ( nth_a @ v_s @ I ) ) ) ) ).
% mat_rep_num_N_row
thf(fact_1116_dual__sys_Oblock__sizes__non__empty,axiom,
( ( ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s )
!= zero_z3157962936165190495et_nat )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( design8152002643121538447es_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ) ).
% dual_sys.block_sizes_non_empty
thf(fact_1117_dual__sys_Ofinite__block__sizes,axiom,
finite_finite_nat @ ( design8152002643121538447es_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).
% dual_sys.finite_block_sizes
thf(fact_1118_dual__sys_Osys__block__sizes__in,axiom,
! [Bl2: set_nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( member_nat @ ( finite_card_nat @ Bl2 ) @ ( design8152002643121538447es_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).
% dual_sys.sys_block_sizes_in
thf(fact_1119_dual__sys_Osys__block__sizes__obtain__bl,axiom,
! [X: nat] :
( ( member_nat @ X @ ( design8152002643121538447es_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
& ( ( finite_card_nat @ X3 )
= X ) ) ) ).
% dual_sys.sys_block_sizes_obtain_bl
thf(fact_1120_dual__sys_Oblock__sizes__non__empty__set,axiom,
( ( ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s )
!= zero_z3157962936165190495et_nat )
=> ( ( design8152002643121538447es_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
!= bot_bot_set_nat ) ) ).
% dual_sys.block_sizes_non_empty_set
thf(fact_1121_dual__sys_Odel__point__block__count,axiom,
! [P3: nat] :
( ( size_s7462436076474991978et_nat @ ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) )
= ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).
% dual_sys.del_point_block_count
thf(fact_1122_dual__sys_Odelete__point__blocks__sub,axiom,
! [B2: set_nat,P3: nat] :
( ( member_set_nat @ B2 @ ( set_mset_set_nat @ ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) ) )
=> ~ ! [Bl: set_nat] :
~ ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
& ( ord_less_eq_set_nat @ B2 @ Bl ) ) ) ).
% dual_sys.delete_point_blocks_sub
thf(fact_1123_dual__sys_Odelete__point__p__not__in__bl__blocks,axiom,
! [P3: nat] :
( ! [Bl: set_nat] :
( ( member_set_nat @ Bl @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ~ ( member_nat @ P3 @ Bl ) )
=> ( ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 )
= ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).
% dual_sys.delete_point_p_not_in_bl_blocks
thf(fact_1124_dual__sys_Ostr__del__block__del__point,axiom,
! [X: nat] :
( ~ ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ( design3550126062406151447ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X ) ) ) ).
% dual_sys.str_del_block_del_point
thf(fact_1125_dual__sys_Oadd__block__index__in,axiom,
! [Ps: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ Ps @ B2 )
=> ( ( design6574611146354332593ex_nat @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) @ Ps )
= ( plus_plus_nat @ ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Ps ) @ one_one_nat ) ) ) ).
% dual_sys.add_block_index_in
thf(fact_1126_dual__sys_Odelete__point__strong__block__not__in,axiom,
! [P3: nat,Bl2: set_nat] :
( ( member_nat @ P3 @ Bl2 )
=> ~ ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) ) ) ) ).
% dual_sys.delete_point_strong_block_not_in
thf(fact_1127_dual__sys_Oadd__block__rep__number__not__in,axiom,
! [X: nat,B2: set_nat] :
( ~ ( member_nat @ X @ B2 )
=> ( ( design3571518413069006949er_nat @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) @ X )
= ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X ) ) ) ).
% dual_sys.add_block_rep_number_not_in
thf(fact_1128_dual__sys_Odelete__point__strong__block__in,axiom,
! [P3: nat,Bl2: set_nat] :
( ~ ( member_nat @ P3 @ Bl2 )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) ) ) ) ) ).
% dual_sys.delete_point_strong_block_in
thf(fact_1129_dual__sys_Odelete__point__strong__block__in__iff,axiom,
! [Bl2: set_nat,P3: nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) ) )
= ( ~ ( member_nat @ P3 @ Bl2 ) ) ) ) ).
% dual_sys.delete_point_strong_block_in_iff
thf(fact_1130_dual__sys_Odelete__point__strong__block__in__orig,axiom,
! [Bl2: set_nat,P3: nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) ) )
=> ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).
% dual_sys.delete_point_strong_block_in_orig
thf(fact_1131_dual__sys_Oadd__block__rep__number__in,axiom,
! [X: nat,B2: set_nat] :
( ( member_nat @ X @ B2 )
=> ( ( design3571518413069006949er_nat @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) @ X )
= ( plus_plus_nat @ ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X ) @ one_one_nat ) ) ) ).
% dual_sys.add_block_rep_number_in
thf(fact_1132_dual__sys_Oadd__block__index__not__in,axiom,
! [Ps: set_nat,B2: set_nat] :
( ~ ( ord_less_eq_set_nat @ Ps @ B2 )
=> ( ( design6574611146354332593ex_nat @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) @ Ps )
= ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Ps ) ) ) ).
% dual_sys.add_block_index_not_in
thf(fact_1133_dual__sys_Odel__add__block__inv,axiom,
! [Bl2: set_nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ( design4725324266511619850ck_nat @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Bl2 ) @ Bl2 )
= ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).
% dual_sys.del_add_block_inv
thf(fact_1134_dual__sys_Odel__invalid__add__block__eq,axiom,
! [Bl2: set_nat] :
( ~ ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ( design4725324266511619850ck_nat @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Bl2 ) @ Bl2 )
= ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Bl2 ) ) ) ).
% dual_sys.del_invalid_add_block_eq
thf(fact_1135_dual__sys_Odel__block__b_I2_J,axiom,
! [Bl2: set_nat] :
( ~ ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ( size_s7462436076474991978et_nat @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Bl2 ) )
= ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).
% dual_sys.del_block_b(2)
thf(fact_1136_dual__sys_Odelete__invalid__block__eq,axiom,
! [B2: set_nat] :
( ~ ( member_set_nat @ B2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 )
= ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).
% dual_sys.delete_invalid_block_eq
thf(fact_1137_delete__point__blocks__wf,axiom,
! [B2: set_a,P3: a] :
( ( member_set_a @ B2 @ ( set_mset_set_a @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P3 ) ) )
=> ( ord_less_eq_set_a @ B2 @ ( minus_minus_set_a @ ( set_a2 @ v_s ) @ ( insert_a @ P3 @ bot_bot_set_a ) ) ) ) ).
% delete_point_blocks_wf
thf(fact_1138_strong__del__block__des,axiom,
! [B2: set_a] :
( ! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ~ ( ord_less_set_a @ Bl @ B2 ) )
=> ( design_design_a @ ( minus_minus_set_a @ ( set_a2 @ v_s ) @ B2 ) @ ( design4241783006516448631lock_a @ ( mset_set_a @ b_s ) @ B2 ) ) ) ).
% strong_del_block_des
thf(fact_1139_dual__sys_Oadd__block__fin,axiom,
! [B2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( design5426232790142929158em_nat @ ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B2 ) @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) ) ) ).
% dual_sys.add_block_fin
thf(fact_1140_block__complement__def,axiom,
! [B2: set_a] :
( ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ B2 )
= ( minus_minus_set_a @ ( set_a2 @ v_s ) @ B2 ) ) ).
% block_complement_def
thf(fact_1141_dual__sys_Ofinite__sets,axiom,
finite_finite_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ).
% dual_sys.finite_sets
thf(fact_1142_del__point__def,axiom,
! [P3: a] :
( ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P3 )
= ( minus_minus_set_a @ ( set_a2 @ v_s ) @ ( insert_a @ P3 @ bot_bot_set_a ) ) ) ).
% del_point_def
thf(fact_1143_dual__blocks__v,axiom,
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% dual_blocks_v
thf(fact_1144_dual__sys_Oincidence__system__axioms,axiom,
design3753904077504641269em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ).
% dual_sys.incidence_system_axioms
thf(fact_1145_dual__sys_Owf__invalid__point,axiom,
! [X: nat,B2: set_nat] :
( ~ ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( member_set_nat @ B2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ~ ( member_nat @ X @ B2 ) ) ) ).
% dual_sys.wf_invalid_point
thf(fact_1146_dual__sys_Oincidence__alt__def,axiom,
! [P3: nat,B2: set_nat] :
( ( member_nat @ P3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( member_set_nat @ B2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ( design8502206366797944887nt_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 @ B2 )
= ( member_nat @ P3 @ B2 ) ) ) ) ).
% dual_sys.incidence_alt_def
thf(fact_1147_strong__del__block__wf,axiom,
! [B2: set_a] : ( design1863209521793301785stem_a @ ( minus_minus_set_a @ ( set_a2 @ v_s ) @ B2 ) @ ( design4241783006516448631lock_a @ ( mset_set_a @ b_s ) @ B2 ) ) ).
% strong_del_block_wf
thf(fact_1148_strong__del__block__fin,axiom,
! [B2: set_a] : ( design9187838744727572296stem_a @ ( minus_minus_set_a @ ( set_a2 @ v_s ) @ B2 ) @ ( design4241783006516448631lock_a @ ( mset_set_a @ b_s ) @ B2 ) ) ).
% strong_del_block_fin
thf(fact_1149_dual__sys_Odelete__block__wf,axiom,
! [B2: set_nat] : ( design3753904077504641269em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) ) ).
% dual_sys.delete_block_wf
thf(fact_1150_dual__sys_Ofinite__incidence__system__axioms,axiom,
design5426232790142929158em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ).
% dual_sys.finite_incidence_system_axioms
thf(fact_1151_dual__sys_Odelete__invalid__pt__strong__eq,axiom,
! [P3: nat] :
( ~ ( member_nat @ P3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s )
= ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) ) ) ).
% dual_sys.delete_invalid_pt_strong_eq
thf(fact_1152_dual__sys_Odel__invalid__point__blocks,axiom,
! [P3: nat] :
( ~ ( member_nat @ P3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 )
= ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).
% dual_sys.del_invalid_point_blocks
thf(fact_1153_remove__invalid__point__block,axiom,
! [P3: a,Bl2: set_a] :
( ~ ( member_a @ P3 @ ( set_a2 @ v_s ) )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( minus_minus_set_a @ Bl2 @ ( insert_a @ P3 @ bot_bot_set_a ) )
= Bl2 ) ) ) ).
% remove_invalid_point_block
thf(fact_1154_dual__sys_Orep__number__non__zero__system__point,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X ) )
=> ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).
% dual_sys.rep_number_non_zero_system_point
thf(fact_1155_dual__sys_Ocomplete__block__size__eq__points,axiom,
! [Bl2: set_nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ( ( finite_card_nat @ Bl2 )
= ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
=> ( Bl2
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ) ).
% dual_sys.complete_block_size_eq_points
thf(fact_1156_dual__sys_Owellformed,axiom,
! [B2: set_nat] :
( ( member_set_nat @ B2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ord_less_eq_set_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).
% dual_sys.wellformed
thf(fact_1157_dual__sys_Opoints__index__ps__nin,axiom,
! [Ps: set_nat] :
( ~ ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Ps )
= zero_zero_nat ) ) ).
% dual_sys.points_index_ps_nin
thf(fact_1158_dual__sys_Oadd__block__wf__cond,axiom,
! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( design3753904077504641269em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) ) ) ).
% dual_sys.add_block_wf_cond
thf(fact_1159_dual__sys_Odelete__block__fin__incidence__sys,axiom,
! [B2: set_nat] : ( design5426232790142929158em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) ) ).
% dual_sys.delete_block_fin_incidence_sys
thf(fact_1160_dual__sys_Oadd__block__wf,axiom,
! [B2: set_nat] : ( design3753904077504641269em_nat @ ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B2 ) @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) ) ).
% dual_sys.add_block_wf
thf(fact_1161_dual__is__proper__design,axiom,
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ v_s ) )
=> ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X3 ) ) )
=> ( design435815215503836206gn_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).
% dual_is_proper_design
thf(fact_1162_dual__is__design,axiom,
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ v_s ) )
=> ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X3 ) ) )
=> ( design_design_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).
% dual_is_design
thf(fact_1163_dual__sys_Oincomplete__block__proper__subset,axiom,
! [Bl2: set_nat] :
( ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
& ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
=> ( ord_less_set_nat @ Bl2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).
% dual_sys.incomplete_block_proper_subset
thf(fact_1164_dual__sys_Oincomplete__alt__in,axiom,
! [Bl2: set_nat] :
( ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
& ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
=> ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).
% dual_sys.incomplete_alt_in
thf(fact_1165_dual__sys_Oincomplete__alt__size,axiom,
! [Bl2: set_nat] :
( ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
& ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
=> ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ) ).
% dual_sys.incomplete_alt_size
thf(fact_1166_dual__sys_Oblock__size__lt__order,axiom,
! [Bl2: set_nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ) ).
% dual_sys.block_size_lt_order
thf(fact_1167_dual__sys_Opoints__index__zero,axiom,
! [Ps: set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) @ ( finite_card_nat @ Ps ) )
=> ( ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Ps )
= zero_zero_nat ) ) ).
% dual_sys.points_index_zero
thf(fact_1168_dual__sys_Opoint__in__block__rep__min__iff,axiom,
! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ? [Bl: set_nat] :
( ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
& ( member_nat @ X @ Bl ) )
= ( ord_less_nat @ zero_zero_nat @ ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X ) ) ) ) ).
% dual_sys.point_in_block_rep_min_iff
thf(fact_1169_dual__sys_Ocomplete__block__all__subsets,axiom,
! [Bl2: set_nat,Ps: set_nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ( ( finite_card_nat @ Bl2 )
= ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
=> ( ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ord_less_eq_set_nat @ Ps @ Bl2 ) ) ) ) ).
% dual_sys.complete_block_all_subsets
thf(fact_1170_dual__sys_Oadd__del__block__inv,axiom,
! [Bl2: set_nat] :
( ( ord_less_eq_set_nat @ Bl2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( design755385109423264192ck_nat @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Bl2 ) @ Bl2 )
= ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).
% dual_sys.add_del_block_inv
thf(fact_1171_dual__sys_Oadd__block__fin__cond,axiom,
! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( design5426232790142929158em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) ) ) ).
% dual_sys.add_block_fin_cond
thf(fact_1172_dual__sys_Opoint__rep__non__existance,axiom,
! [X: nat] :
( ~ ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X )
= zero_zero_nat ) ) ).
% dual_sys.point_rep_non_existance
thf(fact_1173_dual__sys_Ofinite__sysI,axiom,
( ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( design5426232790142929158em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).
% dual_sys.finite_sysI
thf(fact_1174_dual__sys_Oincomplete__alt__imp,axiom,
! [Bl2: set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
& ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ) ) ).
% dual_sys.incomplete_alt_imp
thf(fact_1175_dual__sys_OdesignI,axiom,
( ! [B5: set_nat] :
( ( member_set_nat @ B5 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( B5 != bot_bot_set_nat ) )
=> ( ( ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s )
!= zero_z3157962936165190495et_nat )
=> ( ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) )
!= bot_bot_set_nat )
=> ( design_design_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ) ).
% dual_sys.designI
thf(fact_1176_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M4: nat] :
( ( ord_less_nat @ M4 @ N )
& ( P @ M4 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1177_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( P @ M4 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_1178_dual__sys_Opoint__indices__elem__in,axiom,
! [Ps: set_nat,T: nat] :
( ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( ( finite_card_nat @ Ps )
= T )
=> ( member_nat @ ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Ps ) @ ( design1227534709319296284es_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ T ) ) ) ) ).
% dual_sys.point_indices_elem_in
thf(fact_1179_dual__sys_Oobtain__point__with__rep,axiom,
! [R: nat] :
( ( member_nat @ R @ ( design3853898657598026467rs_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
& ( ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X3 )
= R ) ) ) ).
% dual_sys.obtain_point_with_rep
thf(fact_1180_dual__sys_Opoint__rep__number__in__set,axiom,
! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( member_nat @ ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X ) @ ( design3853898657598026467rs_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).
% dual_sys.point_rep_number_in_set
thf(fact_1181_del__point__order,axiom,
! [P3: a] :
( ( member_a @ P3 @ ( set_a2 @ v_s ) )
=> ( ( finite_card_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P3 ) )
= ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) ) ).
% del_point_order
thf(fact_1182_block__complement__size,axiom,
! [B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( set_a2 @ v_s ) )
=> ( ( finite_card_a @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ B2 ) )
= ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ ( finite_card_a @ B2 ) ) ) ) ).
% block_complement_size
thf(fact_1183_del__block__b_I1_J,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) @ one_one_nat ) ) ) ).
% del_block_b(1)
thf(fact_1184_dual__sys_Odel__block__b_I1_J,axiom,
! [Bl2: set_nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ( size_s7462436076474991978et_nat @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Bl2 ) )
= ( minus_minus_nat @ ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) @ one_one_nat ) ) ) ).
% dual_sys.del_block_b(1)
thf(fact_1185_dual__sys_Oreplication__numbers__non__empty,axiom,
( ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) )
!= bot_bot_set_nat )
=> ( ( design3853898657598026467rs_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
!= bot_bot_set_nat ) ) ).
% dual_sys.replication_numbers_non_empty
thf(fact_1186_dual__sys_Oreplication__numbers__finite,axiom,
finite_finite_nat @ ( design3853898657598026467rs_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).
% dual_sys.replication_numbers_finite
thf(fact_1187_dual__sys_Ostrong__del__block__wf,axiom,
! [B2: set_nat] : ( design3753904077504641269em_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B2 ) @ ( design3550126062406151447ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) ) ).
% dual_sys.strong_del_block_wf
thf(fact_1188_dual__sys_Ostrong__del__block__fin,axiom,
! [B2: set_nat] : ( design5426232790142929158em_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B2 ) @ ( design3550126062406151447ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) ) ).
% dual_sys.strong_del_block_fin
thf(fact_1189_dual__sys_Oremove__invalid__point__block,axiom,
! [P3: nat,Bl2: set_nat] :
( ~ ( member_nat @ P3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ( minus_minus_set_nat @ Bl2 @ ( insert_nat @ P3 @ bot_bot_set_nat ) )
= Bl2 ) ) ) ).
% dual_sys.remove_invalid_point_block
thf(fact_1190_dual__sys_Odelete__point__blocks__wf,axiom,
! [B2: set_nat,P3: nat] :
( ( member_set_nat @ B2 @ ( set_mset_set_nat @ ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) ) )
=> ( ord_less_eq_set_nat @ B2 @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( insert_nat @ P3 @ bot_bot_set_nat ) ) ) ) ).
% dual_sys.delete_point_blocks_wf
thf(fact_1191_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_1192_int__distrib_I3_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1193_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(4)
thf(fact_1194_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1195_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1196_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1197_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1198_dual__sys_Oadd__point__sys__rep__numbers,axiom,
! [P3: nat] :
( ( design3853898657598026467rs_nat @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
= ( sup_sup_set_nat @ ( design3853898657598026467rs_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) @ ( insert_nat @ ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) @ bot_bot_set_nat ) ) ) ).
% dual_sys.add_point_sys_rep_numbers
thf(fact_1199_dual__sys_Oadd__point__blocks__wf,axiom,
! [P3: nat,Bs4: set_set_nat] : ( design3753904077504641269em_nat @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) @ ( design5698312687278145166ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 @ Bs4 ) ) ).
% dual_sys.add_point_blocks_wf
thf(fact_1200_dual__sys_Oadd__point__def,axiom,
! [P3: nat] :
( ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 )
= ( insert_nat @ P3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).
% dual_sys.add_point_def
thf(fact_1201_dual__sys_Oadd__point__finite,axiom,
! [P3: nat] : ( design5426232790142929158em_nat @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).
% dual_sys.add_point_finite
thf(fact_1202_dual__sys_Oadd__point__wf,axiom,
! [P3: nat] : ( design3753904077504641269em_nat @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).
% dual_sys.add_point_wf
thf(fact_1203_dual__sys_Oadd__point__to__blocks__finite,axiom,
! [P3: nat,Bs4: set_set_nat] : ( design5426232790142929158em_nat @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) @ ( design5698312687278145166ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 @ Bs4 ) ) ).
% dual_sys.add_point_to_blocks_finite
thf(fact_1204_dual__sys_Oadd__existing__point,axiom,
! [P3: nat] :
( ( member_nat @ P3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).
% dual_sys.add_existing_point
thf(fact_1205_dual__sys_Odelete__point__index__eq,axiom,
! [Ps: set_nat,P3: nat] :
( ( ord_less_eq_set_nat @ Ps @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) )
=> ( ( design6574611146354332593ex_nat @ ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) @ Ps )
= ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Ps ) ) ) ).
% dual_sys.delete_point_index_eq
thf(fact_1206_dual__sys_Oadd__block__sub__sys,axiom,
! [B2: set_nat] : ( sub_su5953739893325741765em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B2 ) @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) ) ).
% dual_sys.add_block_sub_sys
thf(fact_1207_dual__sys_Odel__invalid__point,axiom,
! [P3: nat] :
( ~ ( member_nat @ P3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).
% dual_sys.del_invalid_point
thf(fact_1208_dual__sys_Oadd__delete__point__inv,axiom,
! [P3: nat] :
( ~ ( member_nat @ P3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( design4269233978287968195nt_nat @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) @ P3 )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).
% dual_sys.add_delete_point_inv
thf(fact_1209_dual__sys_Odel__point__order,axiom,
! [P3: nat] :
( ( member_nat @ P3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( finite_card_nat @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) @ one_one_nat ) ) ) ).
% dual_sys.del_point_order
thf(fact_1210_dual__sys_Odel__point__def,axiom,
! [P3: nat] :
( ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 )
= ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( insert_nat @ P3 @ bot_bot_set_nat ) ) ) ).
% dual_sys.del_point_def
thf(fact_1211_dual__sys_Odelete__block__sub__sys,axiom,
! [B2: set_nat] : ( sub_su5953739893325741765em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).
% dual_sys.delete_block_sub_sys
thf(fact_1212_dual__sys_Oadd__point__sub__sys,axiom,
! [P3: nat] : ( sub_su5953739893325741765em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).
% dual_sys.add_point_sub_sys
thf(fact_1213_dual__sys_Ostrong__del__point__incidence__wf,axiom,
! [P3: nat] : ( design3753904077504641269em_nat @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) ) ).
% dual_sys.strong_del_point_incidence_wf
thf(fact_1214_dual__sys_Ostrong__del__point__finite,axiom,
! [P3: nat] : ( design5426232790142929158em_nat @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) ) ).
% dual_sys.strong_del_point_finite
thf(fact_1215_dual__sys_Odelete__point__wf,axiom,
! [P3: nat] : ( design3753904077504641269em_nat @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) @ ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) ) ).
% dual_sys.delete_point_wf
thf(fact_1216_dual__sys_Odelete__point__finite,axiom,
! [P3: nat] : ( design5426232790142929158em_nat @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) @ ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) ) ).
% dual_sys.delete_point_finite
thf(fact_1217_dual__sys_Ostrong__del__point__sub__sys,axiom,
! [P3: nat] : ( sub_su5953739893325741765em_nat @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P3 ) @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P3 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).
% dual_sys.strong_del_point_sub_sys
thf(fact_1218_dual__sys_Ocomplement__rep__number,axiom,
! [X: nat,R: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X )
= R )
=> ( ( design3571518413069006949er_nat @ ( design5569578106646884273ks_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) @ X )
= ( minus_minus_nat @ ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) @ R ) ) ) ) ).
% dual_sys.complement_rep_number
thf(fact_1219_dual__sys_Ocomplement__blocks__wf,axiom,
! [Bl2: set_nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( design5569578106646884273ks_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
=> ( ord_less_eq_set_nat @ Bl2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).
% dual_sys.complement_blocks_wf
thf(fact_1220_dual__sys_Ocomplement__finite,axiom,
design5426232790142929158em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design5569578106646884273ks_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).
% dual_sys.complement_finite
thf(fact_1221_dual__sys_Ocomplement__same__b,axiom,
( ( size_s7462436076474991978et_nat @ ( design5569578106646884273ks_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
= ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).
% dual_sys.complement_same_b
thf(fact_1222_dual__sys_Ocomplement__wf,axiom,
design3753904077504641269em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design5569578106646884273ks_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).
% dual_sys.complement_wf
thf(fact_1223_dual__sys_Oblock__comp__incomplete__nempty,axiom,
! [Bl2: set_nat] :
( ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
& ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
=> ( ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 )
!= bot_bot_set_nat ) ) ).
% dual_sys.block_comp_incomplete_nempty
thf(fact_1224_dual__sys_Oblock__comp__incomplete,axiom,
! [Bl2: set_nat] :
( ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
& ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) ) ) ) ).
% dual_sys.block_comp_incomplete
thf(fact_1225_dual__sys_Oblock__comp__elem__alt__left,axiom,
! [X: nat,Bl2: set_nat,Ps: set_nat] :
( ( member_nat @ X @ Bl2 )
=> ( ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) )
=> ~ ( member_nat @ X @ Ps ) ) ) ).
% dual_sys.block_comp_elem_alt_left
thf(fact_1226_dual__sys_Oblock__comp__elem__alt__right,axiom,
! [Ps: set_nat,Bl2: set_nat] :
( ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ Ps )
=> ~ ( member_nat @ X3 @ Bl2 ) )
=> ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) ) ) ) ).
% dual_sys.block_comp_elem_alt_right
thf(fact_1227_dual__sys_Oblock__complement__elem__iff,axiom,
! [Ps: set_nat,Bl2: set_nat] :
( ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ Ps )
=> ~ ( member_nat @ X2 @ Bl2 ) ) ) ) ) ).
% dual_sys.block_complement_elem_iff
thf(fact_1228_dual__sys_Oblock__complement__subset__points,axiom,
! [Ps: set_nat,Bl2: set_nat] :
( ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) )
=> ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).
% dual_sys.block_complement_subset_points
thf(fact_1229_dual__sys_Oblock__complement__def,axiom,
! [B2: set_nat] :
( ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B2 )
= ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B2 ) ) ).
% dual_sys.block_complement_def
thf(fact_1230_dual__sys_Oblock__complement__size,axiom,
! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( finite_card_nat @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% dual_sys.block_complement_size
thf(fact_1231_dual__sys_Oblock__complement__inv,axiom,
! [Bl2: set_nat,Bl23: set_nat] :
( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( ( ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 )
= Bl23 )
=> ( ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl23 )
= Bl2 ) ) ) ).
% dual_sys.block_complement_inv
thf(fact_1232_dual__sys_Oobtain__comp__block__orig,axiom,
! [Bl1: set_nat] :
( ( member_set_nat @ Bl1 @ ( set_mset_set_nat @ ( design5569578106646884273ks_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
=> ~ ! [Bl22: set_nat] :
( ( member_set_nat @ Bl22 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( Bl1
!= ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl22 ) ) ) ) ).
% dual_sys.obtain_comp_block_orig
thf(fact_1233_map__col__to__block__wf,axiom,
! [C2: vec_int] :
( ( member_vec_int @ C2 @ ( set_vec_int2 @ ( cols_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) )
=> ( ord_less_eq_set_nat @ ( incide3973235006681262014ck_int @ C2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ).
% map_col_to_block_wf
thf(fact_1234_dual__blocks__elem__iff,axiom,
! [J: nat,X: nat] :
( ( ord_less_nat @ J @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( member_nat @ X @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J ) )
= ( ( member_a @ ( nth_a @ v_s @ J ) @ ( nth_set_a @ b_s @ X ) )
& ( ord_less_nat @ X @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ) ).
% dual_blocks_elem_iff
thf(fact_1235_dual__blocks__len,axiom,
( ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) )
= ( size_size_list_a @ v_s ) ) ).
% dual_blocks_len
thf(fact_1236_dual__blocks__ne__index__ne,axiom,
! [J1: nat,J22: nat] :
( ( ord_less_nat @ J1 @ ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
=> ( ( ord_less_nat @ J22 @ ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
=> ( ( ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J1 )
!= ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J22 ) )
=> ( J1 != J22 ) ) ) ) ).
% dual_blocks_ne_index_ne
thf(fact_1237_dual__blocks__ordered__eq,axiom,
( ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s )
= ( mset_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) ) ).
% dual_blocks_ordered_eq
thf(fact_1238_ord__dual__blocks__b,axiom,
( ( size_s7462436076474991978et_nat @ ( mset_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% ord_dual_blocks_b
thf(fact_1239_dual__blocks__size__is__rep,axiom,
! [J: nat] :
( ( ord_less_nat @ J @ ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
=> ( ( finite_card_nat @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J ) )
= ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ ( nth_a @ v_s @ J ) ) ) ) ).
% dual_blocks_size_is_rep
thf(fact_1240_dual__blocks__rep__is__size,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ b_s ) )
=> ( ( design3571518413069006949er_nat @ ( mset_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) @ I )
= ( finite_card_a @ ( nth_set_a @ b_s @ I ) ) ) ) ).
% dual_blocks_rep_is_size
thf(fact_1241_dual__incidence__iff2,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_a @ ( nth_a @ v_s @ I ) @ ( nth_set_a @ b_s @ J ) )
= ( member_nat @ J @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ I ) ) ) ) ) ).
% dual_incidence_iff2
thf(fact_1242_dual__incidence__iff,axiom,
! [I: nat,J: nat,Bl2: set_a,X: a] :
( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ( nth_set_a @ b_s @ J )
= Bl2 )
=> ( ( ( nth_a @ v_s @ I )
= X )
=> ( ( member_a @ X @ Bl2 )
= ( member_nat @ J @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ I ) ) ) ) ) ) ) ).
% dual_incidence_iff
thf(fact_1243_dual__blocks__points__index__inter,axiom,
! [I1: nat,I22: nat] :
( ( ord_less_nat @ I1 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ I22 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design6574611146354332593ex_nat @ ( mset_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) @ ( insert_nat @ I1 @ ( insert_nat @ I22 @ bot_bot_set_nat ) ) )
= ( design7842873109100088828mber_a @ ( nth_set_a @ b_s @ I1 ) @ ( nth_set_a @ b_s @ I22 ) ) ) ) ) ).
% dual_blocks_points_index_inter
thf(fact_1244_dual__blocks__inter__index,axiom,
! [J1: nat,J22: nat] :
( ( ord_less_nat @ J1 @ ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
=> ( ( ord_less_nat @ J22 @ ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
=> ( ( design7485525362727208274er_nat @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J1 ) @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J22 ) )
= ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ ( insert_a @ ( nth_a @ v_s @ J1 ) @ ( insert_a @ ( nth_a @ v_s @ J22 ) @ bot_bot_set_a ) ) ) ) ) ) ).
% dual_blocks_inter_index
thf(fact_1245_mat__inter__num__conv,axiom,
! [J1: nat,J22: nat] :
( ( ord_less_nat @ J1 @ ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( ord_less_nat @ J22 @ ( dim_col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( design7842873109100088828mber_a @ ( nth_set_a @ b_s @ J1 ) @ ( nth_set_a @ b_s @ J22 ) )
= ( incide903612199826827877um_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ J1 @ J22 ) ) ) ) ).
% mat_inter_num_conv
thf(fact_1246_complement__mat__entries,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ~ ( member_a @ ( nth_a @ v_s @ I ) @ ( nth_set_a @ b_s @ J ) ) )
= ( member_a @ ( nth_a @ v_s @ I ) @ ( nth_set_a @ ( map_set_a_set_a @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) ) @ b_s ) @ J ) ) ) ) ) ).
% complement_mat_entries
thf(fact_1247_dim__vec__N__col,axiom,
! [J: nat] :
( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( dim_vec_int @ ( nth_vec_int @ ( cols_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) @ J ) )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ).
% dim_vec_N_col
thf(fact_1248_ordered__complement,axiom,
incide1624170830610365509stem_a @ v_s @ ( map_set_a_set_a @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) ) @ b_s ) ).
% ordered_complement
thf(fact_1249_length__blocks__complement,axiom,
( ( size_size_list_set_a @ ( map_set_a_set_a @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) ) @ b_s ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% length_blocks_complement
thf(fact_1250_map__block__complement__entry,axiom,
! [J: nat] :
( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( nth_set_a @ ( map_set_a_set_a @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) ) @ b_s ) @ J )
= ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ ( nth_set_a @ b_s @ J ) ) ) ) ).
% map_block_complement_entry
thf(fact_1251_map__col__to__block__elem__not,axiom,
! [C2: vec_int,I: nat] :
( ( member_vec_int @ C2 @ ( set_vec_int2 @ ( cols_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) )
=> ( ( ord_less_nat @ I @ ( dim_vec_int @ C2 ) )
=> ( ( ~ ( member_nat @ I @ ( incide3973235006681262014ck_int @ C2 ) ) )
= ( ( vec_index_int @ C2 @ I )
= zero_zero_int ) ) ) ) ).
% map_col_to_block_elem_not
thf(fact_1252_map__col__block__eq,axiom,
! [C2: vec_int] :
( ( member_vec_int @ C2 @ ( set_vec_int2 @ ( cols_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) )
=> ( ( incide8105300093900788317at_int @ ( upt @ zero_zero_nat @ ( dim_vec_int @ C2 ) ) @ ( incide3973235006681262014ck_int @ C2 ) )
= C2 ) ) ).
% map_col_block_eq
thf(fact_1253_generated__is__dual,axiom,
dual_o7311038347201774518_a_nat @ v_s @ b_s @ ( upt @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ).
% generated_is_dual
thf(fact_1254_is__dual__rev,axiom,
dual_o2119597811525285140_nat_a @ ( upt @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ v_s @ b_s ).
% is_dual_rev
thf(fact_1255_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_1256_dual__is__ordered__inc__sys,axiom,
incide6998539924841383625em_nat @ ( upt @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ).
% dual_is_ordered_inc_sys
thf(fact_1257_ord__dual__blocks__v,axiom,
( ( finite_card_nat @ ( set_nat2 @ ( upt @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% ord_dual_blocks_v
thf(fact_1258_dual__incidence__mat__eq__trans,axiom,
( ( incide1177982898701834729at_int @ ( upt @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) )
= ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ).
% dual_incidence_mat_eq_trans
thf(fact_1259_dual__incidence__mat__eq__trans__rev,axiom,
( ( transpose_mat_int @ ( incide1177982898701834729at_int @ ( upt @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
= ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ).
% dual_incidence_mat_eq_trans_rev
thf(fact_1260_inc__mat__of__map__rev,axiom,
( ( incide1177982898701834729at_int @ ( upt @ zero_zero_nat @ ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) @ ( map_vec_int_set_nat @ incide3973235006681262014ck_int @ ( cols_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) )
= ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ).
% inc_mat_of_map_rev
thf(fact_1261_mat__ord__inc__sys__N,axiom,
( ( incide1177982898701834729at_int @ ( upt @ zero_zero_nat @ ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) @ ( map_vec_int_set_nat @ incide3973235006681262014ck_int @ ( cols_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) )
= ( matrix323868623736973467nt_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ).
% mat_ord_inc_sys_N
thf(fact_1262_mat__is__ordered__incidence__sys,axiom,
incide6998539924841383625em_nat @ ( upt @ zero_zero_nat @ ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) @ ( map_vec_int_set_nat @ incide3973235006681262014ck_int @ ( cols_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ).
% mat_is_ordered_incidence_sys
thf(fact_1263_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_1264_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_1265_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_1266_mat__ord__inc__sys__point,axiom,
! [X: nat] :
( ( ord_less_nat @ X @ ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) )
=> ( ( nth_nat @ ( upt @ zero_zero_nat @ ( dim_row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) @ X )
= X ) ) ).
% mat_ord_inc_sys_point
thf(fact_1267_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_1268_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).
% distinct_upt
thf(fact_1269_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I3: nat,J3: nat] : ( set_nat2 @ ( upt @ I3 @ J3 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_1270_N__col__def,axiom,
! [J: nat,I: nat] :
( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( ( member_a @ ( nth_a @ v_s @ I ) @ ( nth_set_a @ b_s @ J ) )
=> ( ( vec_index_int @ ( col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ J ) @ I )
= one_one_int ) )
& ( ~ ( member_a @ ( nth_a @ v_s @ I ) @ ( nth_set_a @ b_s @ J ) )
=> ( ( vec_index_int @ ( col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ J ) @ I )
= zero_zero_int ) ) ) ) ) ).
% N_col_def
thf(fact_1271_N__row__def,axiom,
! [J: nat,I: nat] :
( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( ( ( member_a @ ( nth_a @ v_s @ I ) @ ( nth_set_a @ b_s @ J ) )
=> ( ( vec_index_int @ ( row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ I ) @ J )
= one_one_int ) )
& ( ~ ( member_a @ ( nth_a @ v_s @ I ) @ ( nth_set_a @ b_s @ J ) )
=> ( ( vec_index_int @ ( row_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ I ) @ J )
= zero_zero_int ) ) ) ) ) ).
% N_row_def
thf(fact_1272_col__inc__vec__of,axiom,
! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_set_a @ b_s ) )
=> ( ( incide4681487840832596491_a_int @ v_s @ ( nth_set_a @ b_s @ J ) )
= ( col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ J ) ) ) ).
% col_inc_vec_of
thf(fact_1273_dim__vec__col__N,axiom,
! [I: nat] :
( ( dim_vec_int @ ( col_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ I ) )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% dim_vec_col_N
% Conjectures (1)
thf(conj_0,conjecture,
( ( map_mat_int_real @ ring_1_of_int_real @ ( times_times_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) @ ( transpose_mat_int @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) )
= ( times_times_mat_real @ ( map_mat_int_real @ ring_1_of_int_real @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) @ ( transpose_mat_real @ ( map_mat_int_real @ ring_1_of_int_real @ ( incide7016682120514301311_a_int @ v_s @ b_s ) ) ) ) ) ).
%------------------------------------------------------------------------------