TPTP Problem File: SLH0776^1.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 : Commuting_Hermitian/0002_Commuting_Hermitian/prob_01046_042650__19430990_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1335 ( 483 unt; 287 typ; 0 def)
% Number of atoms : 2731 (1176 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 8946 ( 253 ~; 56 |; 115 &;7185 @)
% ( 0 <=>;1337 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 40 ( 39 usr)
% Number of type conns : 692 ( 692 >; 0 *; 0 +; 0 <<)
% Number of symbols : 251 ( 248 usr; 12 con; 0-5 aty)
% Number of variables : 2870 ( 152 ^;2628 !; 90 ?;2870 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:37:10.303
%------------------------------------------------------------------------------
% Could-be-implicit typings (39)
thf(ty_n_t__List__Olist_It__Matrix__Ovec_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
list_vec_mat_complex: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
list_l5436439031154120755omplex: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_vec_mat_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_mat_mat_complex: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_list_mat_complex: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Ovec_It__Matrix__Omat_Itf__a_J_J_J,type,
list_vec_mat_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Matrix__Omat_Itf__a_J_J_J,type,
list_list_mat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Matrix__Omat_Itf__a_J_J_J,type,
set_vec_mat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Matrix__Omat_Itf__a_J_J_J,type,
set_mat_mat_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Matrix__Omat_Itf__a_J_J_J,type,
set_list_mat_a: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
vec_mat_complex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Matrix__Ovec_It__Complex__Ocomplex_J_J,type,
mat_vec_complex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
mat_mat_complex: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Ovec_It__Complex__Ocomplex_J_J,type,
list_vec_complex: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
list_mat_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Complex__Ocomplex_J_J,type,
set_vec_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
set_mat_complex: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Matrix__Omat_Itf__a_J_J,type,
vec_mat_a: $tType ).
thf(ty_n_t__Matrix__Omat_It__Matrix__Ovec_Itf__a_J_J,type,
mat_vec_a: $tType ).
thf(ty_n_t__Matrix__Omat_It__Matrix__Omat_Itf__a_J_J,type,
mat_mat_a: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Ovec_Itf__a_J_J,type,
list_vec_a: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
list_mat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
set_vec_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
set_mat_a: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
vec_complex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Complex__Ocomplex_J,type,
mat_complex: $tType ).
thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_Itf__a_J,type,
vec_a: $tType ).
thf(ty_n_t__Matrix__Omat_Itf__a_J,type,
mat_a: $tType ).
thf(ty_n_t__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__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (248)
thf(sy_c_Commuting__Hermitian__Misc_On__sum_001t__Nat__Onat,type,
commut2019222099004354946um_nat: nat > list_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
minus_2412168080157227406omplex: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
minus_6391593812940525058omplex: vec_complex > vec_complex > vec_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
times_8009071140041733218omplex: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_Itf__a_J,type,
times_times_mat_a: mat_a > mat_a > mat_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
uminus467866341702955550omplex: mat_complex > mat_complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Omat_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
uminus9210244920068684493omplex: mat_mat_complex > mat_mat_complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
uminus4447292074486253202omplex: vec_complex > vec_complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Ovec_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
uminus1815796379637173593omplex: vec_mat_complex > vec_mat_complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
uminus5491753114148463108omplex: set_list_mat_complex > set_list_mat_complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__Matrix__Omat_Itf__a_J_J_J,type,
uminus1627440288842321386_mat_a: set_list_mat_a > set_list_mat_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
uminus5815530220087396478omplex: set_mat_complex > set_mat_complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
uminus1296375033039821146_mat_a: set_mat_a > set_mat_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Matrix__Ovec_It__Complex__Ocomplex_J_J,type,
uminus1004567299294339826omplex: set_vec_complex > set_vec_complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
uminus2769705506071317478_vec_a: set_vec_a > set_vec_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Jordan__Normal__Form__Existence_Oinv__all_H_001t__Complex__Ocomplex,type,
jordan5032732407113867375omplex: ( mat_complex > nat > nat > $o ) > mat_complex > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__1_001t__Complex__Ocomplex,type,
jordan2017415923357163885omplex: mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__1__main_001t__Complex__Ocomplex,type,
jordan9130142659770429862omplex: nat > nat > nat > mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__2_001t__Complex__Ocomplex,type,
jordan7871273693253786478omplex: mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__2__main_001t__Complex__Ocomplex,type,
jordan6916311984355858983omplex: nat > nat > mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__3_001t__Complex__Ocomplex,type,
jordan4501759426295633263omplex: mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__3__a_001t__Complex__Ocomplex,type,
jordan2858886415929732048omplex: nat > nat > mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__3__c_001t__Complex__Ocomplex,type,
jordan5343229918868201426omplex: complex > nat > nat > list_P6011104703257516679at_nat > mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__3__c__inner__loop_001t__Complex__Ocomplex,type,
jordan7656109678144820486omplex: complex > nat > nat > nat > mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__3__main_001t__Complex__Ocomplex,type,
jordan4702481308941288104omplex: nat > nat > mat_complex > mat_complex ).
thf(sy_c_Linear__Algebra__Complements_Orank__1__proj_001t__Complex__Ocomplex,type,
linear1949544614684794075omplex: vec_complex > mat_complex ).
thf(sy_c_List_Obutlast_001t__Complex__Ocomplex,type,
butlast_complex: list_complex > list_complex ).
thf(sy_c_List_Obutlast_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
butlas2964118825291935103omplex: list_l5436439031154120755omplex > list_l5436439031154120755omplex ).
thf(sy_c_List_Obutlast_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
butlast_list_mat_a: list_list_mat_a > list_list_mat_a ).
thf(sy_c_List_Obutlast_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
butlast_mat_complex: list_mat_complex > list_mat_complex ).
thf(sy_c_List_Obutlast_001t__Matrix__Omat_Itf__a_J,type,
butlast_mat_a: list_mat_a > list_mat_a ).
thf(sy_c_List_Obutlast_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
butlast_vec_complex: list_vec_complex > list_vec_complex ).
thf(sy_c_List_Obutlast_001t__Matrix__Ovec_Itf__a_J,type,
butlast_vec_a: list_vec_a > list_vec_a ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Ogen__length_001t__Complex__Ocomplex,type,
gen_length_complex: nat > list_complex > nat ).
thf(sy_c_List_Ogen__length_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
gen_le107826107610854458omplex: nat > list_mat_complex > nat ).
thf(sy_c_List_Ogen__length_001t__Matrix__Omat_Itf__a_J,type,
gen_length_mat_a: nat > list_mat_a > nat ).
thf(sy_c_List_Ogen__length_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
gen_le4087251840394152110omplex: nat > list_vec_complex > nat ).
thf(sy_c_List_Ogen__length_001t__Matrix__Ovec_Itf__a_J,type,
gen_length_vec_a: nat > list_vec_a > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
thf(sy_c_List_Olist_OCons_001t__Complex__Ocomplex,type,
cons_complex: complex > list_complex > list_complex ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
cons_l4198107141827137507omplex: list_mat_complex > list_l5436439031154120755omplex > list_l5436439031154120755omplex ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
cons_list_mat_a: list_mat_a > list_list_mat_a > list_list_mat_a ).
thf(sy_c_List_Olist_OCons_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
cons_mat_complex: mat_complex > list_mat_complex > list_mat_complex ).
thf(sy_c_List_Olist_OCons_001t__Matrix__Omat_Itf__a_J,type,
cons_mat_a: mat_a > list_mat_a > list_mat_a ).
thf(sy_c_List_Olist_OCons_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
cons_vec_complex: vec_complex > list_vec_complex > list_vec_complex ).
thf(sy_c_List_Olist_OCons_001t__Matrix__Ovec_Itf__a_J,type,
cons_vec_a: vec_a > list_vec_a > list_vec_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
nil_mat_complex: list_mat_complex ).
thf(sy_c_List_Olist_ONil_001t__Matrix__Omat_Itf__a_J,type,
nil_mat_a: list_mat_a ).
thf(sy_c_List_Olist_ONil_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
nil_vec_complex: list_vec_complex ).
thf(sy_c_List_Olist_ONil_001t__Matrix__Ovec_Itf__a_J,type,
nil_vec_a: list_vec_a ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_Ohd_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
hd_mat_complex: list_mat_complex > mat_complex ).
thf(sy_c_List_Olist_Ohd_001t__Matrix__Omat_Itf__a_J,type,
hd_mat_a: list_mat_a > mat_a ).
thf(sy_c_List_Olist_Ohd_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
hd_vec_complex: list_vec_complex > vec_complex ).
thf(sy_c_List_Olist_Ohd_001t__Matrix__Ovec_Itf__a_J,type,
hd_vec_a: list_vec_a > vec_a ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Olist__all_001t__Complex__Ocomplex,type,
list_all_complex: ( complex > $o ) > list_complex > $o ).
thf(sy_c_List_Olist_Olist__all_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
list_a2530308013472304865omplex: ( list_mat_complex > $o ) > list_l5436439031154120755omplex > $o ).
thf(sy_c_List_Olist_Olist__all_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
list_all_list_mat_a: ( list_mat_a > $o ) > list_list_mat_a > $o ).
thf(sy_c_List_Olist_Olist__all_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
list_all_mat_complex: ( mat_complex > $o ) > list_mat_complex > $o ).
thf(sy_c_List_Olist_Olist__all_001t__Matrix__Omat_Itf__a_J,type,
list_all_mat_a: ( mat_a > $o ) > list_mat_a > $o ).
thf(sy_c_List_Olist_Olist__all_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
list_all_vec_complex: ( vec_complex > $o ) > list_vec_complex > $o ).
thf(sy_c_List_Olist_Olist__all_001t__Matrix__Ovec_Itf__a_J,type,
list_all_vec_a: ( vec_a > $o ) > list_vec_a > $o ).
thf(sy_c_List_Olist_Olist__all_001t__Nat__Onat,type,
list_all_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist_Olist__all_001tf__a,type,
list_all_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
set_complex2: list_complex > set_complex ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
set_list_mat_complex2: list_l5436439031154120755omplex > set_list_mat_complex ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
set_list_mat_a2: list_list_mat_a > set_list_mat_a ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
set_mat_complex2: list_mat_complex > set_mat_complex ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Omat_Itf__a_J,type,
set_mat_a2: list_mat_a > set_mat_a ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
set_vec_complex2: list_vec_complex > set_vec_complex ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Ovec_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
set_vec_mat_complex2: list_vec_mat_complex > set_vec_mat_complex ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Ovec_It__Matrix__Omat_Itf__a_J_J,type,
set_vec_mat_a2: list_vec_mat_a > set_vec_mat_a ).
thf(sy_c_List_Olist_Oset_001t__Matrix__Ovec_Itf__a_J,type,
set_vec_a2: list_vec_a > set_vec_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Otl_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
tl_mat_complex: list_mat_complex > list_mat_complex ).
thf(sy_c_List_Olist_Otl_001t__Matrix__Omat_Itf__a_J,type,
tl_mat_a: list_mat_a > list_mat_a ).
thf(sy_c_List_Olist_Otl_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
tl_vec_complex: list_vec_complex > list_vec_complex ).
thf(sy_c_List_Olist_Otl_001t__Matrix__Ovec_Itf__a_J,type,
tl_vec_a: list_vec_a > list_vec_a ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist__ex_001t__Complex__Ocomplex,type,
list_ex_complex: ( complex > $o ) > list_complex > $o ).
thf(sy_c_List_Olist__ex_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
list_e422197382367757169omplex: ( list_mat_complex > $o ) > list_l5436439031154120755omplex > $o ).
thf(sy_c_List_Olist__ex_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
list_ex_list_mat_a: ( list_mat_a > $o ) > list_list_mat_a > $o ).
thf(sy_c_List_Olist__ex_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
list_ex_mat_complex: ( mat_complex > $o ) > list_mat_complex > $o ).
thf(sy_c_List_Olist__ex_001t__Matrix__Omat_Itf__a_J,type,
list_ex_mat_a: ( mat_a > $o ) > list_mat_a > $o ).
thf(sy_c_List_Olist__ex_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
list_ex_vec_complex: ( vec_complex > $o ) > list_vec_complex > $o ).
thf(sy_c_List_Olist__ex_001t__Matrix__Ovec_Itf__a_J,type,
list_ex_vec_a: ( vec_a > $o ) > list_vec_a > $o ).
thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex_001tf__a,type,
list_ex_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_On__lists_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
n_lists_mat_complex: nat > list_mat_complex > list_l5436439031154120755omplex ).
thf(sy_c_List_On__lists_001t__Matrix__Omat_Itf__a_J,type,
n_lists_mat_a: nat > list_mat_a > list_list_mat_a ).
thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
nth_complex: list_complex > nat > complex ).
thf(sy_c_List_Onth_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
nth_list_mat_complex: list_l5436439031154120755omplex > nat > list_mat_complex ).
thf(sy_c_List_Onth_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
nth_list_mat_a: list_list_mat_a > nat > list_mat_a ).
thf(sy_c_List_Onth_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
nth_mat_complex: list_mat_complex > nat > mat_complex ).
thf(sy_c_List_Onth_001t__Matrix__Omat_Itf__a_J,type,
nth_mat_a: list_mat_a > nat > mat_a ).
thf(sy_c_List_Onth_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
nth_vec_complex: list_vec_complex > nat > vec_complex ).
thf(sy_c_List_Onth_001t__Matrix__Ovec_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
nth_vec_mat_complex: list_vec_mat_complex > nat > vec_mat_complex ).
thf(sy_c_List_Onth_001t__Matrix__Ovec_It__Matrix__Omat_Itf__a_J_J,type,
nth_vec_mat_a: list_vec_mat_a > nat > vec_mat_a ).
thf(sy_c_List_Onth_001t__Matrix__Ovec_Itf__a_J,type,
nth_vec_a: list_vec_a > nat > vec_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Complex__Ocomplex,type,
carrier_mat_complex: nat > nat > set_mat_complex ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
carrie8442657464762054641omplex: nat > nat > set_mat_mat_complex ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Matrix__Omat_Itf__a_J,type,
carrier_mat_mat_a: nat > nat > set_mat_mat_a ).
thf(sy_c_Matrix_Ocarrier__mat_001tf__a,type,
carrier_mat_a: nat > nat > set_mat_a ).
thf(sy_c_Matrix_Ocarrier__vec_001t__Complex__Ocomplex,type,
carrier_vec_complex: nat > set_vec_complex ).
thf(sy_c_Matrix_Ocarrier__vec_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
carrie1048208924330543741omplex: nat > set_vec_mat_complex ).
thf(sy_c_Matrix_Ocarrier__vec_001t__Matrix__Omat_Itf__a_J,type,
carrier_vec_mat_a: nat > set_vec_mat_a ).
thf(sy_c_Matrix_Ocarrier__vec_001tf__a,type,
carrier_vec_a: nat > set_vec_a ).
thf(sy_c_Matrix_Ocol_001t__Complex__Ocomplex,type,
col_complex: mat_complex > nat > vec_complex ).
thf(sy_c_Matrix_Ocol_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
col_mat_complex: mat_mat_complex > nat > vec_mat_complex ).
thf(sy_c_Matrix_Ocol_001t__Matrix__Omat_Itf__a_J,type,
col_mat_a: mat_mat_a > nat > vec_mat_a ).
thf(sy_c_Matrix_Ocol_001tf__a,type,
col_a: mat_a > nat > vec_a ).
thf(sy_c_Matrix_Ocols_001t__Complex__Ocomplex,type,
cols_complex: mat_complex > list_vec_complex ).
thf(sy_c_Matrix_Ocols_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
cols_mat_complex: mat_mat_complex > list_vec_mat_complex ).
thf(sy_c_Matrix_Ocols_001t__Matrix__Omat_Itf__a_J,type,
cols_mat_a: mat_mat_a > list_vec_mat_a ).
thf(sy_c_Matrix_Ocols_001tf__a,type,
cols_a: mat_a > list_vec_a ).
thf(sy_c_Matrix_Odiag__block__mat_001t__Complex__Ocomplex,type,
diag_b9145358668110806138omplex: list_mat_complex > mat_complex ).
thf(sy_c_Matrix_Odiag__block__mat_001tf__a,type,
diag_block_mat_a: list_mat_a > mat_a ).
thf(sy_c_Matrix_Odiag__mat_001t__Complex__Ocomplex,type,
diag_mat_complex: mat_complex > list_complex ).
thf(sy_c_Matrix_Odiag__mat_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
diag_mat_mat_complex: mat_mat_complex > list_mat_complex ).
thf(sy_c_Matrix_Odiag__mat_001t__Matrix__Omat_Itf__a_J,type,
diag_mat_mat_a: mat_mat_a > list_mat_a ).
thf(sy_c_Matrix_Odiag__mat_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
diag_mat_vec_complex: mat_vec_complex > list_vec_complex ).
thf(sy_c_Matrix_Odiag__mat_001t__Matrix__Ovec_Itf__a_J,type,
diag_mat_vec_a: mat_vec_a > list_vec_a ).
thf(sy_c_Matrix_Odiag__mat_001t__Nat__Onat,type,
diag_mat_nat: mat_nat > list_nat ).
thf(sy_c_Matrix_Odiag__mat_001tf__a,type,
diag_mat_a: mat_a > list_a ).
thf(sy_c_Matrix_Odim__col_001t__Complex__Ocomplex,type,
dim_col_complex: mat_complex > nat ).
thf(sy_c_Matrix_Odim__col_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
dim_col_mat_complex: mat_mat_complex > nat ).
thf(sy_c_Matrix_Odim__col_001t__Matrix__Omat_Itf__a_J,type,
dim_col_mat_a: mat_mat_a > nat ).
thf(sy_c_Matrix_Odim__col_001t__Nat__Onat,type,
dim_col_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__col_001tf__a,type,
dim_col_a: mat_a > nat ).
thf(sy_c_Matrix_Odim__row_001t__Complex__Ocomplex,type,
dim_row_complex: mat_complex > nat ).
thf(sy_c_Matrix_Odim__row_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
dim_row_mat_complex: mat_mat_complex > nat ).
thf(sy_c_Matrix_Odim__row_001t__Matrix__Omat_Itf__a_J,type,
dim_row_mat_a: mat_mat_a > nat ).
thf(sy_c_Matrix_Odim__row_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
dim_row_vec_complex: mat_vec_complex > nat ).
thf(sy_c_Matrix_Odim__row_001t__Matrix__Ovec_Itf__a_J,type,
dim_row_vec_a: mat_vec_a > nat ).
thf(sy_c_Matrix_Odim__row_001t__Nat__Onat,type,
dim_row_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__row_001tf__a,type,
dim_row_a: mat_a > nat ).
thf(sy_c_Matrix_Omat__of__cols_001t__Complex__Ocomplex,type,
mat_of_cols_complex: nat > list_vec_complex > mat_complex ).
thf(sy_c_Matrix_Omat__of__cols_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
mat_of4965844074629950985omplex: nat > list_vec_mat_complex > mat_mat_complex ).
thf(sy_c_Matrix_Omat__of__cols_001t__Matrix__Omat_Itf__a_J,type,
mat_of_cols_mat_a: nat > list_vec_mat_a > mat_mat_a ).
thf(sy_c_Matrix_Omat__of__cols_001tf__a,type,
mat_of_cols_a: nat > list_vec_a > mat_a ).
thf(sy_c_Matrix_Omat__of__row_001t__Complex__Ocomplex,type,
mat_of_row_complex: vec_complex > mat_complex ).
thf(sy_c_Matrix_Omat__of__row_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
mat_of8530337284418210486omplex: vec_mat_complex > mat_mat_complex ).
thf(sy_c_Matrix_Omat__of__row_001t__Matrix__Omat_Itf__a_J,type,
mat_of_row_mat_a: vec_mat_a > mat_mat_a ).
thf(sy_c_Matrix_Omat__of__row_001tf__a,type,
mat_of_row_a: vec_a > mat_a ).
thf(sy_c_Matrix_Omat__of__rows_001t__Complex__Ocomplex,type,
mat_of_rows_complex: nat > list_vec_complex > mat_complex ).
thf(sy_c_Matrix_Omat__of__rows_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
mat_of2504328217136560291omplex: nat > list_vec_mat_complex > mat_mat_complex ).
thf(sy_c_Matrix_Omat__of__rows_001t__Matrix__Omat_Itf__a_J,type,
mat_of_rows_mat_a: nat > list_vec_mat_a > mat_mat_a ).
thf(sy_c_Matrix_Omat__of__rows_001tf__a,type,
mat_of_rows_a: nat > list_vec_a > mat_a ).
thf(sy_c_Matrix_Omk__diagonal_001t__Complex__Ocomplex,type,
mk_diagonal_complex: list_complex > mat_complex ).
thf(sy_c_Matrix_Omk__diagonal_001t__Nat__Onat,type,
mk_diagonal_nat: list_nat > mat_nat ).
thf(sy_c_Matrix_Omk__diagonal_001tf__a,type,
mk_diagonal_a: list_a > mat_a ).
thf(sy_c_Matrix_Omult__mat__vec_001t__Complex__Ocomplex,type,
mult_mat_vec_complex: mat_complex > vec_complex > vec_complex ).
thf(sy_c_Matrix_Omult__mat__vec_001tf__a,type,
mult_mat_vec_a: mat_a > vec_a > vec_a ).
thf(sy_c_Matrix_Orow_001t__Complex__Ocomplex,type,
row_complex: mat_complex > nat > vec_complex ).
thf(sy_c_Matrix_Orow_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
row_mat_complex: mat_mat_complex > nat > vec_mat_complex ).
thf(sy_c_Matrix_Orow_001t__Matrix__Omat_Itf__a_J,type,
row_mat_a: mat_mat_a > nat > vec_mat_a ).
thf(sy_c_Matrix_Orow_001tf__a,type,
row_a: mat_a > nat > vec_a ).
thf(sy_c_Matrix_Orows_001t__Complex__Ocomplex,type,
rows_complex: mat_complex > list_vec_complex ).
thf(sy_c_Matrix_Orows_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
rows_mat_complex: mat_mat_complex > list_vec_mat_complex ).
thf(sy_c_Matrix_Orows_001t__Matrix__Omat_Itf__a_J,type,
rows_mat_a: mat_mat_a > list_vec_mat_a ).
thf(sy_c_Matrix_Orows_001tf__a,type,
rows_a: mat_a > list_vec_a ).
thf(sy_c_Matrix_Osquare__mat_001t__Complex__Ocomplex,type,
square_mat_complex: mat_complex > $o ).
thf(sy_c_Matrix_Osquare__mat_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
square7091239463517965744omplex: mat_mat_complex > $o ).
thf(sy_c_Matrix_Osquare__mat_001t__Matrix__Omat_Itf__a_J,type,
square_mat_mat_a: mat_mat_a > $o ).
thf(sy_c_Matrix_Osquare__mat_001tf__a,type,
square_mat_a: mat_a > $o ).
thf(sy_c_Matrix_Otranspose__mat_001t__Complex__Ocomplex,type,
transp3074176993011536131omplex: mat_complex > mat_complex ).
thf(sy_c_Matrix_Otranspose__mat_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
transp4906945491372815122omplex: mat_mat_complex > mat_mat_complex ).
thf(sy_c_Matrix_Otranspose__mat_001t__Matrix__Omat_Itf__a_J,type,
transpose_mat_mat_a: mat_mat_a > mat_mat_a ).
thf(sy_c_Matrix_Otranspose__mat_001tf__a,type,
transpose_mat_a: mat_a > mat_a ).
thf(sy_c_Matrix_Oundef__vec_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
undef_2495355514574404529omplex: nat > mat_complex ).
thf(sy_c_Matrix_Oundef__vec_001t__Matrix__Omat_Itf__a_J,type,
undef_vec_mat_a: nat > mat_a ).
thf(sy_c_Matrix_Ounit__vec_001t__Complex__Ocomplex,type,
unit_vec_complex: nat > nat > vec_complex ).
thf(sy_c_Matrix_Ounit__vecs_001t__Complex__Ocomplex,type,
unit_vecs_complex: nat > list_vec_complex ).
thf(sy_c_Matrix_Ounit__vecs__last_001t__Complex__Ocomplex,type,
unit_v8657589406246362837omplex: nat > nat > list_vec_complex ).
thf(sy_c_Matrix_Ovec__first_001t__Complex__Ocomplex,type,
vec_first_complex: vec_complex > nat > vec_complex ).
thf(sy_c_Matrix_Ovec__first_001tf__a,type,
vec_first_a: vec_a > nat > vec_a ).
thf(sy_c_Matrix_Ovec__last_001t__Complex__Ocomplex,type,
vec_last_complex: vec_complex > nat > vec_complex ).
thf(sy_c_Matrix_Ovec__last_001tf__a,type,
vec_last_a: vec_a > nat > vec_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
size_s479360804472521375omplex: list_l5436439031154120755omplex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Matrix__Omat_Itf__a_J_J_J,type,
size_s6656407794899724303_mat_a: list_list_mat_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
size_s5969786470865220249omplex: list_mat_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
size_size_list_mat_a: list_mat_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Ovec_It__Complex__Ocomplex_J_J,type,
size_s1158823550072163597omplex: list_vec_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Ovec_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
size_s2077990086586600628omplex: list_vec_mat_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Ovec_It__Matrix__Omat_Itf__a_J_J_J,type,
size_s5765634329853218618_mat_a: list_vec_mat_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Ovec_Itf__a_J_J,type,
size_size_list_vec_a: list_vec_a > 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_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
ord_le5598786136212072115omplex: set_mat_complex > set_mat_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Matrix__Ovec_It__Complex__Ocomplex_J_J,type,
ord_le787823215419015463omplex: set_vec_complex > set_vec_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
ord_less_set_vec_a: set_vec_a > set_vec_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J_M_Eo_J,type,
ord_le1186661314528880752plex_o: ( list_mat_complex > $o ) > ( list_mat_complex > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Matrix__Omat_Itf__a_J_J_M_Eo_J,type,
ord_le963178399796136004at_a_o: ( list_mat_a > $o ) > ( list_mat_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Matrix__Omat_It__Complex__Ocomplex_J_M_Eo_J,type,
ord_le2790225379703085046plex_o: ( mat_complex > $o ) > ( mat_complex > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Matrix__Omat_Itf__a_J_M_Eo_J,type,
ord_less_eq_mat_a_o: ( mat_a > $o ) > ( mat_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
ord_le7594668674868021933omplex: set_list_mat_complex > set_list_mat_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Matrix__Omat_Itf__a_J_J_J,type,
ord_le4771995077433322369_mat_a: set_list_mat_a > set_list_mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
ord_le3632134057777142183omplex: set_mat_complex > set_mat_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
ord_le3318621148231462513_mat_a: set_mat_a > set_mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_It__Complex__Ocomplex_J_J,type,
ord_le8044543173838861339omplex: set_vec_complex > set_vec_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
ord_le1193683088243165222omplex: set_vec_mat_complex > set_vec_mat_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_It__Matrix__Omat_Itf__a_J_J_J,type,
ord_le1807819179058128968_mat_a: set_vec_mat_a > set_vec_mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
ord_le4791951621262958845_vec_a: set_vec_a > set_vec_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Projective__Measurements_Odensity__collapse,type,
projec3470689467825365843llapse: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Projective__Measurements_Ohermitian__decomp_001t__Complex__Ocomplex,type,
projec5943904436471448624omplex: mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
collec136165848909103768omplex: ( list_mat_complex > $o ) > set_list_mat_complex ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
collect_list_mat_a: ( list_mat_a > $o ) > set_list_mat_a ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
collect_mat_complex: ( mat_complex > $o ) > set_mat_complex ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_Itf__a_J,type,
collect_mat_a: ( mat_a > $o ) > set_mat_a ).
thf(sy_c_Set_OCollect_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
collect_vec_complex: ( vec_complex > $o ) > set_vec_complex ).
thf(sy_c_Set_OCollect_001t__Matrix__Ovec_Itf__a_J,type,
collect_vec_a: ( vec_a > $o ) > set_vec_a ).
thf(sy_c_Spectral__Theory__Complements_Ounitarily__equiv_001t__Complex__Ocomplex,type,
spectr6340060708231679580omplex: mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
member279434397506102358omplex: list_mat_complex > set_list_mat_complex > $o ).
thf(sy_c_member_001t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
member_list_mat_a: list_mat_a > set_list_mat_a > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
member_mat_complex: mat_complex > set_mat_complex > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
member7752848204589936667omplex: mat_mat_complex > set_mat_mat_complex > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Matrix__Omat_Itf__a_J_J,type,
member_mat_mat_a: mat_mat_a > set_mat_mat_a > $o ).
thf(sy_c_member_001t__Matrix__Omat_Itf__a_J,type,
member_mat_a: mat_a > set_mat_a > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Complex__Ocomplex_J,type,
member_vec_complex: vec_complex > set_vec_complex > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
member358399664158425767omplex: vec_mat_complex > set_vec_mat_complex > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Matrix__Omat_Itf__a_J_J,type,
member_vec_mat_a: vec_mat_a > set_vec_mat_a > $o ).
thf(sy_c_member_001t__Matrix__Ovec_Itf__a_J,type,
member_vec_a: vec_a > set_vec_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_Al,type,
al: list_mat_a ).
thf(sy_v_Bl,type,
bl: list_mat_a ).
thf(sy_v_i,type,
i: nat ).
% Relevant facts (1044)
thf(fact_0_assms_I1_J,axiom,
( ( size_size_list_mat_a @ al )
= ( size_size_list_mat_a @ bl ) ) ).
% assms(1)
thf(fact_1_assms_I5_J,axiom,
( ( times_times_mat_a @ ( diag_block_mat_a @ al ) @ ( diag_block_mat_a @ bl ) )
= ( times_times_mat_a @ ( diag_block_mat_a @ bl ) @ ( diag_block_mat_a @ al ) ) ) ).
% assms(5)
thf(fact_2_assms_I6_J,axiom,
ord_less_nat @ i @ ( size_size_list_mat_a @ al ) ).
% assms(6)
thf(fact_3_assms_I3_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_mat_a @ al ) )
=> ( ( dim_row_a @ ( nth_mat_a @ al @ I ) )
= ( dim_row_a @ ( nth_mat_a @ bl @ I ) ) ) ) ).
% assms(3)
thf(fact_4_assms_I4_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_mat_a @ al ) )
=> ( ( dim_col_a @ ( nth_mat_a @ al @ I ) )
= ( dim_col_a @ ( nth_mat_a @ bl @ I ) ) ) ) ).
% assms(4)
thf(fact_5_Groups_Omult__ac_I3_J,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_6_Groups_Omult__ac_I2_J,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_7_Groups_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_8_assms_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_mat_a @ al ) )
=> ( ( dim_row_a @ ( nth_mat_a @ al @ I ) )
= ( dim_col_a @ ( nth_mat_a @ al @ I ) ) ) ) ).
% assms(2)
thf(fact_9_diag__block__mat__commute,axiom,
! [Al: list_mat_complex,Bl: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Al )
= ( size_s5969786470865220249omplex @ Bl ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5969786470865220249omplex @ Al ) )
=> ( ( times_8009071140041733218omplex @ ( nth_mat_complex @ Al @ I2 ) @ ( nth_mat_complex @ Bl @ I2 ) )
= ( times_8009071140041733218omplex @ ( nth_mat_complex @ Bl @ I2 ) @ ( nth_mat_complex @ Al @ I2 ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5969786470865220249omplex @ Al ) )
=> ( ( dim_col_complex @ ( nth_mat_complex @ Al @ I2 ) )
= ( dim_row_complex @ ( nth_mat_complex @ Bl @ I2 ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5969786470865220249omplex @ Al ) )
=> ( ( dim_col_complex @ ( nth_mat_complex @ Bl @ I2 ) )
= ( dim_row_complex @ ( nth_mat_complex @ Al @ I2 ) ) ) )
=> ( ( times_8009071140041733218omplex @ ( diag_b9145358668110806138omplex @ Al ) @ ( diag_b9145358668110806138omplex @ Bl ) )
= ( times_8009071140041733218omplex @ ( diag_b9145358668110806138omplex @ Bl ) @ ( diag_b9145358668110806138omplex @ Al ) ) ) ) ) ) ) ).
% diag_block_mat_commute
thf(fact_10_diag__block__mat__commute,axiom,
! [Al: list_mat_a,Bl: list_mat_a] :
( ( ( size_size_list_mat_a @ Al )
= ( size_size_list_mat_a @ Bl ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( times_times_mat_a @ ( nth_mat_a @ Al @ I2 ) @ ( nth_mat_a @ Bl @ I2 ) )
= ( times_times_mat_a @ ( nth_mat_a @ Bl @ I2 ) @ ( nth_mat_a @ Al @ I2 ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( dim_col_a @ ( nth_mat_a @ Al @ I2 ) )
= ( dim_row_a @ ( nth_mat_a @ Bl @ I2 ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( dim_col_a @ ( nth_mat_a @ Bl @ I2 ) )
= ( dim_row_a @ ( nth_mat_a @ Al @ I2 ) ) ) )
=> ( ( times_times_mat_a @ ( diag_block_mat_a @ Al ) @ ( diag_block_mat_a @ Bl ) )
= ( times_times_mat_a @ ( diag_block_mat_a @ Bl ) @ ( diag_block_mat_a @ Al ) ) ) ) ) ) ) ).
% diag_block_mat_commute
thf(fact_11_diag__block__mat__cong__comp,axiom,
! [Al: list_mat_a,Bl: list_mat_a,J: nat] :
( ( ( size_size_list_mat_a @ Al )
= ( size_size_list_mat_a @ Bl ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( dim_row_a @ ( nth_mat_a @ Al @ I2 ) )
= ( dim_row_a @ ( nth_mat_a @ Bl @ I2 ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Al ) )
=> ( ( dim_col_a @ ( nth_mat_a @ Al @ I2 ) )
= ( dim_col_a @ ( nth_mat_a @ Bl @ I2 ) ) ) )
=> ( ( ( diag_block_mat_a @ Al )
= ( diag_block_mat_a @ Bl ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_mat_a @ Al ) )
=> ( ( nth_mat_a @ Al @ J )
= ( nth_mat_a @ Bl @ J ) ) ) ) ) ) ) ).
% diag_block_mat_cong_comp
thf(fact_12_diag__block__mat__cong__comp,axiom,
! [Al: list_mat_complex,Bl: list_mat_complex,J: nat] :
( ( ( size_s5969786470865220249omplex @ Al )
= ( size_s5969786470865220249omplex @ Bl ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5969786470865220249omplex @ Al ) )
=> ( ( dim_row_complex @ ( nth_mat_complex @ Al @ I2 ) )
= ( dim_row_complex @ ( nth_mat_complex @ Bl @ I2 ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5969786470865220249omplex @ Al ) )
=> ( ( dim_col_complex @ ( nth_mat_complex @ Al @ I2 ) )
= ( dim_col_complex @ ( nth_mat_complex @ Bl @ I2 ) ) ) )
=> ( ( ( diag_b9145358668110806138omplex @ Al )
= ( diag_b9145358668110806138omplex @ Bl ) )
=> ( ( ord_less_nat @ J @ ( size_s5969786470865220249omplex @ Al ) )
=> ( ( nth_mat_complex @ Al @ J )
= ( nth_mat_complex @ Bl @ J ) ) ) ) ) ) ) ).
% diag_block_mat_cong_comp
thf(fact_13_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_14_nth__equalityI,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( ( nth_complex @ Xs @ I2 )
= ( nth_complex @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_15_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_16_nth__equalityI,axiom,
! [Xs: list_vec_complex,Ys: list_vec_complex] :
( ( ( size_s1158823550072163597omplex @ Xs )
= ( size_s1158823550072163597omplex @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1158823550072163597omplex @ Xs ) )
=> ( ( nth_vec_complex @ Xs @ I2 )
= ( nth_vec_complex @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_17_nth__equalityI,axiom,
! [Xs: list_vec_a,Ys: list_vec_a] :
( ( ( size_size_list_vec_a @ Xs )
= ( size_size_list_vec_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_vec_a @ Xs ) )
=> ( ( nth_vec_a @ Xs @ I2 )
= ( nth_vec_a @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_18_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_19_nth__equalityI,axiom,
! [Xs: list_mat_a,Ys: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Xs ) )
=> ( ( nth_mat_a @ Xs @ I2 )
= ( nth_mat_a @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_20_nth__equalityI,axiom,
! [Xs: list_mat_complex,Ys: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ( nth_mat_complex @ Xs @ I2 )
= ( nth_mat_complex @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_21_Skolem__list__nth,axiom,
! [K: nat,P: nat > complex > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X: complex] : ( P @ I3 @ X ) ) )
= ( ? [Xs2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_complex @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_22_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X: a] : ( P @ I3 @ X ) ) )
= ( ? [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_a @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_23_Skolem__list__nth,axiom,
! [K: nat,P: nat > vec_complex > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X: vec_complex] : ( P @ I3 @ X ) ) )
= ( ? [Xs2: list_vec_complex] :
( ( ( size_s1158823550072163597omplex @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_vec_complex @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_24_Skolem__list__nth,axiom,
! [K: nat,P: nat > vec_a > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X: vec_a] : ( P @ I3 @ X ) ) )
= ( ? [Xs2: list_vec_a] :
( ( ( size_size_list_vec_a @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_vec_a @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_25_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X: nat] : ( P @ I3 @ X ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_26_Skolem__list__nth,axiom,
! [K: nat,P: nat > mat_a > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X: mat_a] : ( P @ I3 @ X ) ) )
= ( ? [Xs2: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_mat_a @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_27_Skolem__list__nth,axiom,
! [K: nat,P: nat > mat_complex > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X: mat_complex] : ( P @ I3 @ X ) ) )
= ( ? [Xs2: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_mat_complex @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_28_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_complex,Z: list_complex] : ( Y = Z ) )
= ( ^ [Xs2: list_complex,Ys2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( ( nth_complex @ Xs2 @ I3 )
= ( nth_complex @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_29_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_a,Z: list_a] : ( Y = Z ) )
= ( ^ [Xs2: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ Xs2 @ I3 )
= ( nth_a @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_30_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_vec_complex,Z: list_vec_complex] : ( Y = Z ) )
= ( ^ [Xs2: list_vec_complex,Ys2: list_vec_complex] :
( ( ( size_s1158823550072163597omplex @ Xs2 )
= ( size_s1158823550072163597omplex @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1158823550072163597omplex @ Xs2 ) )
=> ( ( nth_vec_complex @ Xs2 @ I3 )
= ( nth_vec_complex @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_31_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_vec_a,Z: list_vec_a] : ( Y = Z ) )
= ( ^ [Xs2: list_vec_a,Ys2: list_vec_a] :
( ( ( size_size_list_vec_a @ Xs2 )
= ( size_size_list_vec_a @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_vec_a @ Xs2 ) )
=> ( ( nth_vec_a @ Xs2 @ I3 )
= ( nth_vec_a @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_32_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z ) )
= ( ^ [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_33_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_mat_a,Z: list_mat_a] : ( Y = Z ) )
= ( ^ [Xs2: list_mat_a,Ys2: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs2 )
= ( size_size_list_mat_a @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_mat_a @ Xs2 ) )
=> ( ( nth_mat_a @ Xs2 @ I3 )
= ( nth_mat_a @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_34_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_mat_complex,Z: list_mat_complex] : ( Y = Z ) )
= ( ^ [Xs2: list_mat_complex,Ys2: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs2 )
= ( size_s5969786470865220249omplex @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5969786470865220249omplex @ Xs2 ) )
=> ( ( nth_mat_complex @ Xs2 @ I3 )
= ( nth_mat_complex @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_35_index__mult__mat_I3_J,axiom,
! [A3: mat_a,B3: mat_a] :
( ( dim_col_a @ ( times_times_mat_a @ A3 @ B3 ) )
= ( dim_col_a @ B3 ) ) ).
% index_mult_mat(3)
thf(fact_36_index__mult__mat_I3_J,axiom,
! [A3: mat_complex,B3: mat_complex] :
( ( dim_col_complex @ ( times_8009071140041733218omplex @ A3 @ B3 ) )
= ( dim_col_complex @ B3 ) ) ).
% index_mult_mat(3)
thf(fact_37_index__mult__mat_I2_J,axiom,
! [A3: mat_a,B3: mat_a] :
( ( dim_row_a @ ( times_times_mat_a @ A3 @ B3 ) )
= ( dim_row_a @ A3 ) ) ).
% index_mult_mat(2)
thf(fact_38_index__mult__mat_I2_J,axiom,
! [A3: mat_complex,B3: mat_complex] :
( ( dim_row_complex @ ( times_8009071140041733218omplex @ A3 @ B3 ) )
= ( dim_row_complex @ A3 ) ) ).
% index_mult_mat(2)
thf(fact_39_length__induct,axiom,
! [P: list_complex > $o,Xs: list_complex] :
( ! [Xs3: list_complex] :
( ! [Ys3: list_complex] :
( ( ord_less_nat @ ( size_s3451745648224563538omplex @ Ys3 ) @ ( size_s3451745648224563538omplex @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_40_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs3: list_a] :
( ! [Ys3: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys3 ) @ ( size_size_list_a @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_41_length__induct,axiom,
! [P: list_vec_complex > $o,Xs: list_vec_complex] :
( ! [Xs3: list_vec_complex] :
( ! [Ys3: list_vec_complex] :
( ( ord_less_nat @ ( size_s1158823550072163597omplex @ Ys3 ) @ ( size_s1158823550072163597omplex @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_42_length__induct,axiom,
! [P: list_vec_a > $o,Xs: list_vec_a] :
( ! [Xs3: list_vec_a] :
( ! [Ys3: list_vec_a] :
( ( ord_less_nat @ ( size_size_list_vec_a @ Ys3 ) @ ( size_size_list_vec_a @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_43_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_44_length__induct,axiom,
! [P: list_mat_a > $o,Xs: list_mat_a] :
( ! [Xs3: list_mat_a] :
( ! [Ys3: list_mat_a] :
( ( ord_less_nat @ ( size_size_list_mat_a @ Ys3 ) @ ( size_size_list_mat_a @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_45_length__induct,axiom,
! [P: list_mat_complex > $o,Xs: list_mat_complex] :
( ! [Xs3: list_mat_complex] :
( ! [Ys3: list_mat_complex] :
( ( ord_less_nat @ ( size_s5969786470865220249omplex @ Ys3 ) @ ( size_s5969786470865220249omplex @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_46_inv__all_H__def,axiom,
( jordan5032732407113867375omplex
= ( ^ [P2: mat_complex > nat > nat > $o,A4: mat_complex] :
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_complex @ A4 ) )
=> ( ( ord_less_nat @ J2 @ ( dim_row_complex @ A4 ) )
=> ( P2 @ A4 @ I3 @ J2 ) ) ) ) ) ).
% inv_all'_def
thf(fact_47_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_complex] :
( ( size_s3451745648224563538omplex @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_48_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_a] :
( ( size_size_list_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_49_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_vec_complex] :
( ( size_s1158823550072163597omplex @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_50_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_vec_a] :
( ( size_size_list_vec_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_51_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_52_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_mat_a] :
( ( size_size_list_mat_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_53_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_mat_complex] :
( ( size_s5969786470865220249omplex @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_54_neq__if__length__neq,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
!= ( size_s3451745648224563538omplex @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_55_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_56_neq__if__length__neq,axiom,
! [Xs: list_vec_complex,Ys: list_vec_complex] :
( ( ( size_s1158823550072163597omplex @ Xs )
!= ( size_s1158823550072163597omplex @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_57_neq__if__length__neq,axiom,
! [Xs: list_vec_a,Ys: list_vec_a] :
( ( ( size_size_list_vec_a @ Xs )
!= ( size_size_list_vec_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_58_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_59_neq__if__length__neq,axiom,
! [Xs: list_mat_a,Ys: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs )
!= ( size_size_list_mat_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_60_neq__if__length__neq,axiom,
! [Xs: list_mat_complex,Ys: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs )
!= ( size_s5969786470865220249omplex @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_61_size__neq__size__imp__neq,axiom,
! [X2: list_complex,Y2: list_complex] :
( ( ( size_s3451745648224563538omplex @ X2 )
!= ( size_s3451745648224563538omplex @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_62_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_63_size__neq__size__imp__neq,axiom,
! [X2: list_vec_complex,Y2: list_vec_complex] :
( ( ( size_s1158823550072163597omplex @ X2 )
!= ( size_s1158823550072163597omplex @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_64_size__neq__size__imp__neq,axiom,
! [X2: list_vec_a,Y2: list_vec_a] :
( ( ( size_size_list_vec_a @ X2 )
!= ( size_size_list_vec_a @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_65_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_66_size__neq__size__imp__neq,axiom,
! [X2: list_mat_a,Y2: list_mat_a] :
( ( ( size_size_list_mat_a @ X2 )
!= ( size_size_list_mat_a @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_67_size__neq__size__imp__neq,axiom,
! [X2: list_mat_complex,Y2: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ X2 )
!= ( size_s5969786470865220249omplex @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_68_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_69_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_70_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_71_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_72_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_73_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_74_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_75_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_76_mk__diagonal__dim_I2_J,axiom,
! [As: list_nat] :
( ( dim_col_nat @ ( mk_diagonal_nat @ As ) )
= ( size_size_list_nat @ As ) ) ).
% mk_diagonal_dim(2)
thf(fact_77_mk__diagonal__dim_I2_J,axiom,
! [As: list_a] :
( ( dim_col_a @ ( mk_diagonal_a @ As ) )
= ( size_size_list_a @ As ) ) ).
% mk_diagonal_dim(2)
thf(fact_78_mk__diagonal__dim_I2_J,axiom,
! [As: list_complex] :
( ( dim_col_complex @ ( mk_diagonal_complex @ As ) )
= ( size_s3451745648224563538omplex @ As ) ) ).
% mk_diagonal_dim(2)
thf(fact_79_mk__diagonal__dim_I1_J,axiom,
! [As: list_nat] :
( ( dim_row_nat @ ( mk_diagonal_nat @ As ) )
= ( size_size_list_nat @ As ) ) ).
% mk_diagonal_dim(1)
thf(fact_80_mk__diagonal__dim_I1_J,axiom,
! [As: list_a] :
( ( dim_row_a @ ( mk_diagonal_a @ As ) )
= ( size_size_list_a @ As ) ) ).
% mk_diagonal_dim(1)
thf(fact_81_mk__diagonal__dim_I1_J,axiom,
! [As: list_complex] :
( ( dim_row_complex @ ( mk_diagonal_complex @ As ) )
= ( size_s3451745648224563538omplex @ As ) ) ).
% mk_diagonal_dim(1)
thf(fact_82_list__ex__length,axiom,
( list_ex_complex
= ( ^ [P3: complex > $o,Xs2: list_complex] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
& ( P3 @ ( nth_complex @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_83_list__ex__length,axiom,
( list_ex_a
= ( ^ [P3: a > $o,Xs2: list_a] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
& ( P3 @ ( nth_a @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_84_list__ex__length,axiom,
( list_ex_vec_complex
= ( ^ [P3: vec_complex > $o,Xs2: list_vec_complex] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s1158823550072163597omplex @ Xs2 ) )
& ( P3 @ ( nth_vec_complex @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_85_list__ex__length,axiom,
( list_ex_vec_a
= ( ^ [P3: vec_a > $o,Xs2: list_vec_a] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_vec_a @ Xs2 ) )
& ( P3 @ ( nth_vec_a @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_86_list__ex__length,axiom,
( list_ex_nat
= ( ^ [P3: nat > $o,Xs2: list_nat] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
& ( P3 @ ( nth_nat @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_87_list__ex__length,axiom,
( list_ex_mat_a
= ( ^ [P3: mat_a > $o,Xs2: list_mat_a] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_mat_a @ Xs2 ) )
& ( P3 @ ( nth_mat_a @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_88_list__ex__length,axiom,
( list_ex_mat_complex
= ( ^ [P3: mat_complex > $o,Xs2: list_mat_complex] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s5969786470865220249omplex @ Xs2 ) )
& ( P3 @ ( nth_mat_complex @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_89_mem__Collect__eq,axiom,
! [A: vec_complex,P: vec_complex > $o] :
( ( member_vec_complex @ A @ ( collect_vec_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_90_mem__Collect__eq,axiom,
! [A: vec_a,P: vec_a > $o] :
( ( member_vec_a @ A @ ( collect_vec_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
! [A: mat_a,P: mat_a > $o] :
( ( member_mat_a @ A @ ( collect_mat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_92_mem__Collect__eq,axiom,
! [A: list_mat_complex,P: list_mat_complex > $o] :
( ( member279434397506102358omplex @ A @ ( collec136165848909103768omplex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
! [A: list_mat_a,P: list_mat_a > $o] :
( ( member_list_mat_a @ A @ ( collect_list_mat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
! [A: mat_complex,P: mat_complex > $o] :
( ( member_mat_complex @ A @ ( collect_mat_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_95_Collect__mem__eq,axiom,
! [A3: set_vec_complex] :
( ( collect_vec_complex
@ ^ [X3: vec_complex] : ( member_vec_complex @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_96_Collect__mem__eq,axiom,
! [A3: set_vec_a] :
( ( collect_vec_a
@ ^ [X3: vec_a] : ( member_vec_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_97_Collect__mem__eq,axiom,
! [A3: set_mat_a] :
( ( collect_mat_a
@ ^ [X3: mat_a] : ( member_mat_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_98_Collect__mem__eq,axiom,
! [A3: set_list_mat_complex] :
( ( collec136165848909103768omplex
@ ^ [X3: list_mat_complex] : ( member279434397506102358omplex @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_99_Collect__mem__eq,axiom,
! [A3: set_list_mat_a] :
( ( collect_list_mat_a
@ ^ [X3: list_mat_a] : ( member_list_mat_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_100_Collect__mem__eq,axiom,
! [A3: set_mat_complex] :
( ( collect_mat_complex
@ ^ [X3: mat_complex] : ( member_mat_complex @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_101_Collect__cong,axiom,
! [P: mat_complex > $o,Q: mat_complex > $o] :
( ! [X4: mat_complex] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_mat_complex @ P )
= ( collect_mat_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_102_step__2__main__dims__main,axiom,
! [N: nat,J: nat,A3: mat_complex] :
( ( ( dim_row_complex @ ( jordan6916311984355858983omplex @ N @ J @ A3 ) )
= ( dim_row_complex @ A3 ) )
& ( ( dim_col_complex @ ( jordan6916311984355858983omplex @ N @ J @ A3 ) )
= ( dim_col_complex @ A3 ) ) ) ).
% step_2_main_dims_main
thf(fact_103_nth__butlast,axiom,
! [N: nat,Xs: list_complex] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ ( butlast_complex @ Xs ) ) )
=> ( ( nth_complex @ ( butlast_complex @ Xs ) @ N )
= ( nth_complex @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_104_nth__butlast,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ ( butlast_a @ Xs ) ) )
=> ( ( nth_a @ ( butlast_a @ Xs ) @ N )
= ( nth_a @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_105_nth__butlast,axiom,
! [N: nat,Xs: list_vec_complex] :
( ( ord_less_nat @ N @ ( size_s1158823550072163597omplex @ ( butlast_vec_complex @ Xs ) ) )
=> ( ( nth_vec_complex @ ( butlast_vec_complex @ Xs ) @ N )
= ( nth_vec_complex @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_106_nth__butlast,axiom,
! [N: nat,Xs: list_vec_a] :
( ( ord_less_nat @ N @ ( size_size_list_vec_a @ ( butlast_vec_a @ Xs ) ) )
=> ( ( nth_vec_a @ ( butlast_vec_a @ Xs ) @ N )
= ( nth_vec_a @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_107_nth__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_108_nth__butlast,axiom,
! [N: nat,Xs: list_mat_a] :
( ( ord_less_nat @ N @ ( size_size_list_mat_a @ ( butlast_mat_a @ Xs ) ) )
=> ( ( nth_mat_a @ ( butlast_mat_a @ Xs ) @ N )
= ( nth_mat_a @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_109_nth__butlast,axiom,
! [N: nat,Xs: list_mat_complex] :
( ( ord_less_nat @ N @ ( size_s5969786470865220249omplex @ ( butlast_mat_complex @ Xs ) ) )
=> ( ( nth_mat_complex @ ( butlast_mat_complex @ Xs ) @ N )
= ( nth_mat_complex @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_110_list__all__length,axiom,
( list_all_complex
= ( ^ [P3: complex > $o,Xs2: list_complex] :
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( P3 @ ( nth_complex @ Xs2 @ N3 ) ) ) ) ) ).
% list_all_length
thf(fact_111_list__all__length,axiom,
( list_all_a
= ( ^ [P3: a > $o,Xs2: list_a] :
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
=> ( P3 @ ( nth_a @ Xs2 @ N3 ) ) ) ) ) ).
% list_all_length
thf(fact_112_list__all__length,axiom,
( list_all_vec_complex
= ( ^ [P3: vec_complex > $o,Xs2: list_vec_complex] :
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s1158823550072163597omplex @ Xs2 ) )
=> ( P3 @ ( nth_vec_complex @ Xs2 @ N3 ) ) ) ) ) ).
% list_all_length
thf(fact_113_list__all__length,axiom,
( list_all_vec_a
= ( ^ [P3: vec_a > $o,Xs2: list_vec_a] :
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_vec_a @ Xs2 ) )
=> ( P3 @ ( nth_vec_a @ Xs2 @ N3 ) ) ) ) ) ).
% list_all_length
thf(fact_114_list__all__length,axiom,
( list_all_nat
= ( ^ [P3: nat > $o,Xs2: list_nat] :
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
=> ( P3 @ ( nth_nat @ Xs2 @ N3 ) ) ) ) ) ).
% list_all_length
thf(fact_115_list__all__length,axiom,
( list_all_mat_a
= ( ^ [P3: mat_a > $o,Xs2: list_mat_a] :
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_mat_a @ Xs2 ) )
=> ( P3 @ ( nth_mat_a @ Xs2 @ N3 ) ) ) ) ) ).
% list_all_length
thf(fact_116_list__all__length,axiom,
( list_all_mat_complex
= ( ^ [P3: mat_complex > $o,Xs2: list_mat_complex] :
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s5969786470865220249omplex @ Xs2 ) )
=> ( P3 @ ( nth_mat_complex @ Xs2 @ N3 ) ) ) ) ) ).
% list_all_length
thf(fact_117_step__3__c__inner__loop__dims__main,axiom,
! [Val: complex,L: nat,I4: nat,J: nat,A3: mat_complex] :
( ( ( dim_row_complex @ ( jordan7656109678144820486omplex @ Val @ L @ I4 @ J @ A3 ) )
= ( dim_row_complex @ A3 ) )
& ( ( dim_col_complex @ ( jordan7656109678144820486omplex @ Val @ L @ I4 @ J @ A3 ) )
= ( dim_col_complex @ A3 ) ) ) ).
% step_3_c_inner_loop_dims_main
thf(fact_118_step__3__main__dims__main,axiom,
! [N: nat,K: nat,A3: mat_complex] :
( ( ( dim_row_complex @ ( jordan4702481308941288104omplex @ N @ K @ A3 ) )
= ( dim_row_complex @ A3 ) )
& ( ( dim_col_complex @ ( jordan4702481308941288104omplex @ N @ K @ A3 ) )
= ( dim_col_complex @ A3 ) ) ) ).
% step_3_main_dims_main
thf(fact_119_square__mat_Osimps,axiom,
( square7091239463517965744omplex
= ( ^ [A4: mat_mat_complex] :
( ( dim_col_mat_complex @ A4 )
= ( dim_row_mat_complex @ A4 ) ) ) ) ).
% square_mat.simps
thf(fact_120_square__mat_Osimps,axiom,
( square_mat_mat_a
= ( ^ [A4: mat_mat_a] :
( ( dim_col_mat_a @ A4 )
= ( dim_row_mat_a @ A4 ) ) ) ) ).
% square_mat.simps
thf(fact_121_square__mat_Osimps,axiom,
( square_mat_a
= ( ^ [A4: mat_a] :
( ( dim_col_a @ A4 )
= ( dim_row_a @ A4 ) ) ) ) ).
% square_mat.simps
thf(fact_122_square__mat_Osimps,axiom,
( square_mat_complex
= ( ^ [A4: mat_complex] :
( ( dim_col_complex @ A4 )
= ( dim_row_complex @ A4 ) ) ) ) ).
% square_mat.simps
thf(fact_123_square__mat_Oelims_I1_J,axiom,
! [X2: mat_mat_complex,Y2: $o] :
( ( ( square7091239463517965744omplex @ X2 )
= Y2 )
=> ( Y2
= ( ( dim_col_mat_complex @ X2 )
= ( dim_row_mat_complex @ X2 ) ) ) ) ).
% square_mat.elims(1)
thf(fact_124_square__mat_Oelims_I1_J,axiom,
! [X2: mat_mat_a,Y2: $o] :
( ( ( square_mat_mat_a @ X2 )
= Y2 )
=> ( Y2
= ( ( dim_col_mat_a @ X2 )
= ( dim_row_mat_a @ X2 ) ) ) ) ).
% square_mat.elims(1)
thf(fact_125_square__mat_Oelims_I1_J,axiom,
! [X2: mat_a,Y2: $o] :
( ( ( square_mat_a @ X2 )
= Y2 )
=> ( Y2
= ( ( dim_col_a @ X2 )
= ( dim_row_a @ X2 ) ) ) ) ).
% square_mat.elims(1)
thf(fact_126_square__mat_Oelims_I1_J,axiom,
! [X2: mat_complex,Y2: $o] :
( ( ( square_mat_complex @ X2 )
= Y2 )
=> ( Y2
= ( ( dim_col_complex @ X2 )
= ( dim_row_complex @ X2 ) ) ) ) ).
% square_mat.elims(1)
thf(fact_127_square__mat_Oelims_I2_J,axiom,
! [X2: mat_mat_complex] :
( ( square7091239463517965744omplex @ X2 )
=> ( ( dim_col_mat_complex @ X2 )
= ( dim_row_mat_complex @ X2 ) ) ) ).
% square_mat.elims(2)
thf(fact_128_square__mat_Oelims_I2_J,axiom,
! [X2: mat_mat_a] :
( ( square_mat_mat_a @ X2 )
=> ( ( dim_col_mat_a @ X2 )
= ( dim_row_mat_a @ X2 ) ) ) ).
% square_mat.elims(2)
thf(fact_129_square__mat_Oelims_I2_J,axiom,
! [X2: mat_a] :
( ( square_mat_a @ X2 )
=> ( ( dim_col_a @ X2 )
= ( dim_row_a @ X2 ) ) ) ).
% square_mat.elims(2)
thf(fact_130_square__mat_Oelims_I2_J,axiom,
! [X2: mat_complex] :
( ( square_mat_complex @ X2 )
=> ( ( dim_col_complex @ X2 )
= ( dim_row_complex @ X2 ) ) ) ).
% square_mat.elims(2)
thf(fact_131_step__3__main__dims_I1_J,axiom,
! [N: nat,J: nat,A3: mat_complex] :
( ( dim_row_complex @ ( jordan4702481308941288104omplex @ N @ J @ A3 ) )
= ( dim_row_complex @ A3 ) ) ).
% step_3_main_dims(1)
thf(fact_132_step__3__c__inner__loop__dims_I1_J,axiom,
! [Val: complex,L: nat,I4: nat,J: nat,A3: mat_complex] :
( ( dim_row_complex @ ( jordan7656109678144820486omplex @ Val @ L @ I4 @ J @ A3 ) )
= ( dim_row_complex @ A3 ) ) ).
% step_3_c_inner_loop_dims(1)
thf(fact_133_step__3__main__dims_I2_J,axiom,
! [N: nat,J: nat,A3: mat_complex] :
( ( dim_col_complex @ ( jordan4702481308941288104omplex @ N @ J @ A3 ) )
= ( dim_col_complex @ A3 ) ) ).
% step_3_main_dims(2)
thf(fact_134_step__3__c__inner__loop__dims_I2_J,axiom,
! [Val: complex,L: nat,I4: nat,J: nat,A3: mat_complex] :
( ( dim_col_complex @ ( jordan7656109678144820486omplex @ Val @ L @ I4 @ J @ A3 ) )
= ( dim_col_complex @ A3 ) ) ).
% step_3_c_inner_loop_dims(2)
thf(fact_135_step__2__main__dims_I1_J,axiom,
! [N: nat,J: nat,A3: mat_complex] :
( ( dim_row_complex @ ( jordan6916311984355858983omplex @ N @ J @ A3 ) )
= ( dim_row_complex @ A3 ) ) ).
% step_2_main_dims(1)
thf(fact_136_step__2__main__dims_I2_J,axiom,
! [N: nat,J: nat,A3: mat_complex] :
( ( dim_col_complex @ ( jordan6916311984355858983omplex @ N @ J @ A3 ) )
= ( dim_col_complex @ A3 ) ) ).
% step_2_main_dims(2)
thf(fact_137_square__mat_Oelims_I3_J,axiom,
! [X2: mat_mat_complex] :
( ~ ( square7091239463517965744omplex @ X2 )
=> ( ( dim_col_mat_complex @ X2 )
!= ( dim_row_mat_complex @ X2 ) ) ) ).
% square_mat.elims(3)
thf(fact_138_square__mat_Oelims_I3_J,axiom,
! [X2: mat_mat_a] :
( ~ ( square_mat_mat_a @ X2 )
=> ( ( dim_col_mat_a @ X2 )
!= ( dim_row_mat_a @ X2 ) ) ) ).
% square_mat.elims(3)
thf(fact_139_square__mat_Oelims_I3_J,axiom,
! [X2: mat_a] :
( ~ ( square_mat_a @ X2 )
=> ( ( dim_col_a @ X2 )
!= ( dim_row_a @ X2 ) ) ) ).
% square_mat.elims(3)
thf(fact_140_square__mat_Oelims_I3_J,axiom,
! [X2: mat_complex] :
( ~ ( square_mat_complex @ X2 )
=> ( ( dim_col_complex @ X2 )
!= ( dim_row_complex @ X2 ) ) ) ).
% square_mat.elims(3)
thf(fact_141_uminus__mult__left__mat,axiom,
! [A3: mat_complex,B3: mat_complex] :
( ( ( dim_col_complex @ A3 )
= ( dim_row_complex @ B3 ) )
=> ( ( times_8009071140041733218omplex @ ( uminus467866341702955550omplex @ A3 ) @ B3 )
= ( uminus467866341702955550omplex @ ( times_8009071140041733218omplex @ A3 @ B3 ) ) ) ) ).
% uminus_mult_left_mat
thf(fact_142_uminus__mult__right__mat,axiom,
! [A3: mat_complex,B3: mat_complex] :
( ( ( dim_col_complex @ A3 )
= ( dim_row_complex @ B3 ) )
=> ( ( times_8009071140041733218omplex @ A3 @ ( uminus467866341702955550omplex @ B3 ) )
= ( uminus467866341702955550omplex @ ( times_8009071140041733218omplex @ A3 @ B3 ) ) ) ) ).
% uminus_mult_right_mat
thf(fact_143_step__3__a__dims__main,axiom,
! [I4: nat,J: nat,A3: mat_complex] :
( ( ( dim_row_complex @ ( jordan2858886415929732048omplex @ I4 @ J @ A3 ) )
= ( dim_row_complex @ A3 ) )
& ( ( dim_col_complex @ ( jordan2858886415929732048omplex @ I4 @ J @ A3 ) )
= ( dim_col_complex @ A3 ) ) ) ).
% step_3_a_dims_main
thf(fact_144_step__3__c__dims__main,axiom,
! [X2: complex,L: nat,K: nat,I4: list_P6011104703257516679at_nat,A3: mat_complex] :
( ( ( dim_row_complex @ ( jordan5343229918868201426omplex @ X2 @ L @ K @ I4 @ A3 ) )
= ( dim_row_complex @ A3 ) )
& ( ( dim_col_complex @ ( jordan5343229918868201426omplex @ X2 @ L @ K @ I4 @ A3 ) )
= ( dim_col_complex @ A3 ) ) ) ).
% step_3_c_dims_main
thf(fact_145_step__1__main__dims__main,axiom,
! [N: nat,I4: nat,J: nat,A3: mat_complex] :
( ( ( dim_row_complex @ ( jordan9130142659770429862omplex @ N @ I4 @ J @ A3 ) )
= ( dim_row_complex @ A3 ) )
& ( ( dim_col_complex @ ( jordan9130142659770429862omplex @ N @ I4 @ J @ A3 ) )
= ( dim_col_complex @ A3 ) ) ) ).
% step_1_main_dims_main
thf(fact_146_eq__rowI,axiom,
! [B3: mat_mat_complex,A3: mat_mat_complex] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_mat_complex @ B3 ) )
=> ( ( row_mat_complex @ A3 @ I2 )
= ( row_mat_complex @ B3 @ I2 ) ) )
=> ( ( ( dim_row_mat_complex @ A3 )
= ( dim_row_mat_complex @ B3 ) )
=> ( ( ( dim_col_mat_complex @ A3 )
= ( dim_col_mat_complex @ B3 ) )
=> ( A3 = B3 ) ) ) ) ).
% eq_rowI
thf(fact_147_eq__rowI,axiom,
! [B3: mat_mat_a,A3: mat_mat_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_mat_a @ B3 ) )
=> ( ( row_mat_a @ A3 @ I2 )
= ( row_mat_a @ B3 @ I2 ) ) )
=> ( ( ( dim_row_mat_a @ A3 )
= ( dim_row_mat_a @ B3 ) )
=> ( ( ( dim_col_mat_a @ A3 )
= ( dim_col_mat_a @ B3 ) )
=> ( A3 = B3 ) ) ) ) ).
% eq_rowI
thf(fact_148_eq__rowI,axiom,
! [B3: mat_a,A3: mat_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_a @ B3 ) )
=> ( ( row_a @ A3 @ I2 )
= ( row_a @ B3 @ I2 ) ) )
=> ( ( ( dim_row_a @ A3 )
= ( dim_row_a @ B3 ) )
=> ( ( ( dim_col_a @ A3 )
= ( dim_col_a @ B3 ) )
=> ( A3 = B3 ) ) ) ) ).
% eq_rowI
thf(fact_149_eq__rowI,axiom,
! [B3: mat_complex,A3: mat_complex] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_complex @ B3 ) )
=> ( ( row_complex @ A3 @ I2 )
= ( row_complex @ B3 @ I2 ) ) )
=> ( ( ( dim_row_complex @ A3 )
= ( dim_row_complex @ B3 ) )
=> ( ( ( dim_col_complex @ A3 )
= ( dim_col_complex @ B3 ) )
=> ( A3 = B3 ) ) ) ) ).
% eq_rowI
thf(fact_150_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_151_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_152_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_153_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_154_uminus__eq__mat,axiom,
! [A3: mat_complex,B3: mat_complex] :
( ( ( uminus467866341702955550omplex @ A3 )
= ( uminus467866341702955550omplex @ B3 ) )
= ( A3 = B3 ) ) ).
% uminus_eq_mat
thf(fact_155_uminus__uminus__mat,axiom,
! [A3: mat_complex] :
( ( uminus467866341702955550omplex @ ( uminus467866341702955550omplex @ A3 ) )
= A3 ) ).
% uminus_uminus_mat
thf(fact_156_index__uminus__mat_I2_J,axiom,
! [A3: mat_mat_complex] :
( ( dim_row_mat_complex @ ( uminus9210244920068684493omplex @ A3 ) )
= ( dim_row_mat_complex @ A3 ) ) ).
% index_uminus_mat(2)
thf(fact_157_index__uminus__mat_I2_J,axiom,
! [A3: mat_complex] :
( ( dim_row_complex @ ( uminus467866341702955550omplex @ A3 ) )
= ( dim_row_complex @ A3 ) ) ).
% index_uminus_mat(2)
thf(fact_158_index__uminus__mat_I3_J,axiom,
! [A3: mat_complex] :
( ( dim_col_complex @ ( uminus467866341702955550omplex @ A3 ) )
= ( dim_col_complex @ A3 ) ) ).
% index_uminus_mat(3)
thf(fact_159_step__1__main__dims_I1_J,axiom,
! [N: nat,I4: nat,J: nat,A3: mat_complex] :
( ( dim_row_complex @ ( jordan9130142659770429862omplex @ N @ I4 @ J @ A3 ) )
= ( dim_row_complex @ A3 ) ) ).
% step_1_main_dims(1)
thf(fact_160_step__3__c__dims_I1_J,axiom,
! [X2: complex,L: nat,K: nat,I4: list_P6011104703257516679at_nat,A3: mat_complex] :
( ( dim_row_complex @ ( jordan5343229918868201426omplex @ X2 @ L @ K @ I4 @ A3 ) )
= ( dim_row_complex @ A3 ) ) ).
% step_3_c_dims(1)
thf(fact_161_step__3__a__dims_I1_J,axiom,
! [I4: nat,J: nat,A3: mat_complex] :
( ( dim_row_complex @ ( jordan2858886415929732048omplex @ I4 @ J @ A3 ) )
= ( dim_row_complex @ A3 ) ) ).
% step_3_a_dims(1)
thf(fact_162_step__1__main__dims_I2_J,axiom,
! [N: nat,I4: nat,J: nat,A3: mat_complex] :
( ( dim_col_complex @ ( jordan9130142659770429862omplex @ N @ I4 @ J @ A3 ) )
= ( dim_col_complex @ A3 ) ) ).
% step_1_main_dims(2)
thf(fact_163_step__3__c__dims_I2_J,axiom,
! [X2: complex,L: nat,K: nat,I4: list_P6011104703257516679at_nat,A3: mat_complex] :
( ( dim_col_complex @ ( jordan5343229918868201426omplex @ X2 @ L @ K @ I4 @ A3 ) )
= ( dim_col_complex @ A3 ) ) ).
% step_3_c_dims(2)
thf(fact_164_step__3__a__dims_I2_J,axiom,
! [I4: nat,J: nat,A3: mat_complex] :
( ( dim_col_complex @ ( jordan2858886415929732048omplex @ I4 @ J @ A3 ) )
= ( dim_col_complex @ A3 ) ) ).
% step_3_a_dims(2)
thf(fact_165_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_166_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_167_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_168_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_169_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_170_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_171_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_172_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_173_compl__less__compl__iff,axiom,
! [X2: set_mat_complex,Y2: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ ( uminus5815530220087396478omplex @ X2 ) @ ( uminus5815530220087396478omplex @ Y2 ) )
= ( ord_le5598786136212072115omplex @ Y2 @ X2 ) ) ).
% compl_less_compl_iff
thf(fact_174_compl__less__swap2,axiom,
! [Y2: set_mat_complex,X2: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ ( uminus5815530220087396478omplex @ Y2 ) @ X2 )
=> ( ord_le5598786136212072115omplex @ ( uminus5815530220087396478omplex @ X2 ) @ Y2 ) ) ).
% compl_less_swap2
thf(fact_175_compl__less__swap1,axiom,
! [Y2: set_mat_complex,X2: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ Y2 @ ( uminus5815530220087396478omplex @ X2 ) )
=> ( ord_le5598786136212072115omplex @ X2 @ ( uminus5815530220087396478omplex @ Y2 ) ) ) ).
% compl_less_swap1
thf(fact_176_verit__comp__simplify_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify(1)
thf(fact_177_double__compl,axiom,
! [X2: set_mat_complex] :
( ( uminus5815530220087396478omplex @ ( uminus5815530220087396478omplex @ X2 ) )
= X2 ) ).
% double_compl
thf(fact_178_compl__eq__compl__iff,axiom,
! [X2: set_mat_complex,Y2: set_mat_complex] :
( ( ( uminus5815530220087396478omplex @ X2 )
= ( uminus5815530220087396478omplex @ Y2 ) )
= ( X2 = Y2 ) ) ).
% compl_eq_compl_iff
thf(fact_179_row__uminus,axiom,
! [I4: nat,A3: mat_mat_complex] :
( ( ord_less_nat @ I4 @ ( dim_row_mat_complex @ A3 ) )
=> ( ( row_mat_complex @ ( uminus9210244920068684493omplex @ A3 ) @ I4 )
= ( uminus1815796379637173593omplex @ ( row_mat_complex @ A3 @ I4 ) ) ) ) ).
% row_uminus
thf(fact_180_row__uminus,axiom,
! [I4: nat,A3: mat_complex] :
( ( ord_less_nat @ I4 @ ( dim_row_complex @ A3 ) )
=> ( ( row_complex @ ( uminus467866341702955550omplex @ A3 ) @ I4 )
= ( uminus4447292074486253202omplex @ ( row_complex @ A3 @ I4 ) ) ) ) ).
% row_uminus
thf(fact_181_nth__rows,axiom,
! [I4: nat,A3: mat_mat_complex] :
( ( ord_less_nat @ I4 @ ( dim_row_mat_complex @ A3 ) )
=> ( ( nth_vec_mat_complex @ ( rows_mat_complex @ A3 ) @ I4 )
= ( row_mat_complex @ A3 @ I4 ) ) ) ).
% nth_rows
thf(fact_182_nth__rows,axiom,
! [I4: nat,A3: mat_mat_a] :
( ( ord_less_nat @ I4 @ ( dim_row_mat_a @ A3 ) )
=> ( ( nth_vec_mat_a @ ( rows_mat_a @ A3 ) @ I4 )
= ( row_mat_a @ A3 @ I4 ) ) ) ).
% nth_rows
thf(fact_183_nth__rows,axiom,
! [I4: nat,A3: mat_a] :
( ( ord_less_nat @ I4 @ ( dim_row_a @ A3 ) )
=> ( ( nth_vec_a @ ( rows_a @ A3 ) @ I4 )
= ( row_a @ A3 @ I4 ) ) ) ).
% nth_rows
thf(fact_184_nth__rows,axiom,
! [I4: nat,A3: mat_complex] :
( ( ord_less_nat @ I4 @ ( dim_row_complex @ A3 ) )
=> ( ( nth_vec_complex @ ( rows_complex @ A3 ) @ I4 )
= ( row_complex @ A3 @ I4 ) ) ) ).
% nth_rows
thf(fact_185_basic__trans__rules_I28_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% basic_trans_rules(28)
thf(fact_186_basic__trans__rules_I27_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% basic_trans_rules(27)
thf(fact_187_basic__trans__rules_I20_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% basic_trans_rules(20)
thf(fact_188_basic__trans__rules_I19_J,axiom,
! [X2: nat,Y2: nat,Z4: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z4 )
=> ( ord_less_nat @ X2 @ Z4 ) ) ) ).
% basic_trans_rules(19)
thf(fact_189_uminus__eq__vec,axiom,
! [V: vec_complex,W: vec_complex] :
( ( ( uminus4447292074486253202omplex @ V )
= ( uminus4447292074486253202omplex @ W ) )
= ( V = W ) ) ).
% uminus_eq_vec
thf(fact_190_uminus__uminus__vec,axiom,
! [V: vec_complex] :
( ( uminus4447292074486253202omplex @ ( uminus4447292074486253202omplex @ V ) )
= V ) ).
% uminus_uminus_vec
thf(fact_191_length__rows,axiom,
! [A3: mat_mat_complex] :
( ( size_s2077990086586600628omplex @ ( rows_mat_complex @ A3 ) )
= ( dim_row_mat_complex @ A3 ) ) ).
% length_rows
thf(fact_192_length__rows,axiom,
! [A3: mat_mat_a] :
( ( size_s5765634329853218618_mat_a @ ( rows_mat_a @ A3 ) )
= ( dim_row_mat_a @ A3 ) ) ).
% length_rows
thf(fact_193_length__rows,axiom,
! [A3: mat_a] :
( ( size_size_list_vec_a @ ( rows_a @ A3 ) )
= ( dim_row_a @ A3 ) ) ).
% length_rows
thf(fact_194_length__rows,axiom,
! [A3: mat_complex] :
( ( size_s1158823550072163597omplex @ ( rows_complex @ A3 ) )
= ( dim_row_complex @ A3 ) ) ).
% length_rows
thf(fact_195_basic__trans__rules_I1_J,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(1)
thf(fact_196_basic__trans__rules_I2_J,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(2)
thf(fact_197_basic__trans__rules_I11_J,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(11)
thf(fact_198_basic__trans__rules_I12_J,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(12)
thf(fact_199_col__uminus,axiom,
! [I4: nat,A3: mat_complex] :
( ( ord_less_nat @ I4 @ ( dim_col_complex @ A3 ) )
=> ( ( col_complex @ ( uminus467866341702955550omplex @ A3 ) @ I4 )
= ( uminus4447292074486253202omplex @ ( col_complex @ A3 @ I4 ) ) ) ) ).
% col_uminus
thf(fact_200_mat__of__row__uminus,axiom,
! [V: vec_complex] :
( ( mat_of_row_complex @ ( uminus4447292074486253202omplex @ V ) )
= ( uminus467866341702955550omplex @ ( mat_of_row_complex @ V ) ) ) ).
% mat_of_row_uminus
thf(fact_201_mat__of__rows__rows,axiom,
! [A3: mat_a] :
( ( mat_of_rows_a @ ( dim_col_a @ A3 ) @ ( rows_a @ A3 ) )
= A3 ) ).
% mat_of_rows_rows
thf(fact_202_mat__of__rows__rows,axiom,
! [A3: mat_complex] :
( ( mat_of_rows_complex @ ( dim_col_complex @ A3 ) @ ( rows_complex @ A3 ) )
= A3 ) ).
% mat_of_rows_rows
thf(fact_203_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_204_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_205_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_206_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_207_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_208_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_209_mat__of__rows__carrier_I2_J,axiom,
! [N: nat,Vs: list_vec_mat_complex] :
( ( dim_row_mat_complex @ ( mat_of2504328217136560291omplex @ N @ Vs ) )
= ( size_s2077990086586600628omplex @ Vs ) ) ).
% mat_of_rows_carrier(2)
thf(fact_210_mat__of__rows__carrier_I2_J,axiom,
! [N: nat,Vs: list_vec_mat_a] :
( ( dim_row_mat_a @ ( mat_of_rows_mat_a @ N @ Vs ) )
= ( size_s5765634329853218618_mat_a @ Vs ) ) ).
% mat_of_rows_carrier(2)
thf(fact_211_mat__of__rows__carrier_I2_J,axiom,
! [N: nat,Vs: list_vec_a] :
( ( dim_row_a @ ( mat_of_rows_a @ N @ Vs ) )
= ( size_size_list_vec_a @ Vs ) ) ).
% mat_of_rows_carrier(2)
thf(fact_212_mat__of__rows__carrier_I2_J,axiom,
! [N: nat,Vs: list_vec_complex] :
( ( dim_row_complex @ ( mat_of_rows_complex @ N @ Vs ) )
= ( size_s1158823550072163597omplex @ Vs ) ) ).
% mat_of_rows_carrier(2)
thf(fact_213_mat__of__rows__carrier_I3_J,axiom,
! [N: nat,Vs: list_vec_a] :
( ( dim_col_a @ ( mat_of_rows_a @ N @ Vs ) )
= N ) ).
% mat_of_rows_carrier(3)
thf(fact_214_mat__of__rows__carrier_I3_J,axiom,
! [N: nat,Vs: list_vec_complex] :
( ( dim_col_complex @ ( mat_of_rows_complex @ N @ Vs ) )
= N ) ).
% mat_of_rows_carrier(3)
thf(fact_215_Matrix_Omat__col__eqI,axiom,
! [B3: mat_mat_complex,A3: mat_mat_complex] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( dim_col_mat_complex @ B3 ) )
=> ( ( col_mat_complex @ A3 @ I2 )
= ( col_mat_complex @ B3 @ I2 ) ) )
=> ( ( ( dim_row_mat_complex @ A3 )
= ( dim_row_mat_complex @ B3 ) )
=> ( ( ( dim_col_mat_complex @ A3 )
= ( dim_col_mat_complex @ B3 ) )
=> ( A3 = B3 ) ) ) ) ).
% Matrix.mat_col_eqI
thf(fact_216_Matrix_Omat__col__eqI,axiom,
! [B3: mat_mat_a,A3: mat_mat_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( dim_col_mat_a @ B3 ) )
=> ( ( col_mat_a @ A3 @ I2 )
= ( col_mat_a @ B3 @ I2 ) ) )
=> ( ( ( dim_row_mat_a @ A3 )
= ( dim_row_mat_a @ B3 ) )
=> ( ( ( dim_col_mat_a @ A3 )
= ( dim_col_mat_a @ B3 ) )
=> ( A3 = B3 ) ) ) ) ).
% Matrix.mat_col_eqI
thf(fact_217_Matrix_Omat__col__eqI,axiom,
! [B3: mat_a,A3: mat_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( dim_col_a @ B3 ) )
=> ( ( col_a @ A3 @ I2 )
= ( col_a @ B3 @ I2 ) ) )
=> ( ( ( dim_row_a @ A3 )
= ( dim_row_a @ B3 ) )
=> ( ( ( dim_col_a @ A3 )
= ( dim_col_a @ B3 ) )
=> ( A3 = B3 ) ) ) ) ).
% Matrix.mat_col_eqI
thf(fact_218_Matrix_Omat__col__eqI,axiom,
! [B3: mat_complex,A3: mat_complex] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( dim_col_complex @ B3 ) )
=> ( ( col_complex @ A3 @ I2 )
= ( col_complex @ B3 @ I2 ) ) )
=> ( ( ( dim_row_complex @ A3 )
= ( dim_row_complex @ B3 ) )
=> ( ( ( dim_col_complex @ A3 )
= ( dim_col_complex @ B3 ) )
=> ( A3 = B3 ) ) ) ) ).
% Matrix.mat_col_eqI
thf(fact_219_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_220_neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% neqE
thf(fact_221_neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% neq_iff
thf(fact_222_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_223_less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% less_asym
thf(fact_224_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_225_less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% less_irrefl
thf(fact_226_order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% order.irrefl
thf(fact_227_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X4 )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_228_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_229_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_230_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_231_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X6: nat] : ( P5 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_232_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B4: nat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_233_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_234_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_235_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_236_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_237_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_238_cols__nth,axiom,
! [I4: nat,A3: mat_a] :
( ( ord_less_nat @ I4 @ ( dim_col_a @ A3 ) )
=> ( ( nth_vec_a @ ( cols_a @ A3 ) @ I4 )
= ( col_a @ A3 @ I4 ) ) ) ).
% cols_nth
thf(fact_239_cols__nth,axiom,
! [I4: nat,A3: mat_complex] :
( ( ord_less_nat @ I4 @ ( dim_col_complex @ A3 ) )
=> ( ( nth_vec_complex @ ( cols_complex @ A3 ) @ I4 )
= ( col_complex @ A3 @ I4 ) ) ) ).
% cols_nth
thf(fact_240_mat__of__rows__row,axiom,
! [I4: nat,Vs: list_vec_complex,N: nat] :
( ( ord_less_nat @ I4 @ ( size_s1158823550072163597omplex @ Vs ) )
=> ( ( member_vec_complex @ ( nth_vec_complex @ Vs @ I4 ) @ ( carrier_vec_complex @ N ) )
=> ( ( row_complex @ ( mat_of_rows_complex @ N @ Vs ) @ I4 )
= ( nth_vec_complex @ Vs @ I4 ) ) ) ) ).
% mat_of_rows_row
thf(fact_241_mat__of__rows__row,axiom,
! [I4: nat,Vs: list_vec_a,N: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_vec_a @ Vs ) )
=> ( ( member_vec_a @ ( nth_vec_a @ Vs @ I4 ) @ ( carrier_vec_a @ N ) )
=> ( ( row_a @ ( mat_of_rows_a @ N @ Vs ) @ I4 )
= ( nth_vec_a @ Vs @ I4 ) ) ) ) ).
% mat_of_rows_row
thf(fact_242_Matrix_Orow__transpose,axiom,
! [J: nat,A3: mat_a] :
( ( ord_less_nat @ J @ ( dim_col_a @ A3 ) )
=> ( ( row_a @ ( transpose_mat_a @ A3 ) @ J )
= ( col_a @ A3 @ J ) ) ) ).
% Matrix.row_transpose
thf(fact_243_Matrix_Orow__transpose,axiom,
! [J: nat,A3: mat_complex] :
( ( ord_less_nat @ J @ ( dim_col_complex @ A3 ) )
=> ( ( row_complex @ ( transp3074176993011536131omplex @ A3 ) @ J )
= ( col_complex @ A3 @ J ) ) ) ).
% Matrix.row_transpose
thf(fact_244_col__transpose,axiom,
! [I4: nat,A3: mat_mat_complex] :
( ( ord_less_nat @ I4 @ ( dim_row_mat_complex @ A3 ) )
=> ( ( col_mat_complex @ ( transp4906945491372815122omplex @ A3 ) @ I4 )
= ( row_mat_complex @ A3 @ I4 ) ) ) ).
% col_transpose
thf(fact_245_col__transpose,axiom,
! [I4: nat,A3: mat_mat_a] :
( ( ord_less_nat @ I4 @ ( dim_row_mat_a @ A3 ) )
=> ( ( col_mat_a @ ( transpose_mat_mat_a @ A3 ) @ I4 )
= ( row_mat_a @ A3 @ I4 ) ) ) ).
% col_transpose
thf(fact_246_col__transpose,axiom,
! [I4: nat,A3: mat_a] :
( ( ord_less_nat @ I4 @ ( dim_row_a @ A3 ) )
=> ( ( col_a @ ( transpose_mat_a @ A3 ) @ I4 )
= ( row_a @ A3 @ I4 ) ) ) ).
% col_transpose
thf(fact_247_col__transpose,axiom,
! [I4: nat,A3: mat_complex] :
( ( ord_less_nat @ I4 @ ( dim_row_complex @ A3 ) )
=> ( ( col_complex @ ( transp3074176993011536131omplex @ A3 ) @ I4 )
= ( row_complex @ A3 @ I4 ) ) ) ).
% col_transpose
thf(fact_248_cols__length,axiom,
! [A3: mat_a] :
( ( size_size_list_vec_a @ ( cols_a @ A3 ) )
= ( dim_col_a @ A3 ) ) ).
% cols_length
thf(fact_249_cols__length,axiom,
! [A3: mat_complex] :
( ( size_s1158823550072163597omplex @ ( cols_complex @ A3 ) )
= ( dim_col_complex @ A3 ) ) ).
% cols_length
thf(fact_250_Quantum_Orow__transpose,axiom,
! [M4: mat_a,N: nat,I4: nat] :
( ( ( dim_col_a @ M4 )
= N )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( row_a @ ( transpose_mat_a @ M4 ) @ I4 )
= ( col_a @ M4 @ I4 ) ) ) ) ).
% Quantum.row_transpose
thf(fact_251_Quantum_Orow__transpose,axiom,
! [M4: mat_complex,N: nat,I4: nat] :
( ( ( dim_col_complex @ M4 )
= N )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( row_complex @ ( transp3074176993011536131omplex @ M4 ) @ I4 )
= ( col_complex @ M4 @ I4 ) ) ) ) ).
% Quantum.row_transpose
thf(fact_252_col__tranpose,axiom,
! [M4: mat_mat_complex,N: nat,I4: nat] :
( ( ( dim_row_mat_complex @ M4 )
= N )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( col_mat_complex @ ( transp4906945491372815122omplex @ M4 ) @ I4 )
= ( row_mat_complex @ M4 @ I4 ) ) ) ) ).
% col_tranpose
thf(fact_253_col__tranpose,axiom,
! [M4: mat_mat_a,N: nat,I4: nat] :
( ( ( dim_row_mat_a @ M4 )
= N )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( col_mat_a @ ( transpose_mat_mat_a @ M4 ) @ I4 )
= ( row_mat_a @ M4 @ I4 ) ) ) ) ).
% col_tranpose
thf(fact_254_col__tranpose,axiom,
! [M4: mat_a,N: nat,I4: nat] :
( ( ( dim_row_a @ M4 )
= N )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( col_a @ ( transpose_mat_a @ M4 ) @ I4 )
= ( row_a @ M4 @ I4 ) ) ) ) ).
% col_tranpose
thf(fact_255_col__tranpose,axiom,
! [M4: mat_complex,N: nat,I4: nat] :
( ( ( dim_row_complex @ M4 )
= N )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( col_complex @ ( transp3074176993011536131omplex @ M4 ) @ I4 )
= ( row_complex @ M4 @ I4 ) ) ) ) ).
% col_tranpose
thf(fact_256_transpose__mat__eq,axiom,
! [A3: mat_complex,B3: mat_complex] :
( ( ( transp3074176993011536131omplex @ A3 )
= ( transp3074176993011536131omplex @ B3 ) )
= ( A3 = B3 ) ) ).
% transpose_mat_eq
thf(fact_257_transpose__mat__eq,axiom,
! [A3: mat_a,B3: mat_a] :
( ( ( transpose_mat_a @ A3 )
= ( transpose_mat_a @ B3 ) )
= ( A3 = B3 ) ) ).
% transpose_mat_eq
thf(fact_258_Matrix_Otranspose__transpose,axiom,
! [A3: mat_complex] :
( ( transp3074176993011536131omplex @ ( transp3074176993011536131omplex @ A3 ) )
= A3 ) ).
% Matrix.transpose_transpose
thf(fact_259_Matrix_Otranspose__transpose,axiom,
! [A3: mat_a] :
( ( transpose_mat_a @ ( transpose_mat_a @ A3 ) )
= A3 ) ).
% Matrix.transpose_transpose
thf(fact_260_cols__transpose,axiom,
! [A3: mat_complex] :
( ( cols_complex @ ( transp3074176993011536131omplex @ A3 ) )
= ( rows_complex @ A3 ) ) ).
% cols_transpose
thf(fact_261_cols__transpose,axiom,
! [A3: mat_a] :
( ( cols_a @ ( transpose_mat_a @ A3 ) )
= ( rows_a @ A3 ) ) ).
% cols_transpose
thf(fact_262_Matrix_Orows__transpose,axiom,
! [A3: mat_complex] :
( ( rows_complex @ ( transp3074176993011536131omplex @ A3 ) )
= ( cols_complex @ A3 ) ) ).
% Matrix.rows_transpose
thf(fact_263_Matrix_Orows__transpose,axiom,
! [A3: mat_a] :
( ( rows_a @ ( transpose_mat_a @ A3 ) )
= ( cols_a @ A3 ) ) ).
% Matrix.rows_transpose
thf(fact_264_uminus__carrier__vec,axiom,
! [V: vec_complex,N: nat] :
( ( member_vec_complex @ ( uminus4447292074486253202omplex @ V ) @ ( carrier_vec_complex @ N ) )
= ( member_vec_complex @ V @ ( carrier_vec_complex @ N ) ) ) ).
% uminus_carrier_vec
thf(fact_265_transpose__uminus,axiom,
! [A3: mat_complex] :
( ( transp3074176993011536131omplex @ ( uminus467866341702955550omplex @ A3 ) )
= ( uminus467866341702955550omplex @ ( transp3074176993011536131omplex @ A3 ) ) ) ).
% transpose_uminus
thf(fact_266_col__dim,axiom,
! [A3: mat_mat_complex,I4: nat] : ( member358399664158425767omplex @ ( col_mat_complex @ A3 @ I4 ) @ ( carrie1048208924330543741omplex @ ( dim_row_mat_complex @ A3 ) ) ) ).
% col_dim
thf(fact_267_col__dim,axiom,
! [A3: mat_mat_a,I4: nat] : ( member_vec_mat_a @ ( col_mat_a @ A3 @ I4 ) @ ( carrier_vec_mat_a @ ( dim_row_mat_a @ A3 ) ) ) ).
% col_dim
thf(fact_268_col__dim,axiom,
! [A3: mat_a,I4: nat] : ( member_vec_a @ ( col_a @ A3 @ I4 ) @ ( carrier_vec_a @ ( dim_row_a @ A3 ) ) ) ).
% col_dim
thf(fact_269_col__dim,axiom,
! [A3: mat_complex,I4: nat] : ( member_vec_complex @ ( col_complex @ A3 @ I4 ) @ ( carrier_vec_complex @ ( dim_row_complex @ A3 ) ) ) ).
% col_dim
thf(fact_270_row__carrier,axiom,
! [A3: mat_a,I4: nat] : ( member_vec_a @ ( row_a @ A3 @ I4 ) @ ( carrier_vec_a @ ( dim_col_a @ A3 ) ) ) ).
% row_carrier
thf(fact_271_row__carrier,axiom,
! [A3: mat_complex,I4: nat] : ( member_vec_complex @ ( row_complex @ A3 @ I4 ) @ ( carrier_vec_complex @ ( dim_col_complex @ A3 ) ) ) ).
% row_carrier
thf(fact_272_index__transpose__mat_I2_J,axiom,
! [A3: mat_mat_complex] :
( ( dim_row_mat_complex @ ( transp4906945491372815122omplex @ A3 ) )
= ( dim_col_mat_complex @ A3 ) ) ).
% index_transpose_mat(2)
thf(fact_273_index__transpose__mat_I2_J,axiom,
! [A3: mat_mat_a] :
( ( dim_row_mat_a @ ( transpose_mat_mat_a @ A3 ) )
= ( dim_col_mat_a @ A3 ) ) ).
% index_transpose_mat(2)
thf(fact_274_index__transpose__mat_I2_J,axiom,
! [A3: mat_a] :
( ( dim_row_a @ ( transpose_mat_a @ A3 ) )
= ( dim_col_a @ A3 ) ) ).
% index_transpose_mat(2)
thf(fact_275_index__transpose__mat_I2_J,axiom,
! [A3: mat_complex] :
( ( dim_row_complex @ ( transp3074176993011536131omplex @ A3 ) )
= ( dim_col_complex @ A3 ) ) ).
% index_transpose_mat(2)
thf(fact_276_index__transpose__mat_I3_J,axiom,
! [A3: mat_mat_complex] :
( ( dim_col_mat_complex @ ( transp4906945491372815122omplex @ A3 ) )
= ( dim_row_mat_complex @ A3 ) ) ).
% index_transpose_mat(3)
thf(fact_277_index__transpose__mat_I3_J,axiom,
! [A3: mat_mat_a] :
( ( dim_col_mat_a @ ( transpose_mat_mat_a @ A3 ) )
= ( dim_row_mat_a @ A3 ) ) ).
% index_transpose_mat(3)
thf(fact_278_index__transpose__mat_I3_J,axiom,
! [A3: mat_a] :
( ( dim_col_a @ ( transpose_mat_a @ A3 ) )
= ( dim_row_a @ A3 ) ) ).
% index_transpose_mat(3)
thf(fact_279_index__transpose__mat_I3_J,axiom,
! [A3: mat_complex] :
( ( dim_col_complex @ ( transp3074176993011536131omplex @ A3 ) )
= ( dim_row_complex @ A3 ) ) ).
% index_transpose_mat(3)
thf(fact_280_col__mat__of__cols,axiom,
! [J: nat,Vs: list_vec_complex,N: nat] :
( ( ord_less_nat @ J @ ( size_s1158823550072163597omplex @ Vs ) )
=> ( ( member_vec_complex @ ( nth_vec_complex @ Vs @ J ) @ ( carrier_vec_complex @ N ) )
=> ( ( col_complex @ ( mat_of_cols_complex @ N @ Vs ) @ J )
= ( nth_vec_complex @ Vs @ J ) ) ) ) ).
% col_mat_of_cols
thf(fact_281_col__mat__of__cols,axiom,
! [J: nat,Vs: list_vec_a,N: nat] :
( ( ord_less_nat @ J @ ( size_size_list_vec_a @ Vs ) )
=> ( ( member_vec_a @ ( nth_vec_a @ Vs @ J ) @ ( carrier_vec_a @ N ) )
=> ( ( col_a @ ( mat_of_cols_a @ N @ Vs ) @ J )
= ( nth_vec_a @ Vs @ J ) ) ) ) ).
% col_mat_of_cols
thf(fact_282_diag__mat__transpose,axiom,
! [A3: mat_mat_complex] :
( ( ( dim_row_mat_complex @ A3 )
= ( dim_col_mat_complex @ A3 ) )
=> ( ( diag_mat_mat_complex @ ( transp4906945491372815122omplex @ A3 ) )
= ( diag_mat_mat_complex @ A3 ) ) ) ).
% diag_mat_transpose
thf(fact_283_diag__mat__transpose,axiom,
! [A3: mat_mat_a] :
( ( ( dim_row_mat_a @ A3 )
= ( dim_col_mat_a @ A3 ) )
=> ( ( diag_mat_mat_a @ ( transpose_mat_mat_a @ A3 ) )
= ( diag_mat_mat_a @ A3 ) ) ) ).
% diag_mat_transpose
thf(fact_284_diag__mat__transpose,axiom,
! [A3: mat_a] :
( ( ( dim_row_a @ A3 )
= ( dim_col_a @ A3 ) )
=> ( ( diag_mat_a @ ( transpose_mat_a @ A3 ) )
= ( diag_mat_a @ A3 ) ) ) ).
% diag_mat_transpose
thf(fact_285_diag__mat__transpose,axiom,
! [A3: mat_complex] :
( ( ( dim_row_complex @ A3 )
= ( dim_col_complex @ A3 ) )
=> ( ( diag_mat_complex @ ( transp3074176993011536131omplex @ A3 ) )
= ( diag_mat_complex @ A3 ) ) ) ).
% diag_mat_transpose
thf(fact_286_vec__last__carrier,axiom,
! [V: vec_complex,N: nat] : ( member_vec_complex @ ( vec_last_complex @ V @ N ) @ ( carrier_vec_complex @ N ) ) ).
% vec_last_carrier
thf(fact_287_vec__last__carrier,axiom,
! [V: vec_a,N: nat] : ( member_vec_a @ ( vec_last_a @ V @ N ) @ ( carrier_vec_a @ N ) ) ).
% vec_last_carrier
thf(fact_288_vec__first__carrier,axiom,
! [V: vec_complex,N: nat] : ( member_vec_complex @ ( vec_first_complex @ V @ N ) @ ( carrier_vec_complex @ N ) ) ).
% vec_first_carrier
thf(fact_289_vec__first__carrier,axiom,
! [V: vec_a,N: nat] : ( member_vec_a @ ( vec_first_a @ V @ N ) @ ( carrier_vec_a @ N ) ) ).
% vec_first_carrier
thf(fact_290_row__carrier__vec,axiom,
! [I4: nat,Nr: nat,A3: mat_a,Nc: nat] :
( ( ord_less_nat @ I4 @ Nr )
=> ( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_vec_a @ ( row_a @ A3 @ I4 ) @ ( carrier_vec_a @ Nc ) ) ) ) ).
% row_carrier_vec
thf(fact_291_row__carrier__vec,axiom,
! [I4: nat,Nr: nat,A3: mat_complex,Nc: nat] :
( ( ord_less_nat @ I4 @ Nr )
=> ( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_vec_complex @ ( row_complex @ A3 @ I4 ) @ ( carrier_vec_complex @ Nc ) ) ) ) ).
% row_carrier_vec
thf(fact_292_transpose__mat__of__cols,axiom,
! [N: nat,Vs: list_vec_complex] :
( ( transp3074176993011536131omplex @ ( mat_of_cols_complex @ N @ Vs ) )
= ( mat_of_rows_complex @ N @ Vs ) ) ).
% transpose_mat_of_cols
thf(fact_293_transpose__mat__of__cols,axiom,
! [N: nat,Vs: list_vec_a] :
( ( transpose_mat_a @ ( mat_of_cols_a @ N @ Vs ) )
= ( mat_of_rows_a @ N @ Vs ) ) ).
% transpose_mat_of_cols
thf(fact_294_transpose__mat__of__rows,axiom,
! [N: nat,Vs: list_vec_complex] :
( ( transp3074176993011536131omplex @ ( mat_of_rows_complex @ N @ Vs ) )
= ( mat_of_cols_complex @ N @ Vs ) ) ).
% transpose_mat_of_rows
thf(fact_295_transpose__mat__of__rows,axiom,
! [N: nat,Vs: list_vec_a] :
( ( transpose_mat_a @ ( mat_of_rows_a @ N @ Vs ) )
= ( mat_of_cols_a @ N @ Vs ) ) ).
% transpose_mat_of_rows
thf(fact_296_mat__of__cols__carrier_I1_J,axiom,
! [N: nat,Vs: list_vec_a] : ( member_mat_a @ ( mat_of_cols_a @ N @ Vs ) @ ( carrier_mat_a @ N @ ( size_size_list_vec_a @ Vs ) ) ) ).
% mat_of_cols_carrier(1)
thf(fact_297_mat__of__cols__carrier_I1_J,axiom,
! [N: nat,Vs: list_vec_complex] : ( member_mat_complex @ ( mat_of_cols_complex @ N @ Vs ) @ ( carrier_mat_complex @ N @ ( size_s1158823550072163597omplex @ Vs ) ) ) ).
% mat_of_cols_carrier(1)
thf(fact_298_carrier__matD_I1_J,axiom,
! [A3: mat_mat_complex,Nr: nat,Nc: nat] :
( ( member7752848204589936667omplex @ A3 @ ( carrie8442657464762054641omplex @ Nr @ Nc ) )
=> ( ( dim_row_mat_complex @ A3 )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_299_carrier__matD_I1_J,axiom,
! [A3: mat_mat_a,Nr: nat,Nc: nat] :
( ( member_mat_mat_a @ A3 @ ( carrier_mat_mat_a @ Nr @ Nc ) )
=> ( ( dim_row_mat_a @ A3 )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_300_carrier__matD_I1_J,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( dim_row_a @ A3 )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_301_carrier__matD_I1_J,axiom,
! [A3: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( dim_row_complex @ A3 )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_302_carrier__matD_I2_J,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( dim_col_a @ A3 )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_303_carrier__matD_I2_J,axiom,
! [A3: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( dim_col_complex @ A3 )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_304_mult__carrier__mat,axiom,
! [A3: mat_a,Nr: nat,N: nat,B3: mat_a,Nc: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ N @ Nc ) )
=> ( member_mat_a @ ( times_times_mat_a @ A3 @ B3 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_305_mult__carrier__mat,axiom,
! [A3: mat_complex,Nr: nat,N: nat,B3: mat_complex,Nc: nat] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B3 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A3 @ B3 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_306_assoc__mult__mat,axiom,
! [A3: mat_a,N_1: nat,N_2: nat,B3: mat_a,N_3: nat,C2: mat_a,N_4: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ N_1 @ N_2 ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ N_2 @ N_3 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N_3 @ N_4 ) )
=> ( ( times_times_mat_a @ ( times_times_mat_a @ A3 @ B3 ) @ C2 )
= ( times_times_mat_a @ A3 @ ( times_times_mat_a @ B3 @ C2 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_307_assoc__mult__mat,axiom,
! [A3: mat_complex,N_1: nat,N_2: nat,B3: mat_complex,N_3: nat,C2: mat_complex,N_4: nat] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ N_1 @ N_2 ) )
=> ( ( member_mat_complex @ B3 @ ( carrier_mat_complex @ N_2 @ N_3 ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N_3 @ N_4 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A3 @ B3 ) @ C2 )
= ( times_8009071140041733218omplex @ A3 @ ( times_8009071140041733218omplex @ B3 @ C2 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_308_transpose__carrier__mat,axiom,
! [A3: mat_a,Nc: nat,Nr: nat] :
( ( member_mat_a @ ( transpose_mat_a @ A3 ) @ ( carrier_mat_a @ Nc @ Nr ) )
= ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_309_transpose__carrier__mat,axiom,
! [A3: mat_complex,Nc: nat,Nr: nat] :
( ( member_mat_complex @ ( transp3074176993011536131omplex @ A3 ) @ ( carrier_mat_complex @ Nc @ Nr ) )
= ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_310_uminus__carrier__iff__mat,axiom,
! [A3: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ ( uminus467866341702955550omplex @ A3 ) @ ( carrier_mat_complex @ Nr @ Nc ) )
= ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% uminus_carrier_iff_mat
thf(fact_311_uminus__carrier__mat,axiom,
! [A3: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( uminus467866341702955550omplex @ A3 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% uminus_carrier_mat
thf(fact_312_mat__of__cols__carrier_I2_J,axiom,
! [N: nat,Vs: list_vec_mat_complex] :
( ( dim_row_mat_complex @ ( mat_of4965844074629950985omplex @ N @ Vs ) )
= N ) ).
% mat_of_cols_carrier(2)
thf(fact_313_mat__of__cols__carrier_I2_J,axiom,
! [N: nat,Vs: list_vec_mat_a] :
( ( dim_row_mat_a @ ( mat_of_cols_mat_a @ N @ Vs ) )
= N ) ).
% mat_of_cols_carrier(2)
thf(fact_314_mat__of__cols__carrier_I2_J,axiom,
! [N: nat,Vs: list_vec_a] :
( ( dim_row_a @ ( mat_of_cols_a @ N @ Vs ) )
= N ) ).
% mat_of_cols_carrier(2)
thf(fact_315_mat__of__cols__carrier_I2_J,axiom,
! [N: nat,Vs: list_vec_complex] :
( ( dim_row_complex @ ( mat_of_cols_complex @ N @ Vs ) )
= N ) ).
% mat_of_cols_carrier(2)
thf(fact_316_carrier__matI,axiom,
! [A3: mat_mat_complex,Nr: nat,Nc: nat] :
( ( ( dim_row_mat_complex @ A3 )
= Nr )
=> ( ( ( dim_col_mat_complex @ A3 )
= Nc )
=> ( member7752848204589936667omplex @ A3 @ ( carrie8442657464762054641omplex @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_317_carrier__matI,axiom,
! [A3: mat_mat_a,Nr: nat,Nc: nat] :
( ( ( dim_row_mat_a @ A3 )
= Nr )
=> ( ( ( dim_col_mat_a @ A3 )
= Nc )
=> ( member_mat_mat_a @ A3 @ ( carrier_mat_mat_a @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_318_carrier__matI,axiom,
! [A3: mat_a,Nr: nat,Nc: nat] :
( ( ( dim_row_a @ A3 )
= Nr )
=> ( ( ( dim_col_a @ A3 )
= Nc )
=> ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_319_carrier__matI,axiom,
! [A3: mat_complex,Nr: nat,Nc: nat] :
( ( ( dim_row_complex @ A3 )
= Nr )
=> ( ( ( dim_col_complex @ A3 )
= Nc )
=> ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_320_carrier__mat__triv,axiom,
! [M: mat_mat_complex] : ( member7752848204589936667omplex @ M @ ( carrie8442657464762054641omplex @ ( dim_row_mat_complex @ M ) @ ( dim_col_mat_complex @ M ) ) ) ).
% carrier_mat_triv
thf(fact_321_carrier__mat__triv,axiom,
! [M: mat_mat_a] : ( member_mat_mat_a @ M @ ( carrier_mat_mat_a @ ( dim_row_mat_a @ M ) @ ( dim_col_mat_a @ M ) ) ) ).
% carrier_mat_triv
thf(fact_322_carrier__mat__triv,axiom,
! [M: mat_a] : ( member_mat_a @ M @ ( carrier_mat_a @ ( dim_row_a @ M ) @ ( dim_col_a @ M ) ) ) ).
% carrier_mat_triv
thf(fact_323_carrier__mat__triv,axiom,
! [M: mat_complex] : ( member_mat_complex @ M @ ( carrier_mat_complex @ ( dim_row_complex @ M ) @ ( dim_col_complex @ M ) ) ) ).
% carrier_mat_triv
thf(fact_324_transpose__mult,axiom,
! [A3: mat_complex,Nr: nat,N: nat,B3: mat_complex,Nc: nat] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B3 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( transp3074176993011536131omplex @ ( times_8009071140041733218omplex @ A3 @ B3 ) )
= ( times_8009071140041733218omplex @ ( transp3074176993011536131omplex @ B3 ) @ ( transp3074176993011536131omplex @ A3 ) ) ) ) ) ).
% transpose_mult
thf(fact_325_mat__of__rows__carrier_I1_J,axiom,
! [N: nat,Vs: list_vec_a] : ( member_mat_a @ ( mat_of_rows_a @ N @ Vs ) @ ( carrier_mat_a @ ( size_size_list_vec_a @ Vs ) @ N ) ) ).
% mat_of_rows_carrier(1)
thf(fact_326_mat__of__rows__carrier_I1_J,axiom,
! [N: nat,Vs: list_vec_complex] : ( member_mat_complex @ ( mat_of_rows_complex @ N @ Vs ) @ ( carrier_mat_complex @ ( size_s1158823550072163597omplex @ Vs ) @ N ) ) ).
% mat_of_rows_carrier(1)
thf(fact_327_mat__of__cols__carrier_I3_J,axiom,
! [N: nat,Vs: list_vec_a] :
( ( dim_col_a @ ( mat_of_cols_a @ N @ Vs ) )
= ( size_size_list_vec_a @ Vs ) ) ).
% mat_of_cols_carrier(3)
thf(fact_328_mat__of__cols__carrier_I3_J,axiom,
! [N: nat,Vs: list_vec_complex] :
( ( dim_col_complex @ ( mat_of_cols_complex @ N @ Vs ) )
= ( size_s1158823550072163597omplex @ Vs ) ) ).
% mat_of_cols_carrier(3)
thf(fact_329_mat__of__cols__cols,axiom,
! [A3: mat_mat_complex] :
( ( mat_of4965844074629950985omplex @ ( dim_row_mat_complex @ A3 ) @ ( cols_mat_complex @ A3 ) )
= A3 ) ).
% mat_of_cols_cols
thf(fact_330_mat__of__cols__cols,axiom,
! [A3: mat_mat_a] :
( ( mat_of_cols_mat_a @ ( dim_row_mat_a @ A3 ) @ ( cols_mat_a @ A3 ) )
= A3 ) ).
% mat_of_cols_cols
thf(fact_331_mat__of__cols__cols,axiom,
! [A3: mat_a] :
( ( mat_of_cols_a @ ( dim_row_a @ A3 ) @ ( cols_a @ A3 ) )
= A3 ) ).
% mat_of_cols_cols
thf(fact_332_mat__of__cols__cols,axiom,
! [A3: mat_complex] :
( ( mat_of_cols_complex @ ( dim_row_complex @ A3 ) @ ( cols_complex @ A3 ) )
= A3 ) ).
% mat_of_cols_cols
thf(fact_333_col__carrier__vec,axiom,
! [J: nat,Nc: nat,A3: mat_a,Nr: nat] :
( ( ord_less_nat @ J @ Nc )
=> ( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_vec_a @ ( col_a @ A3 @ J ) @ ( carrier_vec_a @ Nr ) ) ) ) ).
% col_carrier_vec
thf(fact_334_col__carrier__vec,axiom,
! [J: nat,Nc: nat,A3: mat_complex,Nr: nat] :
( ( ord_less_nat @ J @ Nc )
=> ( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_vec_complex @ ( col_complex @ A3 @ J ) @ ( carrier_vec_complex @ Nr ) ) ) ) ).
% col_carrier_vec
thf(fact_335_diag__mat__length,axiom,
! [A3: mat_vec_complex] :
( ( size_s1158823550072163597omplex @ ( diag_mat_vec_complex @ A3 ) )
= ( dim_row_vec_complex @ A3 ) ) ).
% diag_mat_length
thf(fact_336_diag__mat__length,axiom,
! [A3: mat_vec_a] :
( ( size_size_list_vec_a @ ( diag_mat_vec_a @ A3 ) )
= ( dim_row_vec_a @ A3 ) ) ).
% diag_mat_length
thf(fact_337_diag__mat__length,axiom,
! [A3: mat_nat] :
( ( size_size_list_nat @ ( diag_mat_nat @ A3 ) )
= ( dim_row_nat @ A3 ) ) ).
% diag_mat_length
thf(fact_338_diag__mat__length,axiom,
! [A3: mat_a] :
( ( size_size_list_a @ ( diag_mat_a @ A3 ) )
= ( dim_row_a @ A3 ) ) ).
% diag_mat_length
thf(fact_339_diag__mat__length,axiom,
! [A3: mat_complex] :
( ( size_s3451745648224563538omplex @ ( diag_mat_complex @ A3 ) )
= ( dim_row_complex @ A3 ) ) ).
% diag_mat_length
thf(fact_340_diag__mat__length,axiom,
! [A3: mat_mat_a] :
( ( size_size_list_mat_a @ ( diag_mat_mat_a @ A3 ) )
= ( dim_row_mat_a @ A3 ) ) ).
% diag_mat_length
thf(fact_341_diag__mat__length,axiom,
! [A3: mat_mat_complex] :
( ( size_s5969786470865220249omplex @ ( diag_mat_mat_complex @ A3 ) )
= ( dim_row_mat_complex @ A3 ) ) ).
% diag_mat_length
thf(fact_342_rank__1__proj__col__carrier,axiom,
! [I4: nat,A3: mat_complex] :
( ( ord_less_nat @ I4 @ ( dim_col_complex @ A3 ) )
=> ( member_mat_complex @ ( linear1949544614684794075omplex @ ( col_complex @ A3 @ I4 ) ) @ ( carrier_mat_complex @ ( dim_row_complex @ A3 ) @ ( dim_row_complex @ A3 ) ) ) ) ).
% rank_1_proj_col_carrier
thf(fact_343_mat__of__row__carrier_I1_J,axiom,
! [Y2: vec_a,N: nat] :
( ( member_vec_a @ Y2 @ ( carrier_vec_a @ N ) )
=> ( member_mat_a @ ( mat_of_row_a @ Y2 ) @ ( carrier_mat_a @ one_one_nat @ N ) ) ) ).
% mat_of_row_carrier(1)
thf(fact_344_mat__of__row__carrier_I1_J,axiom,
! [Y2: vec_complex,N: nat] :
( ( member_vec_complex @ Y2 @ ( carrier_vec_complex @ N ) )
=> ( member_mat_complex @ ( mat_of_row_complex @ Y2 ) @ ( carrier_mat_complex @ one_one_nat @ N ) ) ) ).
% mat_of_row_carrier(1)
thf(fact_345_col__mult2,axiom,
! [A3: mat_a,Nr: nat,N: nat,B3: mat_a,Nc: nat,J: nat] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( col_a @ ( times_times_mat_a @ A3 @ B3 ) @ J )
= ( mult_mat_vec_a @ A3 @ ( col_a @ B3 @ J ) ) ) ) ) ) ).
% col_mult2
thf(fact_346_col__mult2,axiom,
! [A3: mat_complex,Nr: nat,N: nat,B3: mat_complex,Nc: nat,J: nat] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B3 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( col_complex @ ( times_8009071140041733218omplex @ A3 @ B3 ) @ J )
= ( mult_mat_vec_complex @ A3 @ ( col_complex @ B3 @ J ) ) ) ) ) ) ).
% col_mult2
thf(fact_347_hermitian__decomp__dim__carrier,axiom,
! [A3: mat_complex,B3: mat_complex,U: mat_complex] :
( ( projec5943904436471448624omplex @ A3 @ B3 @ U )
=> ( member_mat_complex @ B3 @ ( carrier_mat_complex @ ( dim_row_complex @ A3 ) @ ( dim_col_complex @ A3 ) ) ) ) ).
% hermitian_decomp_dim_carrier
thf(fact_348_set__rows__carrier,axiom,
! [A3: mat_a,M: nat,N: nat,V: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ M @ N ) )
=> ( ( member_vec_a @ V @ ( set_vec_a2 @ ( rows_a @ A3 ) ) )
=> ( member_vec_a @ V @ ( carrier_vec_a @ N ) ) ) ) ).
% set_rows_carrier
thf(fact_349_set__rows__carrier,axiom,
! [A3: mat_complex,M: nat,N: nat,V: vec_complex] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ M @ N ) )
=> ( ( member_vec_complex @ V @ ( set_vec_complex2 @ ( rows_complex @ A3 ) ) )
=> ( member_vec_complex @ V @ ( carrier_vec_complex @ N ) ) ) ) ).
% set_rows_carrier
thf(fact_350_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_351_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_352_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_353_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_354_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_355_more__arith__simps_I6_J,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% more_arith_simps(6)
thf(fact_356_more__arith__simps_I5_J,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% more_arith_simps(5)
thf(fact_357_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_358_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_359_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_360_list_Opred__cong,axiom,
! [X2: list_vec_complex,Ya: list_vec_complex,P: vec_complex > $o,Pa: vec_complex > $o] :
( ( X2 = Ya )
=> ( ! [Z2: vec_complex] :
( ( member_vec_complex @ Z2 @ ( set_vec_complex2 @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( list_all_vec_complex @ P @ X2 )
= ( list_all_vec_complex @ Pa @ Ya ) ) ) ) ).
% list.pred_cong
thf(fact_361_list_Opred__cong,axiom,
! [X2: list_vec_a,Ya: list_vec_a,P: vec_a > $o,Pa: vec_a > $o] :
( ( X2 = Ya )
=> ( ! [Z2: vec_a] :
( ( member_vec_a @ Z2 @ ( set_vec_a2 @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( list_all_vec_a @ P @ X2 )
= ( list_all_vec_a @ Pa @ Ya ) ) ) ) ).
% list.pred_cong
thf(fact_362_list_Opred__cong,axiom,
! [X2: list_l5436439031154120755omplex,Ya: list_l5436439031154120755omplex,P: list_mat_complex > $o,Pa: list_mat_complex > $o] :
( ( X2 = Ya )
=> ( ! [Z2: list_mat_complex] :
( ( member279434397506102358omplex @ Z2 @ ( set_list_mat_complex2 @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( list_a2530308013472304865omplex @ P @ X2 )
= ( list_a2530308013472304865omplex @ Pa @ Ya ) ) ) ) ).
% list.pred_cong
thf(fact_363_list_Opred__cong,axiom,
! [X2: list_list_mat_a,Ya: list_list_mat_a,P: list_mat_a > $o,Pa: list_mat_a > $o] :
( ( X2 = Ya )
=> ( ! [Z2: list_mat_a] :
( ( member_list_mat_a @ Z2 @ ( set_list_mat_a2 @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( list_all_list_mat_a @ P @ X2 )
= ( list_all_list_mat_a @ Pa @ Ya ) ) ) ) ).
% list.pred_cong
thf(fact_364_list_Opred__cong,axiom,
! [X2: list_mat_a,Ya: list_mat_a,P: mat_a > $o,Pa: mat_a > $o] :
( ( X2 = Ya )
=> ( ! [Z2: mat_a] :
( ( member_mat_a @ Z2 @ ( set_mat_a2 @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( list_all_mat_a @ P @ X2 )
= ( list_all_mat_a @ Pa @ Ya ) ) ) ) ).
% list.pred_cong
thf(fact_365_list_Opred__cong,axiom,
! [X2: list_mat_complex,Ya: list_mat_complex,P: mat_complex > $o,Pa: mat_complex > $o] :
( ( X2 = Ya )
=> ( ! [Z2: mat_complex] :
( ( member_mat_complex @ Z2 @ ( set_mat_complex2 @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( list_all_mat_complex @ P @ X2 )
= ( list_all_mat_complex @ Pa @ Ya ) ) ) ) ).
% list.pred_cong
thf(fact_366_list_Opred__mono__strong,axiom,
! [P: vec_complex > $o,X2: list_vec_complex,Pa: vec_complex > $o] :
( ( list_all_vec_complex @ P @ X2 )
=> ( ! [Z2: vec_complex] :
( ( member_vec_complex @ Z2 @ ( set_vec_complex2 @ X2 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( list_all_vec_complex @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_367_list_Opred__mono__strong,axiom,
! [P: vec_a > $o,X2: list_vec_a,Pa: vec_a > $o] :
( ( list_all_vec_a @ P @ X2 )
=> ( ! [Z2: vec_a] :
( ( member_vec_a @ Z2 @ ( set_vec_a2 @ X2 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( list_all_vec_a @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_368_list_Opred__mono__strong,axiom,
! [P: list_mat_complex > $o,X2: list_l5436439031154120755omplex,Pa: list_mat_complex > $o] :
( ( list_a2530308013472304865omplex @ P @ X2 )
=> ( ! [Z2: list_mat_complex] :
( ( member279434397506102358omplex @ Z2 @ ( set_list_mat_complex2 @ X2 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( list_a2530308013472304865omplex @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_369_list_Opred__mono__strong,axiom,
! [P: list_mat_a > $o,X2: list_list_mat_a,Pa: list_mat_a > $o] :
( ( list_all_list_mat_a @ P @ X2 )
=> ( ! [Z2: list_mat_a] :
( ( member_list_mat_a @ Z2 @ ( set_list_mat_a2 @ X2 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( list_all_list_mat_a @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_370_list_Opred__mono__strong,axiom,
! [P: mat_a > $o,X2: list_mat_a,Pa: mat_a > $o] :
( ( list_all_mat_a @ P @ X2 )
=> ( ! [Z2: mat_a] :
( ( member_mat_a @ Z2 @ ( set_mat_a2 @ X2 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( list_all_mat_a @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_371_list_Opred__mono__strong,axiom,
! [P: mat_complex > $o,X2: list_mat_complex,Pa: mat_complex > $o] :
( ( list_all_mat_complex @ P @ X2 )
=> ( ! [Z2: mat_complex] :
( ( member_mat_complex @ Z2 @ ( set_mat_complex2 @ X2 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( list_all_mat_complex @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_372_in__set__butlastD,axiom,
! [X2: vec_complex,Xs: list_vec_complex] :
( ( member_vec_complex @ X2 @ ( set_vec_complex2 @ ( butlast_vec_complex @ Xs ) ) )
=> ( member_vec_complex @ X2 @ ( set_vec_complex2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_373_in__set__butlastD,axiom,
! [X2: vec_a,Xs: list_vec_a] :
( ( member_vec_a @ X2 @ ( set_vec_a2 @ ( butlast_vec_a @ Xs ) ) )
=> ( member_vec_a @ X2 @ ( set_vec_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_374_in__set__butlastD,axiom,
! [X2: mat_a,Xs: list_mat_a] :
( ( member_mat_a @ X2 @ ( set_mat_a2 @ ( butlast_mat_a @ Xs ) ) )
=> ( member_mat_a @ X2 @ ( set_mat_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_375_in__set__butlastD,axiom,
! [X2: list_mat_complex,Xs: list_l5436439031154120755omplex] :
( ( member279434397506102358omplex @ X2 @ ( set_list_mat_complex2 @ ( butlas2964118825291935103omplex @ Xs ) ) )
=> ( member279434397506102358omplex @ X2 @ ( set_list_mat_complex2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_376_in__set__butlastD,axiom,
! [X2: list_mat_a,Xs: list_list_mat_a] :
( ( member_list_mat_a @ X2 @ ( set_list_mat_a2 @ ( butlast_list_mat_a @ Xs ) ) )
=> ( member_list_mat_a @ X2 @ ( set_list_mat_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_377_in__set__butlastD,axiom,
! [X2: mat_complex,Xs: list_mat_complex] :
( ( member_mat_complex @ X2 @ ( set_mat_complex2 @ ( butlast_mat_complex @ Xs ) ) )
=> ( member_mat_complex @ X2 @ ( set_mat_complex2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_378_list__ex__cong,axiom,
! [Xs: list_vec_complex,Ys: list_vec_complex,F: vec_complex > $o,G: vec_complex > $o] :
( ( Xs = Ys )
=> ( ! [X4: vec_complex] :
( ( member_vec_complex @ X4 @ ( set_vec_complex2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( list_ex_vec_complex @ F @ Xs )
= ( list_ex_vec_complex @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_379_list__ex__cong,axiom,
! [Xs: list_vec_a,Ys: list_vec_a,F: vec_a > $o,G: vec_a > $o] :
( ( Xs = Ys )
=> ( ! [X4: vec_a] :
( ( member_vec_a @ X4 @ ( set_vec_a2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( list_ex_vec_a @ F @ Xs )
= ( list_ex_vec_a @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_380_list__ex__cong,axiom,
! [Xs: list_mat_a,Ys: list_mat_a,F: mat_a > $o,G: mat_a > $o] :
( ( Xs = Ys )
=> ( ! [X4: mat_a] :
( ( member_mat_a @ X4 @ ( set_mat_a2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( list_ex_mat_a @ F @ Xs )
= ( list_ex_mat_a @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_381_list__ex__cong,axiom,
! [Xs: list_l5436439031154120755omplex,Ys: list_l5436439031154120755omplex,F: list_mat_complex > $o,G: list_mat_complex > $o] :
( ( Xs = Ys )
=> ( ! [X4: list_mat_complex] :
( ( member279434397506102358omplex @ X4 @ ( set_list_mat_complex2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( list_e422197382367757169omplex @ F @ Xs )
= ( list_e422197382367757169omplex @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_382_list__ex__cong,axiom,
! [Xs: list_list_mat_a,Ys: list_list_mat_a,F: list_mat_a > $o,G: list_mat_a > $o] :
( ( Xs = Ys )
=> ( ! [X4: list_mat_a] :
( ( member_list_mat_a @ X4 @ ( set_list_mat_a2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( list_ex_list_mat_a @ F @ Xs )
= ( list_ex_list_mat_a @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_383_list__ex__cong,axiom,
! [Xs: list_mat_complex,Ys: list_mat_complex,F: mat_complex > $o,G: mat_complex > $o] :
( ( Xs = Ys )
=> ( ! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ ( set_mat_complex2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( list_ex_mat_complex @ F @ Xs )
= ( list_ex_mat_complex @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_384_mult__mat__vec__carrier,axiom,
! [A3: mat_a,Nr: nat,N: nat,V: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( member_vec_a @ ( mult_mat_vec_a @ A3 @ V ) @ ( carrier_vec_a @ Nr ) ) ) ) ).
% mult_mat_vec_carrier
thf(fact_385_mult__mat__vec__carrier,axiom,
! [A3: mat_complex,Nr: nat,N: nat,V: vec_complex] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_vec_complex @ V @ ( carrier_vec_complex @ N ) )
=> ( member_vec_complex @ ( mult_mat_vec_complex @ A3 @ V ) @ ( carrier_vec_complex @ Nr ) ) ) ) ).
% mult_mat_vec_carrier
thf(fact_386_assoc__mat__mult__vec_H,axiom,
! [A3: mat_a,N: nat,B3: mat_a,C2: mat_a,V: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( mult_mat_vec_a @ ( times_times_mat_a @ ( times_times_mat_a @ A3 @ B3 ) @ C2 ) @ V )
= ( mult_mat_vec_a @ A3 @ ( mult_mat_vec_a @ B3 @ ( mult_mat_vec_a @ C2 @ V ) ) ) ) ) ) ) ) ).
% assoc_mat_mult_vec'
thf(fact_387_assoc__mat__mult__vec_H,axiom,
! [A3: mat_complex,N: nat,B3: mat_complex,C2: mat_complex,V: vec_complex] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_vec_complex @ V @ ( carrier_vec_complex @ N ) )
=> ( ( mult_mat_vec_complex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A3 @ B3 ) @ C2 ) @ V )
= ( mult_mat_vec_complex @ A3 @ ( mult_mat_vec_complex @ B3 @ ( mult_mat_vec_complex @ C2 @ V ) ) ) ) ) ) ) ) ).
% assoc_mat_mult_vec'
thf(fact_388_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_389_mat__of__row__dim_I1_J,axiom,
! [Y2: vec_mat_complex] :
( ( dim_row_mat_complex @ ( mat_of8530337284418210486omplex @ Y2 ) )
= one_one_nat ) ).
% mat_of_row_dim(1)
thf(fact_390_mat__of__row__dim_I1_J,axiom,
! [Y2: vec_mat_a] :
( ( dim_row_mat_a @ ( mat_of_row_mat_a @ Y2 ) )
= one_one_nat ) ).
% mat_of_row_dim(1)
thf(fact_391_mat__of__row__dim_I1_J,axiom,
! [Y2: vec_a] :
( ( dim_row_a @ ( mat_of_row_a @ Y2 ) )
= one_one_nat ) ).
% mat_of_row_dim(1)
thf(fact_392_mat__of__row__dim_I1_J,axiom,
! [Y2: vec_complex] :
( ( dim_row_complex @ ( mat_of_row_complex @ Y2 ) )
= one_one_nat ) ).
% mat_of_row_dim(1)
thf(fact_393_assoc__mult__mat__vec,axiom,
! [A3: mat_a,N_1: nat,N_2: nat,B3: mat_a,N_3: nat,V: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ N_1 @ N_2 ) )
=> ( ( member_mat_a @ B3 @ ( carrier_mat_a @ N_2 @ N_3 ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ N_3 ) )
=> ( ( mult_mat_vec_a @ ( times_times_mat_a @ A3 @ B3 ) @ V )
= ( mult_mat_vec_a @ A3 @ ( mult_mat_vec_a @ B3 @ V ) ) ) ) ) ) ).
% assoc_mult_mat_vec
thf(fact_394_assoc__mult__mat__vec,axiom,
! [A3: mat_complex,N_1: nat,N_2: nat,B3: mat_complex,N_3: nat,V: vec_complex] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ N_1 @ N_2 ) )
=> ( ( member_mat_complex @ B3 @ ( carrier_mat_complex @ N_2 @ N_3 ) )
=> ( ( member_vec_complex @ V @ ( carrier_vec_complex @ N_3 ) )
=> ( ( mult_mat_vec_complex @ ( times_8009071140041733218omplex @ A3 @ B3 ) @ V )
= ( mult_mat_vec_complex @ A3 @ ( mult_mat_vec_complex @ B3 @ V ) ) ) ) ) ) ).
% assoc_mult_mat_vec
thf(fact_395_all__set__conv__all__nth,axiom,
! [Xs: list_l5436439031154120755omplex,P: list_mat_complex > $o] :
( ( ! [X3: list_mat_complex] :
( ( member279434397506102358omplex @ X3 @ ( set_list_mat_complex2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s479360804472521375omplex @ Xs ) )
=> ( P @ ( nth_list_mat_complex @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_396_all__set__conv__all__nth,axiom,
! [Xs: list_list_mat_a,P: list_mat_a > $o] :
( ( ! [X3: list_mat_a] :
( ( member_list_mat_a @ X3 @ ( set_list_mat_a2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6656407794899724303_mat_a @ Xs ) )
=> ( P @ ( nth_list_mat_a @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_397_all__set__conv__all__nth,axiom,
! [Xs: list_complex,P: complex > $o] :
( ( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( P @ ( nth_complex @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_398_all__set__conv__all__nth,axiom,
! [Xs: list_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_399_all__set__conv__all__nth,axiom,
! [Xs: list_vec_complex,P: vec_complex > $o] :
( ( ! [X3: vec_complex] :
( ( member_vec_complex @ X3 @ ( set_vec_complex2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1158823550072163597omplex @ Xs ) )
=> ( P @ ( nth_vec_complex @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_400_all__set__conv__all__nth,axiom,
! [Xs: list_vec_a,P: vec_a > $o] :
( ( ! [X3: vec_a] :
( ( member_vec_a @ X3 @ ( set_vec_a2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_vec_a @ Xs ) )
=> ( P @ ( nth_vec_a @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_401_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_402_all__set__conv__all__nth,axiom,
! [Xs: list_mat_a,P: mat_a > $o] :
( ( ! [X3: mat_a] :
( ( member_mat_a @ X3 @ ( set_mat_a2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_mat_a @ Xs ) )
=> ( P @ ( nth_mat_a @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_403_all__set__conv__all__nth,axiom,
! [Xs: list_mat_complex,P: mat_complex > $o] :
( ( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ ( set_mat_complex2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( P @ ( nth_mat_complex @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_404_all__nth__imp__all__set,axiom,
! [Xs: list_l5436439031154120755omplex,P: list_mat_complex > $o,X2: list_mat_complex] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s479360804472521375omplex @ Xs ) )
=> ( P @ ( nth_list_mat_complex @ Xs @ I2 ) ) )
=> ( ( member279434397506102358omplex @ X2 @ ( set_list_mat_complex2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_405_all__nth__imp__all__set,axiom,
! [Xs: list_list_mat_a,P: list_mat_a > $o,X2: list_mat_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s6656407794899724303_mat_a @ Xs ) )
=> ( P @ ( nth_list_mat_a @ Xs @ I2 ) ) )
=> ( ( member_list_mat_a @ X2 @ ( set_list_mat_a2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_406_all__nth__imp__all__set,axiom,
! [Xs: list_complex,P: complex > $o,X2: complex] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( P @ ( nth_complex @ Xs @ I2 ) ) )
=> ( ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_407_all__nth__imp__all__set,axiom,
! [Xs: list_a,P: a > $o,X2: a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I2 ) ) )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_408_all__nth__imp__all__set,axiom,
! [Xs: list_vec_complex,P: vec_complex > $o,X2: vec_complex] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1158823550072163597omplex @ Xs ) )
=> ( P @ ( nth_vec_complex @ Xs @ I2 ) ) )
=> ( ( member_vec_complex @ X2 @ ( set_vec_complex2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_409_all__nth__imp__all__set,axiom,
! [Xs: list_vec_a,P: vec_a > $o,X2: vec_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_vec_a @ Xs ) )
=> ( P @ ( nth_vec_a @ Xs @ I2 ) ) )
=> ( ( member_vec_a @ X2 @ ( set_vec_a2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_410_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X2: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_411_all__nth__imp__all__set,axiom,
! [Xs: list_mat_a,P: mat_a > $o,X2: mat_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_mat_a @ Xs ) )
=> ( P @ ( nth_mat_a @ Xs @ I2 ) ) )
=> ( ( member_mat_a @ X2 @ ( set_mat_a2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_412_all__nth__imp__all__set,axiom,
! [Xs: list_mat_complex,P: mat_complex > $o,X2: mat_complex] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( P @ ( nth_mat_complex @ Xs @ I2 ) ) )
=> ( ( member_mat_complex @ X2 @ ( set_mat_complex2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_413_in__set__conv__nth,axiom,
! [X2: list_mat_complex,Xs: list_l5436439031154120755omplex] :
( ( member279434397506102358omplex @ X2 @ ( set_list_mat_complex2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s479360804472521375omplex @ Xs ) )
& ( ( nth_list_mat_complex @ Xs @ I3 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_414_in__set__conv__nth,axiom,
! [X2: list_mat_a,Xs: list_list_mat_a] :
( ( member_list_mat_a @ X2 @ ( set_list_mat_a2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6656407794899724303_mat_a @ Xs ) )
& ( ( nth_list_mat_a @ Xs @ I3 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_415_in__set__conv__nth,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
& ( ( nth_complex @ Xs @ I3 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_416_in__set__conv__nth,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I3 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_417_in__set__conv__nth,axiom,
! [X2: vec_complex,Xs: list_vec_complex] :
( ( member_vec_complex @ X2 @ ( set_vec_complex2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1158823550072163597omplex @ Xs ) )
& ( ( nth_vec_complex @ Xs @ I3 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_418_in__set__conv__nth,axiom,
! [X2: vec_a,Xs: list_vec_a] :
( ( member_vec_a @ X2 @ ( set_vec_a2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_vec_a @ Xs ) )
& ( ( nth_vec_a @ Xs @ I3 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_419_in__set__conv__nth,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_420_in__set__conv__nth,axiom,
! [X2: mat_a,Xs: list_mat_a] :
( ( member_mat_a @ X2 @ ( set_mat_a2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_mat_a @ Xs ) )
& ( ( nth_mat_a @ Xs @ I3 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_421_in__set__conv__nth,axiom,
! [X2: mat_complex,Xs: list_mat_complex] :
( ( member_mat_complex @ X2 @ ( set_mat_complex2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5969786470865220249omplex @ Xs ) )
& ( ( nth_mat_complex @ Xs @ I3 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_422_list__ball__nth,axiom,
! [N: nat,Xs: list_l5436439031154120755omplex,P: list_mat_complex > $o] :
( ( ord_less_nat @ N @ ( size_s479360804472521375omplex @ Xs ) )
=> ( ! [X4: list_mat_complex] :
( ( member279434397506102358omplex @ X4 @ ( set_list_mat_complex2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_list_mat_complex @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_423_list__ball__nth,axiom,
! [N: nat,Xs: list_list_mat_a,P: list_mat_a > $o] :
( ( ord_less_nat @ N @ ( size_s6656407794899724303_mat_a @ Xs ) )
=> ( ! [X4: list_mat_a] :
( ( member_list_mat_a @ X4 @ ( set_list_mat_a2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_list_mat_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_424_list__ball__nth,axiom,
! [N: nat,Xs: list_complex,P: complex > $o] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_complex @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_425_list__ball__nth,axiom,
! [N: nat,Xs: list_a,P: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_426_list__ball__nth,axiom,
! [N: nat,Xs: list_vec_complex,P: vec_complex > $o] :
( ( ord_less_nat @ N @ ( size_s1158823550072163597omplex @ Xs ) )
=> ( ! [X4: vec_complex] :
( ( member_vec_complex @ X4 @ ( set_vec_complex2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_vec_complex @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_427_list__ball__nth,axiom,
! [N: nat,Xs: list_vec_a,P: vec_a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_vec_a @ Xs ) )
=> ( ! [X4: vec_a] :
( ( member_vec_a @ X4 @ ( set_vec_a2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_vec_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_428_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_429_list__ball__nth,axiom,
! [N: nat,Xs: list_mat_a,P: mat_a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_mat_a @ Xs ) )
=> ( ! [X4: mat_a] :
( ( member_mat_a @ X4 @ ( set_mat_a2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_mat_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_430_list__ball__nth,axiom,
! [N: nat,Xs: list_mat_complex,P: mat_complex > $o] :
( ( ord_less_nat @ N @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ ( set_mat_complex2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_mat_complex @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_431_nth__mem,axiom,
! [N: nat,Xs: list_l5436439031154120755omplex] :
( ( ord_less_nat @ N @ ( size_s479360804472521375omplex @ Xs ) )
=> ( member279434397506102358omplex @ ( nth_list_mat_complex @ Xs @ N ) @ ( set_list_mat_complex2 @ Xs ) ) ) ).
% nth_mem
thf(fact_432_nth__mem,axiom,
! [N: nat,Xs: list_list_mat_a] :
( ( ord_less_nat @ N @ ( size_s6656407794899724303_mat_a @ Xs ) )
=> ( member_list_mat_a @ ( nth_list_mat_a @ Xs @ N ) @ ( set_list_mat_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_433_nth__mem,axiom,
! [N: nat,Xs: list_complex] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( member_complex @ ( nth_complex @ Xs @ N ) @ ( set_complex2 @ Xs ) ) ) ).
% nth_mem
thf(fact_434_nth__mem,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_435_nth__mem,axiom,
! [N: nat,Xs: list_vec_complex] :
( ( ord_less_nat @ N @ ( size_s1158823550072163597omplex @ Xs ) )
=> ( member_vec_complex @ ( nth_vec_complex @ Xs @ N ) @ ( set_vec_complex2 @ Xs ) ) ) ).
% nth_mem
thf(fact_436_nth__mem,axiom,
! [N: nat,Xs: list_vec_a] :
( ( ord_less_nat @ N @ ( size_size_list_vec_a @ Xs ) )
=> ( member_vec_a @ ( nth_vec_a @ Xs @ N ) @ ( set_vec_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_437_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_438_nth__mem,axiom,
! [N: nat,Xs: list_mat_a] :
( ( ord_less_nat @ N @ ( size_size_list_mat_a @ Xs ) )
=> ( member_mat_a @ ( nth_mat_a @ Xs @ N ) @ ( set_mat_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_439_nth__mem,axiom,
! [N: nat,Xs: list_mat_complex] :
( ( ord_less_nat @ N @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( member_mat_complex @ ( nth_mat_complex @ Xs @ N ) @ ( set_mat_complex2 @ Xs ) ) ) ).
% nth_mem
thf(fact_440_rank__1__proj__square__mat,axiom,
! [V: vec_complex] : ( square_mat_complex @ ( linear1949544614684794075omplex @ V ) ) ).
% rank_1_proj_square_mat
thf(fact_441_step__3__def,axiom,
( jordan4501759426295633263omplex
= ( ^ [A4: mat_complex] : ( jordan4702481308941288104omplex @ ( dim_row_complex @ A4 ) @ one_one_nat @ A4 ) ) ) ).
% step_3_def
thf(fact_442_triangular__to__jnf__steps__dims_I5_J,axiom,
! [A3: mat_complex] :
( ( dim_row_complex @ ( jordan4501759426295633263omplex @ A3 ) )
= ( dim_row_complex @ A3 ) ) ).
% triangular_to_jnf_steps_dims(5)
thf(fact_443_triangular__to__jnf__steps__dims_I6_J,axiom,
! [A3: mat_complex] :
( ( dim_col_complex @ ( jordan4501759426295633263omplex @ A3 ) )
= ( dim_col_complex @ A3 ) ) ).
% triangular_to_jnf_steps_dims(6)
thf(fact_444_diag__block__mat__length__1,axiom,
! [Al: list_mat_a] :
( ( ( size_size_list_mat_a @ Al )
= one_one_nat )
=> ( ( diag_block_mat_a @ Al )
= ( nth_mat_a @ Al @ zero_zero_nat ) ) ) ).
% diag_block_mat_length_1
thf(fact_445_diag__block__mat__length__1,axiom,
! [Al: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Al )
= one_one_nat )
=> ( ( diag_b9145358668110806138omplex @ Al )
= ( nth_mat_complex @ Al @ zero_zero_nat ) ) ) ).
% diag_block_mat_length_1
thf(fact_446_rows__mat__of__rows,axiom,
! [Vs: list_vec_complex,N: nat] :
( ( ord_le8044543173838861339omplex @ ( set_vec_complex2 @ Vs ) @ ( carrier_vec_complex @ N ) )
=> ( ( rows_complex @ ( mat_of_rows_complex @ N @ Vs ) )
= Vs ) ) ).
% rows_mat_of_rows
thf(fact_447_rows__mat__of__rows,axiom,
! [Vs: list_vec_a,N: nat] :
( ( ord_le4791951621262958845_vec_a @ ( set_vec_a2 @ Vs ) @ ( carrier_vec_a @ N ) )
=> ( ( rows_a @ ( mat_of_rows_a @ N @ Vs ) )
= Vs ) ) ).
% rows_mat_of_rows
thf(fact_448_unitarily__equiv__rank__1__proj__col__carrier,axiom,
! [A3: mat_complex,N: nat,B3: mat_complex,U: mat_complex,I4: nat] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A3 @ B3 @ U )
=> ( ( ord_less_nat @ I4 @ N )
=> ( member_mat_complex @ ( linear1949544614684794075omplex @ ( col_complex @ U @ I4 ) ) @ ( carrier_mat_complex @ N @ N ) ) ) ) ) ).
% unitarily_equiv_rank_1_proj_col_carrier
thf(fact_449_cols__mat__of__cols,axiom,
! [Vs: list_vec_complex,N: nat] :
( ( ord_le8044543173838861339omplex @ ( set_vec_complex2 @ Vs ) @ ( carrier_vec_complex @ N ) )
=> ( ( cols_complex @ ( mat_of_cols_complex @ N @ Vs ) )
= Vs ) ) ).
% cols_mat_of_cols
thf(fact_450_cols__mat__of__cols,axiom,
! [Vs: list_vec_a,N: nat] :
( ( ord_le4791951621262958845_vec_a @ ( set_vec_a2 @ Vs ) @ ( carrier_vec_a @ N ) )
=> ( ( cols_a @ ( mat_of_cols_a @ N @ Vs ) )
= Vs ) ) ).
% cols_mat_of_cols
thf(fact_451_verit__comp__simplify_I29_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% verit_comp_simplify(29)
thf(fact_452_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_453_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_454_arithmetic__simps_I63_J,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% arithmetic_simps(63)
thf(fact_455_arithmetic__simps_I62_J,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% arithmetic_simps(62)
thf(fact_456_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_457_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_458_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_459_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_460_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_461_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_462_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_463_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_464_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_465_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_466_verit__eq__simplify_I6_J,axiom,
! [X2: set_mat_complex,Y2: set_mat_complex] :
( ( X2 = Y2 )
=> ( ord_le3632134057777142183omplex @ X2 @ Y2 ) ) ).
% verit_eq_simplify(6)
thf(fact_467_verit__eq__simplify_I6_J,axiom,
! [X2: set_vec_complex,Y2: set_vec_complex] :
( ( X2 = Y2 )
=> ( ord_le8044543173838861339omplex @ X2 @ Y2 ) ) ).
% verit_eq_simplify(6)
thf(fact_468_verit__eq__simplify_I6_J,axiom,
! [X2: set_vec_a,Y2: set_vec_a] :
( ( X2 = Y2 )
=> ( ord_le4791951621262958845_vec_a @ X2 @ Y2 ) ) ).
% verit_eq_simplify(6)
thf(fact_469_verit__eq__simplify_I6_J,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% verit_eq_simplify(6)
thf(fact_470_verit__comp__simplify_I2_J,axiom,
! [A: set_mat_complex] : ( ord_le3632134057777142183omplex @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_471_verit__comp__simplify_I2_J,axiom,
! [A: set_vec_complex] : ( ord_le8044543173838861339omplex @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_472_verit__comp__simplify_I2_J,axiom,
! [A: set_vec_a] : ( ord_le4791951621262958845_vec_a @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_473_verit__comp__simplify_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_474_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_475_rel__simps_I46_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% rel_simps(46)
thf(fact_476_zero__order_I2_J,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% zero_order(2)
thf(fact_477_zero__order_I1_J,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_order(1)
thf(fact_478_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_479_Compl__anti__mono,axiom,
! [A3: set_mat_complex,B3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A3 @ B3 )
=> ( ord_le3632134057777142183omplex @ ( uminus5815530220087396478omplex @ B3 ) @ ( uminus5815530220087396478omplex @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_480_Compl__anti__mono,axiom,
! [A3: set_vec_complex,B3: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ A3 @ B3 )
=> ( ord_le8044543173838861339omplex @ ( uminus1004567299294339826omplex @ B3 ) @ ( uminus1004567299294339826omplex @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_481_Compl__anti__mono,axiom,
! [A3: set_vec_a,B3: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A3 @ B3 )
=> ( ord_le4791951621262958845_vec_a @ ( uminus2769705506071317478_vec_a @ B3 ) @ ( uminus2769705506071317478_vec_a @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_482_Compl__subset__Compl__iff,axiom,
! [A3: set_mat_complex,B3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ ( uminus5815530220087396478omplex @ A3 ) @ ( uminus5815530220087396478omplex @ B3 ) )
= ( ord_le3632134057777142183omplex @ B3 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_483_Compl__subset__Compl__iff,axiom,
! [A3: set_vec_complex,B3: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ ( uminus1004567299294339826omplex @ A3 ) @ ( uminus1004567299294339826omplex @ B3 ) )
= ( ord_le8044543173838861339omplex @ B3 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_484_Compl__subset__Compl__iff,axiom,
! [A3: set_vec_a,B3: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ ( uminus2769705506071317478_vec_a @ A3 ) @ ( uminus2769705506071317478_vec_a @ B3 ) )
= ( ord_le4791951621262958845_vec_a @ B3 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_485_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_486_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_487_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_488_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_489_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_490_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_491_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_492_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_493_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_494_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_495_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D2: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_496_basic__trans__rules_I3_J,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_497_basic__trans__rules_I3_J,axiom,
! [A: nat,B: nat,F: nat > set_mat_complex,C: set_mat_complex] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le5598786136212072115omplex @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le3632134057777142183omplex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le5598786136212072115omplex @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_498_basic__trans__rules_I3_J,axiom,
! [A: nat,B: nat,F: nat > set_vec_complex,C: set_vec_complex] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le787823215419015463omplex @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le8044543173838861339omplex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le787823215419015463omplex @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_499_basic__trans__rules_I3_J,axiom,
! [A: nat,B: nat,F: nat > set_vec_a,C: set_vec_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_vec_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le4791951621262958845_vec_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_500_basic__trans__rules_I3_J,axiom,
! [A: set_mat_complex,B: set_mat_complex,F: set_mat_complex > nat,C: nat] :
( ( ord_le3632134057777142183omplex @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_mat_complex,Y3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_501_basic__trans__rules_I3_J,axiom,
! [A: set_vec_complex,B: set_vec_complex,F: set_vec_complex > nat,C: nat] :
( ( ord_le8044543173838861339omplex @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_vec_complex,Y3: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_502_basic__trans__rules_I3_J,axiom,
! [A: set_vec_a,B: set_vec_a,F: set_vec_a > nat,C: nat] :
( ( ord_le4791951621262958845_vec_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_vec_a,Y3: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_503_basic__trans__rules_I3_J,axiom,
! [A: set_mat_complex,B: set_mat_complex,F: set_mat_complex > set_mat_complex,C: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ B )
=> ( ( ord_le5598786136212072115omplex @ ( F @ B ) @ C )
=> ( ! [X4: set_mat_complex,Y3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X4 @ Y3 )
=> ( ord_le3632134057777142183omplex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le5598786136212072115omplex @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_504_basic__trans__rules_I3_J,axiom,
! [A: set_mat_complex,B: set_mat_complex,F: set_mat_complex > set_vec_complex,C: set_vec_complex] :
( ( ord_le3632134057777142183omplex @ A @ B )
=> ( ( ord_le787823215419015463omplex @ ( F @ B ) @ C )
=> ( ! [X4: set_mat_complex,Y3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X4 @ Y3 )
=> ( ord_le8044543173838861339omplex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le787823215419015463omplex @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_505_basic__trans__rules_I3_J,axiom,
! [A: set_mat_complex,B: set_mat_complex,F: set_mat_complex > set_vec_a,C: set_vec_a] :
( ( ord_le3632134057777142183omplex @ A @ B )
=> ( ( ord_less_set_vec_a @ ( F @ B ) @ C )
=> ( ! [X4: set_mat_complex,Y3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X4 @ Y3 )
=> ( ord_le4791951621262958845_vec_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_506_basic__trans__rules_I4_J,axiom,
! [A: set_mat_complex,F: nat > set_mat_complex,B: nat,C: nat] :
( ( ord_le3632134057777142183omplex @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le5598786136212072115omplex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le5598786136212072115omplex @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(4)
thf(fact_507_basic__trans__rules_I4_J,axiom,
! [A: set_vec_complex,F: nat > set_vec_complex,B: nat,C: nat] :
( ( ord_le8044543173838861339omplex @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le787823215419015463omplex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le787823215419015463omplex @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(4)
thf(fact_508_basic__trans__rules_I4_J,axiom,
! [A: set_vec_a,F: nat > set_vec_a,B: nat,C: nat] :
( ( ord_le4791951621262958845_vec_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_vec_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(4)
thf(fact_509_basic__trans__rules_I4_J,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(4)
thf(fact_510_basic__trans__rules_I5_J,axiom,
! [A: nat,B: nat,F: nat > set_mat_complex,C: set_mat_complex] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le3632134057777142183omplex @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le5598786136212072115omplex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le5598786136212072115omplex @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(5)
thf(fact_511_basic__trans__rules_I5_J,axiom,
! [A: nat,B: nat,F: nat > set_vec_complex,C: set_vec_complex] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le8044543173838861339omplex @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le787823215419015463omplex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le787823215419015463omplex @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(5)
thf(fact_512_basic__trans__rules_I5_J,axiom,
! [A: nat,B: nat,F: nat > set_vec_a,C: set_vec_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le4791951621262958845_vec_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_set_vec_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(5)
thf(fact_513_basic__trans__rules_I5_J,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% basic_trans_rules(5)
thf(fact_514_basic__trans__rules_I6_J,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_515_basic__trans__rules_I6_J,axiom,
! [A: set_mat_complex,F: nat > set_mat_complex,B: nat,C: nat] :
( ( ord_le5598786136212072115omplex @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le3632134057777142183omplex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le5598786136212072115omplex @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_516_basic__trans__rules_I6_J,axiom,
! [A: set_vec_complex,F: nat > set_vec_complex,B: nat,C: nat] :
( ( ord_le787823215419015463omplex @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le8044543173838861339omplex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le787823215419015463omplex @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_517_basic__trans__rules_I6_J,axiom,
! [A: set_vec_a,F: nat > set_vec_a,B: nat,C: nat] :
( ( ord_less_set_vec_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le4791951621262958845_vec_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_518_basic__trans__rules_I6_J,axiom,
! [A: nat,F: set_mat_complex > nat,B: set_mat_complex,C: set_mat_complex] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le3632134057777142183omplex @ B @ C )
=> ( ! [X4: set_mat_complex,Y3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_519_basic__trans__rules_I6_J,axiom,
! [A: nat,F: set_vec_complex > nat,B: set_vec_complex,C: set_vec_complex] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le8044543173838861339omplex @ B @ C )
=> ( ! [X4: set_vec_complex,Y3: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_520_basic__trans__rules_I6_J,axiom,
! [A: nat,F: set_vec_a > nat,B: set_vec_a,C: set_vec_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le4791951621262958845_vec_a @ B @ C )
=> ( ! [X4: set_vec_a,Y3: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_521_basic__trans__rules_I6_J,axiom,
! [A: set_mat_complex,F: set_mat_complex > set_mat_complex,B: set_mat_complex,C: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ A @ ( F @ B ) )
=> ( ( ord_le3632134057777142183omplex @ B @ C )
=> ( ! [X4: set_mat_complex,Y3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X4 @ Y3 )
=> ( ord_le3632134057777142183omplex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le5598786136212072115omplex @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_522_basic__trans__rules_I6_J,axiom,
! [A: set_vec_complex,F: set_mat_complex > set_vec_complex,B: set_mat_complex,C: set_mat_complex] :
( ( ord_le787823215419015463omplex @ A @ ( F @ B ) )
=> ( ( ord_le3632134057777142183omplex @ B @ C )
=> ( ! [X4: set_mat_complex,Y3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X4 @ Y3 )
=> ( ord_le8044543173838861339omplex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le787823215419015463omplex @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_523_basic__trans__rules_I6_J,axiom,
! [A: set_vec_a,F: set_mat_complex > set_vec_a,B: set_mat_complex,C: set_mat_complex] :
( ( ord_less_set_vec_a @ A @ ( F @ B ) )
=> ( ( ord_le3632134057777142183omplex @ B @ C )
=> ( ! [X4: set_mat_complex,Y3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X4 @ Y3 )
=> ( ord_le4791951621262958845_vec_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_524_basic__trans__rules_I17_J,axiom,
! [A: set_mat_complex,B: set_mat_complex] :
( ( A != B )
=> ( ( ord_le3632134057777142183omplex @ A @ B )
=> ( ord_le5598786136212072115omplex @ A @ B ) ) ) ).
% basic_trans_rules(17)
thf(fact_525_basic__trans__rules_I17_J,axiom,
! [A: set_vec_complex,B: set_vec_complex] :
( ( A != B )
=> ( ( ord_le8044543173838861339omplex @ A @ B )
=> ( ord_le787823215419015463omplex @ A @ B ) ) ) ).
% basic_trans_rules(17)
thf(fact_526_basic__trans__rules_I17_J,axiom,
! [A: set_vec_a,B: set_vec_a] :
( ( A != B )
=> ( ( ord_le4791951621262958845_vec_a @ A @ B )
=> ( ord_less_set_vec_a @ A @ B ) ) ) ).
% basic_trans_rules(17)
thf(fact_527_basic__trans__rules_I17_J,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% basic_trans_rules(17)
thf(fact_528_basic__trans__rules_I18_J,axiom,
! [A: set_mat_complex,B: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ B )
=> ( ( A != B )
=> ( ord_le5598786136212072115omplex @ A @ B ) ) ) ).
% basic_trans_rules(18)
thf(fact_529_basic__trans__rules_I18_J,axiom,
! [A: set_vec_complex,B: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ A @ B )
=> ( ( A != B )
=> ( ord_le787823215419015463omplex @ A @ B ) ) ) ).
% basic_trans_rules(18)
thf(fact_530_basic__trans__rules_I18_J,axiom,
! [A: set_vec_a,B: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_vec_a @ A @ B ) ) ) ).
% basic_trans_rules(18)
thf(fact_531_basic__trans__rules_I18_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% basic_trans_rules(18)
thf(fact_532_basic__trans__rules_I21_J,axiom,
! [X2: set_mat_complex,Y2: set_mat_complex,Z4: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X2 @ Y2 )
=> ( ( ord_le5598786136212072115omplex @ Y2 @ Z4 )
=> ( ord_le5598786136212072115omplex @ X2 @ Z4 ) ) ) ).
% basic_trans_rules(21)
thf(fact_533_basic__trans__rules_I21_J,axiom,
! [X2: set_vec_complex,Y2: set_vec_complex,Z4: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ X2 @ Y2 )
=> ( ( ord_le787823215419015463omplex @ Y2 @ Z4 )
=> ( ord_le787823215419015463omplex @ X2 @ Z4 ) ) ) ).
% basic_trans_rules(21)
thf(fact_534_basic__trans__rules_I21_J,axiom,
! [X2: set_vec_a,Y2: set_vec_a,Z4: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X2 @ Y2 )
=> ( ( ord_less_set_vec_a @ Y2 @ Z4 )
=> ( ord_less_set_vec_a @ X2 @ Z4 ) ) ) ).
% basic_trans_rules(21)
thf(fact_535_basic__trans__rules_I21_J,axiom,
! [X2: nat,Y2: nat,Z4: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z4 )
=> ( ord_less_nat @ X2 @ Z4 ) ) ) ).
% basic_trans_rules(21)
thf(fact_536_basic__trans__rules_I22_J,axiom,
! [X2: set_mat_complex,Y2: set_mat_complex,Z4: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ X2 @ Y2 )
=> ( ( ord_le3632134057777142183omplex @ Y2 @ Z4 )
=> ( ord_le5598786136212072115omplex @ X2 @ Z4 ) ) ) ).
% basic_trans_rules(22)
thf(fact_537_basic__trans__rules_I22_J,axiom,
! [X2: set_vec_complex,Y2: set_vec_complex,Z4: set_vec_complex] :
( ( ord_le787823215419015463omplex @ X2 @ Y2 )
=> ( ( ord_le8044543173838861339omplex @ Y2 @ Z4 )
=> ( ord_le787823215419015463omplex @ X2 @ Z4 ) ) ) ).
% basic_trans_rules(22)
thf(fact_538_basic__trans__rules_I22_J,axiom,
! [X2: set_vec_a,Y2: set_vec_a,Z4: set_vec_a] :
( ( ord_less_set_vec_a @ X2 @ Y2 )
=> ( ( ord_le4791951621262958845_vec_a @ Y2 @ Z4 )
=> ( ord_less_set_vec_a @ X2 @ Z4 ) ) ) ).
% basic_trans_rules(22)
thf(fact_539_basic__trans__rules_I22_J,axiom,
! [X2: nat,Y2: nat,Z4: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z4 )
=> ( ord_less_nat @ X2 @ Z4 ) ) ) ).
% basic_trans_rules(22)
thf(fact_540_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_541_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_542_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_543_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_544_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A6 ) )
= ( ord_less_nat @ A6 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_545_leD,axiom,
! [Y2: set_mat_complex,X2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ Y2 @ X2 )
=> ~ ( ord_le5598786136212072115omplex @ X2 @ Y2 ) ) ).
% leD
thf(fact_546_leD,axiom,
! [Y2: set_vec_complex,X2: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ Y2 @ X2 )
=> ~ ( ord_le787823215419015463omplex @ X2 @ Y2 ) ) ).
% leD
thf(fact_547_leD,axiom,
! [Y2: set_vec_a,X2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ Y2 @ X2 )
=> ~ ( ord_less_set_vec_a @ X2 @ Y2 ) ) ).
% leD
thf(fact_548_leD,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_549_leI,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% leI
thf(fact_550_le__less,axiom,
( ord_le3632134057777142183omplex
= ( ^ [X3: set_mat_complex,Y5: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% le_less
thf(fact_551_le__less,axiom,
( ord_le8044543173838861339omplex
= ( ^ [X3: set_vec_complex,Y5: set_vec_complex] :
( ( ord_le787823215419015463omplex @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% le_less
thf(fact_552_le__less,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [X3: set_vec_a,Y5: set_vec_a] :
( ( ord_less_set_vec_a @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% le_less
thf(fact_553_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% le_less
thf(fact_554_less__le,axiom,
( ord_le5598786136212072115omplex
= ( ^ [X3: set_mat_complex,Y5: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% less_le
thf(fact_555_less__le,axiom,
( ord_le787823215419015463omplex
= ( ^ [X3: set_vec_complex,Y5: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% less_le
thf(fact_556_less__le,axiom,
( ord_less_set_vec_a
= ( ^ [X3: set_vec_a,Y5: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% less_le
thf(fact_557_less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% less_le
thf(fact_558_nless__le,axiom,
! [A: set_mat_complex,B: set_mat_complex] :
( ( ~ ( ord_le5598786136212072115omplex @ A @ B ) )
= ( ~ ( ord_le3632134057777142183omplex @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_559_nless__le,axiom,
! [A: set_vec_complex,B: set_vec_complex] :
( ( ~ ( ord_le787823215419015463omplex @ A @ B ) )
= ( ~ ( ord_le8044543173838861339omplex @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_560_nless__le,axiom,
! [A: set_vec_a,B: set_vec_a] :
( ( ~ ( ord_less_set_vec_a @ A @ B ) )
= ( ~ ( ord_le4791951621262958845_vec_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_561_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_562_not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% not_le
thf(fact_563_not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% not_less
thf(fact_564_antisym__conv1,axiom,
! [X2: set_mat_complex,Y2: set_mat_complex] :
( ~ ( ord_le5598786136212072115omplex @ X2 @ Y2 )
=> ( ( ord_le3632134057777142183omplex @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_565_antisym__conv1,axiom,
! [X2: set_vec_complex,Y2: set_vec_complex] :
( ~ ( ord_le787823215419015463omplex @ X2 @ Y2 )
=> ( ( ord_le8044543173838861339omplex @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_566_antisym__conv1,axiom,
! [X2: set_vec_a,Y2: set_vec_a] :
( ~ ( ord_less_set_vec_a @ X2 @ Y2 )
=> ( ( ord_le4791951621262958845_vec_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_567_antisym__conv1,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_568_antisym__conv2,axiom,
! [X2: set_mat_complex,Y2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X2 @ Y2 )
=> ( ( ~ ( ord_le5598786136212072115omplex @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_569_antisym__conv2,axiom,
! [X2: set_vec_complex,Y2: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ X2 @ Y2 )
=> ( ( ~ ( ord_le787823215419015463omplex @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_570_antisym__conv2,axiom,
! [X2: set_vec_a,Y2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X2 @ Y2 )
=> ( ( ~ ( ord_less_set_vec_a @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_571_antisym__conv2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_572_less__imp__le,axiom,
! [X2: set_mat_complex,Y2: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ X2 @ Y2 )
=> ( ord_le3632134057777142183omplex @ X2 @ Y2 ) ) ).
% less_imp_le
thf(fact_573_less__imp__le,axiom,
! [X2: set_vec_complex,Y2: set_vec_complex] :
( ( ord_le787823215419015463omplex @ X2 @ Y2 )
=> ( ord_le8044543173838861339omplex @ X2 @ Y2 ) ) ).
% less_imp_le
thf(fact_574_less__imp__le,axiom,
! [X2: set_vec_a,Y2: set_vec_a] :
( ( ord_less_set_vec_a @ X2 @ Y2 )
=> ( ord_le4791951621262958845_vec_a @ X2 @ Y2 ) ) ).
% less_imp_le
thf(fact_575_less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% less_imp_le
thf(fact_576_le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% le_less_linear
thf(fact_577_le__imp__less__or__eq,axiom,
! [X2: set_mat_complex,Y2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X2 @ Y2 )
=> ( ( ord_le5598786136212072115omplex @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% le_imp_less_or_eq
thf(fact_578_le__imp__less__or__eq,axiom,
! [X2: set_vec_complex,Y2: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ X2 @ Y2 )
=> ( ( ord_le787823215419015463omplex @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% le_imp_less_or_eq
thf(fact_579_le__imp__less__or__eq,axiom,
! [X2: set_vec_a,Y2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X2 @ Y2 )
=> ( ( ord_less_set_vec_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% le_imp_less_or_eq
thf(fact_580_le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% le_imp_less_or_eq
thf(fact_581_less__le__not__le,axiom,
( ord_le5598786136212072115omplex
= ( ^ [X3: set_mat_complex,Y5: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X3 @ Y5 )
& ~ ( ord_le3632134057777142183omplex @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_582_less__le__not__le,axiom,
( ord_le787823215419015463omplex
= ( ^ [X3: set_vec_complex,Y5: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ X3 @ Y5 )
& ~ ( ord_le8044543173838861339omplex @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_583_less__le__not__le,axiom,
( ord_less_set_vec_a
= ( ^ [X3: set_vec_a,Y5: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X3 @ Y5 )
& ~ ( ord_le4791951621262958845_vec_a @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_584_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_585_not__le__imp__less,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_586_order_Oorder__iff__strict,axiom,
( ord_le3632134057777142183omplex
= ( ^ [A2: set_mat_complex,B2: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_587_order_Oorder__iff__strict,axiom,
( ord_le8044543173838861339omplex
= ( ^ [A2: set_vec_complex,B2: set_vec_complex] :
( ( ord_le787823215419015463omplex @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_588_order_Oorder__iff__strict,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [A2: set_vec_a,B2: set_vec_a] :
( ( ord_less_set_vec_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_589_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_590_order_Ostrict__iff__order,axiom,
( ord_le5598786136212072115omplex
= ( ^ [A2: set_mat_complex,B2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_591_order_Ostrict__iff__order,axiom,
( ord_le787823215419015463omplex
= ( ^ [A2: set_vec_complex,B2: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_592_order_Ostrict__iff__order,axiom,
( ord_less_set_vec_a
= ( ^ [A2: set_vec_a,B2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_593_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_594_order_Ostrict__trans1,axiom,
! [A: set_mat_complex,B: set_mat_complex,C: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A @ B )
=> ( ( ord_le5598786136212072115omplex @ B @ C )
=> ( ord_le5598786136212072115omplex @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_595_order_Ostrict__trans1,axiom,
! [A: set_vec_complex,B: set_vec_complex,C: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ A @ B )
=> ( ( ord_le787823215419015463omplex @ B @ C )
=> ( ord_le787823215419015463omplex @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_596_order_Ostrict__trans1,axiom,
! [A: set_vec_a,B: set_vec_a,C: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A @ B )
=> ( ( ord_less_set_vec_a @ B @ C )
=> ( ord_less_set_vec_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_597_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_598_order_Ostrict__trans2,axiom,
! [A: set_mat_complex,B: set_mat_complex,C: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ A @ B )
=> ( ( ord_le3632134057777142183omplex @ B @ C )
=> ( ord_le5598786136212072115omplex @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_599_order_Ostrict__trans2,axiom,
! [A: set_vec_complex,B: set_vec_complex,C: set_vec_complex] :
( ( ord_le787823215419015463omplex @ A @ B )
=> ( ( ord_le8044543173838861339omplex @ B @ C )
=> ( ord_le787823215419015463omplex @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_600_order_Ostrict__trans2,axiom,
! [A: set_vec_a,B: set_vec_a,C: set_vec_a] :
( ( ord_less_set_vec_a @ A @ B )
=> ( ( ord_le4791951621262958845_vec_a @ B @ C )
=> ( ord_less_set_vec_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_601_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_602_order_Ostrict__iff__not,axiom,
( ord_le5598786136212072115omplex
= ( ^ [A2: set_mat_complex,B2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A2 @ B2 )
& ~ ( ord_le3632134057777142183omplex @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_603_order_Ostrict__iff__not,axiom,
( ord_le787823215419015463omplex
= ( ^ [A2: set_vec_complex,B2: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ A2 @ B2 )
& ~ ( ord_le8044543173838861339omplex @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_604_order_Ostrict__iff__not,axiom,
( ord_less_set_vec_a
= ( ^ [A2: set_vec_a,B2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ B2 )
& ~ ( ord_le4791951621262958845_vec_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_605_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_606_dual__order_Oorder__iff__strict,axiom,
( ord_le3632134057777142183omplex
= ( ^ [B2: set_mat_complex,A2: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_607_dual__order_Oorder__iff__strict,axiom,
( ord_le8044543173838861339omplex
= ( ^ [B2: set_vec_complex,A2: set_vec_complex] :
( ( ord_le787823215419015463omplex @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_608_dual__order_Oorder__iff__strict,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [B2: set_vec_a,A2: set_vec_a] :
( ( ord_less_set_vec_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_609_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_610_dual__order_Ostrict__iff__order,axiom,
( ord_le5598786136212072115omplex
= ( ^ [B2: set_mat_complex,A2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_611_dual__order_Ostrict__iff__order,axiom,
( ord_le787823215419015463omplex
= ( ^ [B2: set_vec_complex,A2: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_612_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_vec_a
= ( ^ [B2: set_vec_a,A2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_613_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_614_dual__order_Ostrict__trans1,axiom,
! [B: set_mat_complex,A: set_mat_complex,C: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ B @ A )
=> ( ( ord_le5598786136212072115omplex @ C @ B )
=> ( ord_le5598786136212072115omplex @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_615_dual__order_Ostrict__trans1,axiom,
! [B: set_vec_complex,A: set_vec_complex,C: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ B @ A )
=> ( ( ord_le787823215419015463omplex @ C @ B )
=> ( ord_le787823215419015463omplex @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_616_dual__order_Ostrict__trans1,axiom,
! [B: set_vec_a,A: set_vec_a,C: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ B @ A )
=> ( ( ord_less_set_vec_a @ C @ B )
=> ( ord_less_set_vec_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_617_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_618_dual__order_Ostrict__trans2,axiom,
! [B: set_mat_complex,A: set_mat_complex,C: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ B @ A )
=> ( ( ord_le3632134057777142183omplex @ C @ B )
=> ( ord_le5598786136212072115omplex @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_619_dual__order_Ostrict__trans2,axiom,
! [B: set_vec_complex,A: set_vec_complex,C: set_vec_complex] :
( ( ord_le787823215419015463omplex @ B @ A )
=> ( ( ord_le8044543173838861339omplex @ C @ B )
=> ( ord_le787823215419015463omplex @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_620_dual__order_Ostrict__trans2,axiom,
! [B: set_vec_a,A: set_vec_a,C: set_vec_a] :
( ( ord_less_set_vec_a @ B @ A )
=> ( ( ord_le4791951621262958845_vec_a @ C @ B )
=> ( ord_less_set_vec_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_621_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_622_dual__order_Ostrict__iff__not,axiom,
( ord_le5598786136212072115omplex
= ( ^ [B2: set_mat_complex,A2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ B2 @ A2 )
& ~ ( ord_le3632134057777142183omplex @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_623_dual__order_Ostrict__iff__not,axiom,
( ord_le787823215419015463omplex
= ( ^ [B2: set_vec_complex,A2: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ B2 @ A2 )
& ~ ( ord_le8044543173838861339omplex @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_624_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_vec_a
= ( ^ [B2: set_vec_a,A2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ B2 @ A2 )
& ~ ( ord_le4791951621262958845_vec_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_625_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_626_order_Ostrict__implies__order,axiom,
! [A: set_mat_complex,B: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ A @ B )
=> ( ord_le3632134057777142183omplex @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_627_order_Ostrict__implies__order,axiom,
! [A: set_vec_complex,B: set_vec_complex] :
( ( ord_le787823215419015463omplex @ A @ B )
=> ( ord_le8044543173838861339omplex @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_628_order_Ostrict__implies__order,axiom,
! [A: set_vec_a,B: set_vec_a] :
( ( ord_less_set_vec_a @ A @ B )
=> ( ord_le4791951621262958845_vec_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_629_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_630_dual__order_Ostrict__implies__order,axiom,
! [B: set_mat_complex,A: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ B @ A )
=> ( ord_le3632134057777142183omplex @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_631_dual__order_Ostrict__implies__order,axiom,
! [B: set_vec_complex,A: set_vec_complex] :
( ( ord_le787823215419015463omplex @ B @ A )
=> ( ord_le8044543173838861339omplex @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_632_dual__order_Ostrict__implies__order,axiom,
! [B: set_vec_a,A: set_vec_a] :
( ( ord_less_set_vec_a @ B @ A )
=> ( ord_le4791951621262958845_vec_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_633_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_634_rel__simps_I47_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% rel_simps(47)
thf(fact_635_compl__le__compl__iff,axiom,
! [X2: set_mat_complex,Y2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ ( uminus5815530220087396478omplex @ X2 ) @ ( uminus5815530220087396478omplex @ Y2 ) )
= ( ord_le3632134057777142183omplex @ Y2 @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_636_compl__le__compl__iff,axiom,
! [X2: set_vec_complex,Y2: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ ( uminus1004567299294339826omplex @ X2 ) @ ( uminus1004567299294339826omplex @ Y2 ) )
= ( ord_le8044543173838861339omplex @ Y2 @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_637_compl__le__compl__iff,axiom,
! [X2: set_vec_a,Y2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ ( uminus2769705506071317478_vec_a @ X2 ) @ ( uminus2769705506071317478_vec_a @ Y2 ) )
= ( ord_le4791951621262958845_vec_a @ Y2 @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_638_compl__le__swap2,axiom,
! [Y2: set_mat_complex,X2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ ( uminus5815530220087396478omplex @ Y2 ) @ X2 )
=> ( ord_le3632134057777142183omplex @ ( uminus5815530220087396478omplex @ X2 ) @ Y2 ) ) ).
% compl_le_swap2
thf(fact_639_compl__le__swap2,axiom,
! [Y2: set_vec_complex,X2: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ ( uminus1004567299294339826omplex @ Y2 ) @ X2 )
=> ( ord_le8044543173838861339omplex @ ( uminus1004567299294339826omplex @ X2 ) @ Y2 ) ) ).
% compl_le_swap2
thf(fact_640_compl__le__swap2,axiom,
! [Y2: set_vec_a,X2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ ( uminus2769705506071317478_vec_a @ Y2 ) @ X2 )
=> ( ord_le4791951621262958845_vec_a @ ( uminus2769705506071317478_vec_a @ X2 ) @ Y2 ) ) ).
% compl_le_swap2
thf(fact_641_compl__le__swap1,axiom,
! [Y2: set_mat_complex,X2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ Y2 @ ( uminus5815530220087396478omplex @ X2 ) )
=> ( ord_le3632134057777142183omplex @ X2 @ ( uminus5815530220087396478omplex @ Y2 ) ) ) ).
% compl_le_swap1
thf(fact_642_compl__le__swap1,axiom,
! [Y2: set_vec_complex,X2: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ Y2 @ ( uminus1004567299294339826omplex @ X2 ) )
=> ( ord_le8044543173838861339omplex @ X2 @ ( uminus1004567299294339826omplex @ Y2 ) ) ) ).
% compl_le_swap1
thf(fact_643_compl__le__swap1,axiom,
! [Y2: set_vec_a,X2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ Y2 @ ( uminus2769705506071317478_vec_a @ X2 ) )
=> ( ord_le4791951621262958845_vec_a @ X2 @ ( uminus2769705506071317478_vec_a @ Y2 ) ) ) ).
% compl_le_swap1
thf(fact_644_compl__mono,axiom,
! [X2: set_mat_complex,Y2: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ X2 @ Y2 )
=> ( ord_le3632134057777142183omplex @ ( uminus5815530220087396478omplex @ Y2 ) @ ( uminus5815530220087396478omplex @ X2 ) ) ) ).
% compl_mono
thf(fact_645_compl__mono,axiom,
! [X2: set_vec_complex,Y2: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ X2 @ Y2 )
=> ( ord_le8044543173838861339omplex @ ( uminus1004567299294339826omplex @ Y2 ) @ ( uminus1004567299294339826omplex @ X2 ) ) ) ).
% compl_mono
thf(fact_646_compl__mono,axiom,
! [X2: set_vec_a,Y2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ X2 @ Y2 )
=> ( ord_le4791951621262958845_vec_a @ ( uminus2769705506071317478_vec_a @ Y2 ) @ ( uminus2769705506071317478_vec_a @ X2 ) ) ) ).
% compl_mono
thf(fact_647_subset__code_I1_J,axiom,
! [Xs: list_mat_a,B3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( set_mat_a2 @ Xs ) @ B3 )
= ( ! [X3: mat_a] :
( ( member_mat_a @ X3 @ ( set_mat_a2 @ Xs ) )
=> ( member_mat_a @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_648_subset__code_I1_J,axiom,
! [Xs: list_l5436439031154120755omplex,B3: set_list_mat_complex] :
( ( ord_le7594668674868021933omplex @ ( set_list_mat_complex2 @ Xs ) @ B3 )
= ( ! [X3: list_mat_complex] :
( ( member279434397506102358omplex @ X3 @ ( set_list_mat_complex2 @ Xs ) )
=> ( member279434397506102358omplex @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_649_subset__code_I1_J,axiom,
! [Xs: list_list_mat_a,B3: set_list_mat_a] :
( ( ord_le4771995077433322369_mat_a @ ( set_list_mat_a2 @ Xs ) @ B3 )
= ( ! [X3: list_mat_a] :
( ( member_list_mat_a @ X3 @ ( set_list_mat_a2 @ Xs ) )
=> ( member_list_mat_a @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_650_subset__code_I1_J,axiom,
! [Xs: list_vec_complex,B3: set_vec_complex] :
( ( ord_le8044543173838861339omplex @ ( set_vec_complex2 @ Xs ) @ B3 )
= ( ! [X3: vec_complex] :
( ( member_vec_complex @ X3 @ ( set_vec_complex2 @ Xs ) )
=> ( member_vec_complex @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_651_subset__code_I1_J,axiom,
! [Xs: list_vec_a,B3: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ ( set_vec_a2 @ Xs ) @ B3 )
= ( ! [X3: vec_a] :
( ( member_vec_a @ X3 @ ( set_vec_a2 @ Xs ) )
=> ( member_vec_a @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_652_subset__code_I1_J,axiom,
! [Xs: list_mat_complex,B3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ ( set_mat_complex2 @ Xs ) @ B3 )
= ( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ ( set_mat_complex2 @ Xs ) )
=> ( member_mat_complex @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_653_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_654_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_655_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_656_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_657_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_658_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_659_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_660_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_661_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_662_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_663_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_664_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_665_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_666_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_667_verit__eq__simplify_I24_J,axiom,
one_one_nat != zero_zero_nat ).
% verit_eq_simplify(24)
thf(fact_668_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_669_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_670_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_671_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_672_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_673_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_674_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_675_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_676_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_677_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_678_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_679_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_680_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_681_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_682_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_683_step__3__a_Osimps_I1_J,axiom,
! [J: nat,A3: mat_complex] :
( ( jordan2858886415929732048omplex @ zero_zero_nat @ J @ A3 )
= A3 ) ).
% step_3_a.simps(1)
thf(fact_684_step__3__c__inner__loop_Osimps_I1_J,axiom,
! [Val: complex,L: nat,I4: nat,A3: mat_complex] :
( ( jordan7656109678144820486omplex @ Val @ L @ I4 @ zero_zero_nat @ A3 )
= A3 ) ).
% step_3_c_inner_loop.simps(1)
thf(fact_685_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_686_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_687_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_688_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_689_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_690_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_691_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_692_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_693_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_694_verit__comp__simplify_I28_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% verit_comp_simplify(28)
thf(fact_695_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_696_rel__simps_I68_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% rel_simps(68)
thf(fact_697_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_698_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_699_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_700_mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel1
thf(fact_701_mult__less__mono2,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I4 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_702_mult__less__mono1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_703_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_704_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_705_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_706_unitarily__equiv__carrier_H_I1_J,axiom,
! [A3: mat_complex,B3: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A3 @ B3 @ U )
=> ( member_mat_complex @ A3 @ ( carrier_mat_complex @ ( dim_row_complex @ A3 ) @ ( dim_row_complex @ A3 ) ) ) ) ).
% unitarily_equiv_carrier'(1)
thf(fact_707_unitarily__equiv__carrier_H_I2_J,axiom,
! [A3: mat_complex,B3: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A3 @ B3 @ U )
=> ( member_mat_complex @ B3 @ ( carrier_mat_complex @ ( dim_row_complex @ A3 ) @ ( dim_row_complex @ A3 ) ) ) ) ).
% unitarily_equiv_carrier'(2)
thf(fact_708_unitarily__equiv__carrier_H_I3_J,axiom,
! [A3: mat_complex,B3: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A3 @ B3 @ U )
=> ( member_mat_complex @ U @ ( carrier_mat_complex @ ( dim_row_complex @ A3 ) @ ( dim_row_complex @ A3 ) ) ) ) ).
% unitarily_equiv_carrier'(3)
thf(fact_709_unitarily__equiv__square,axiom,
! [A3: mat_complex,N: nat,B3: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A3 @ B3 @ U )
=> ( spectr6340060708231679580omplex @ ( times_8009071140041733218omplex @ A3 @ A3 ) @ ( times_8009071140041733218omplex @ B3 @ B3 ) @ U ) ) ) ).
% unitarily_equiv_square
thf(fact_710_unitarily__equiv__uminus,axiom,
! [A3: mat_complex,N: nat,B3: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A3 @ B3 @ U )
=> ( spectr6340060708231679580omplex @ ( uminus467866341702955550omplex @ A3 ) @ ( uminus467866341702955550omplex @ B3 ) @ U ) ) ) ).
% unitarily_equiv_uminus
thf(fact_711_row__mat__of__row,axiom,
! [Y2: vec_complex] :
( ( row_complex @ ( mat_of_row_complex @ Y2 ) @ zero_zero_nat )
= Y2 ) ).
% row_mat_of_row
thf(fact_712_row__mat__of__row,axiom,
! [Y2: vec_a] :
( ( row_a @ ( mat_of_row_a @ Y2 ) @ zero_zero_nat )
= Y2 ) ).
% row_mat_of_row
thf(fact_713_length__pos__if__in__set,axiom,
! [X2: list_mat_complex,Xs: list_l5436439031154120755omplex] :
( ( member279434397506102358omplex @ X2 @ ( set_list_mat_complex2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s479360804472521375omplex @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_714_length__pos__if__in__set,axiom,
! [X2: list_mat_a,Xs: list_list_mat_a] :
( ( member_list_mat_a @ X2 @ ( set_list_mat_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6656407794899724303_mat_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_715_length__pos__if__in__set,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_716_length__pos__if__in__set,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_717_length__pos__if__in__set,axiom,
! [X2: vec_complex,Xs: list_vec_complex] :
( ( member_vec_complex @ X2 @ ( set_vec_complex2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s1158823550072163597omplex @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_718_length__pos__if__in__set,axiom,
! [X2: vec_a,Xs: list_vec_a] :
( ( member_vec_a @ X2 @ ( set_vec_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_vec_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_719_length__pos__if__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_720_length__pos__if__in__set,axiom,
! [X2: mat_a,Xs: list_mat_a] :
( ( member_mat_a @ X2 @ ( set_mat_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_mat_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_721_length__pos__if__in__set,axiom,
! [X2: mat_complex,Xs: list_mat_complex] :
( ( member_mat_complex @ X2 @ ( set_mat_complex2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s5969786470865220249omplex @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_722_rows__carrier,axiom,
! [A3: mat_a] : ( ord_le4791951621262958845_vec_a @ ( set_vec_a2 @ ( rows_a @ A3 ) ) @ ( carrier_vec_a @ ( dim_col_a @ A3 ) ) ) ).
% rows_carrier
thf(fact_723_rows__carrier,axiom,
! [A3: mat_complex] : ( ord_le8044543173838861339omplex @ ( set_vec_complex2 @ ( rows_complex @ A3 ) ) @ ( carrier_vec_complex @ ( dim_col_complex @ A3 ) ) ) ).
% rows_carrier
thf(fact_724_cols__dim,axiom,
! [A3: mat_mat_complex] : ( ord_le1193683088243165222omplex @ ( set_vec_mat_complex2 @ ( cols_mat_complex @ A3 ) ) @ ( carrie1048208924330543741omplex @ ( dim_row_mat_complex @ A3 ) ) ) ).
% cols_dim
thf(fact_725_cols__dim,axiom,
! [A3: mat_mat_a] : ( ord_le1807819179058128968_mat_a @ ( set_vec_mat_a2 @ ( cols_mat_a @ A3 ) ) @ ( carrier_vec_mat_a @ ( dim_row_mat_a @ A3 ) ) ) ).
% cols_dim
thf(fact_726_cols__dim,axiom,
! [A3: mat_a] : ( ord_le4791951621262958845_vec_a @ ( set_vec_a2 @ ( cols_a @ A3 ) ) @ ( carrier_vec_a @ ( dim_row_a @ A3 ) ) ) ).
% cols_dim
thf(fact_727_cols__dim,axiom,
! [A3: mat_complex] : ( ord_le8044543173838861339omplex @ ( set_vec_complex2 @ ( cols_complex @ A3 ) ) @ ( carrier_vec_complex @ ( dim_row_complex @ A3 ) ) ) ).
% cols_dim
thf(fact_728_mult__eq__1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= one_one_nat )
= ( ( A = one_one_nat )
& ( B = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_729_list_Opred__mono,axiom,
! [P: mat_complex > $o,Pa: mat_complex > $o] :
( ( ord_le2790225379703085046plex_o @ P @ Pa )
=> ( ord_le1186661314528880752plex_o @ ( list_all_mat_complex @ P ) @ ( list_all_mat_complex @ Pa ) ) ) ).
% list.pred_mono
thf(fact_730_list_Opred__mono,axiom,
! [P: mat_a > $o,Pa: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ P @ Pa )
=> ( ord_le963178399796136004at_a_o @ ( list_all_mat_a @ P ) @ ( list_all_mat_a @ Pa ) ) ) ).
% list.pred_mono
thf(fact_731_Compl__eq__Compl__iff,axiom,
! [A3: set_mat_complex,B3: set_mat_complex] :
( ( ( uminus5815530220087396478omplex @ A3 )
= ( uminus5815530220087396478omplex @ B3 ) )
= ( A3 = B3 ) ) ).
% Compl_eq_Compl_iff
thf(fact_732_double__complement,axiom,
! [A3: set_mat_complex] :
( ( uminus5815530220087396478omplex @ ( uminus5815530220087396478omplex @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_733_Compl__iff,axiom,
! [C: vec_complex,A3: set_vec_complex] :
( ( member_vec_complex @ C @ ( uminus1004567299294339826omplex @ A3 ) )
= ( ~ ( member_vec_complex @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_734_Compl__iff,axiom,
! [C: vec_a,A3: set_vec_a] :
( ( member_vec_a @ C @ ( uminus2769705506071317478_vec_a @ A3 ) )
= ( ~ ( member_vec_a @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_735_Compl__iff,axiom,
! [C: mat_a,A3: set_mat_a] :
( ( member_mat_a @ C @ ( uminus1296375033039821146_mat_a @ A3 ) )
= ( ~ ( member_mat_a @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_736_Compl__iff,axiom,
! [C: list_mat_complex,A3: set_list_mat_complex] :
( ( member279434397506102358omplex @ C @ ( uminus5491753114148463108omplex @ A3 ) )
= ( ~ ( member279434397506102358omplex @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_737_Compl__iff,axiom,
! [C: list_mat_a,A3: set_list_mat_a] :
( ( member_list_mat_a @ C @ ( uminus1627440288842321386_mat_a @ A3 ) )
= ( ~ ( member_list_mat_a @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_738_Compl__iff,axiom,
! [C: mat_complex,A3: set_mat_complex] :
( ( member_mat_complex @ C @ ( uminus5815530220087396478omplex @ A3 ) )
= ( ~ ( member_mat_complex @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_739_ComplI,axiom,
! [C: vec_complex,A3: set_vec_complex] :
( ~ ( member_vec_complex @ C @ A3 )
=> ( member_vec_complex @ C @ ( uminus1004567299294339826omplex @ A3 ) ) ) ).
% ComplI
thf(fact_740_ComplI,axiom,
! [C: vec_a,A3: set_vec_a] :
( ~ ( member_vec_a @ C @ A3 )
=> ( member_vec_a @ C @ ( uminus2769705506071317478_vec_a @ A3 ) ) ) ).
% ComplI
thf(fact_741_ComplI,axiom,
! [C: mat_a,A3: set_mat_a] :
( ~ ( member_mat_a @ C @ A3 )
=> ( member_mat_a @ C @ ( uminus1296375033039821146_mat_a @ A3 ) ) ) ).
% ComplI
thf(fact_742_ComplI,axiom,
! [C: list_mat_complex,A3: set_list_mat_complex] :
( ~ ( member279434397506102358omplex @ C @ A3 )
=> ( member279434397506102358omplex @ C @ ( uminus5491753114148463108omplex @ A3 ) ) ) ).
% ComplI
thf(fact_743_ComplI,axiom,
! [C: list_mat_a,A3: set_list_mat_a] :
( ~ ( member_list_mat_a @ C @ A3 )
=> ( member_list_mat_a @ C @ ( uminus1627440288842321386_mat_a @ A3 ) ) ) ).
% ComplI
thf(fact_744_ComplI,axiom,
! [C: mat_complex,A3: set_mat_complex] :
( ~ ( member_mat_complex @ C @ A3 )
=> ( member_mat_complex @ C @ ( uminus5815530220087396478omplex @ A3 ) ) ) ).
% ComplI
thf(fact_745_ComplD,axiom,
! [C: vec_complex,A3: set_vec_complex] :
( ( member_vec_complex @ C @ ( uminus1004567299294339826omplex @ A3 ) )
=> ~ ( member_vec_complex @ C @ A3 ) ) ).
% ComplD
thf(fact_746_ComplD,axiom,
! [C: vec_a,A3: set_vec_a] :
( ( member_vec_a @ C @ ( uminus2769705506071317478_vec_a @ A3 ) )
=> ~ ( member_vec_a @ C @ A3 ) ) ).
% ComplD
thf(fact_747_ComplD,axiom,
! [C: mat_a,A3: set_mat_a] :
( ( member_mat_a @ C @ ( uminus1296375033039821146_mat_a @ A3 ) )
=> ~ ( member_mat_a @ C @ A3 ) ) ).
% ComplD
thf(fact_748_ComplD,axiom,
! [C: list_mat_complex,A3: set_list_mat_complex] :
( ( member279434397506102358omplex @ C @ ( uminus5491753114148463108omplex @ A3 ) )
=> ~ ( member279434397506102358omplex @ C @ A3 ) ) ).
% ComplD
thf(fact_749_ComplD,axiom,
! [C: list_mat_a,A3: set_list_mat_a] :
( ( member_list_mat_a @ C @ ( uminus1627440288842321386_mat_a @ A3 ) )
=> ~ ( member_list_mat_a @ C @ A3 ) ) ).
% ComplD
thf(fact_750_ComplD,axiom,
! [C: mat_complex,A3: set_mat_complex] :
( ( member_mat_complex @ C @ ( uminus5815530220087396478omplex @ A3 ) )
=> ~ ( member_mat_complex @ C @ A3 ) ) ).
% ComplD
thf(fact_751_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_752_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_753_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_754_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_755_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_756_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_757_le__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% le_simps(1)
thf(fact_758_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_759_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_760_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_761_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_762_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I4: nat,J: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_763_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_764_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_765_mult__le__mono,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_766_mult__le__mono1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_767_mult__le__mono2,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I4 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_768_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K2 )
=> ~ ( P @ I ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_769_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_770_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_771_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_772_mult__if__delta,axiom,
! [P: $o,Q3: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q3 )
= Q3 ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q3 )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_773_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_774_le__trans,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I4 @ K ) ) ) ).
% le_trans
thf(fact_775_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_776_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_777_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_778_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_779_step__2__def,axiom,
( jordan7871273693253786478omplex
= ( ^ [A4: mat_complex] : ( jordan6916311984355858983omplex @ ( dim_row_complex @ A4 ) @ zero_zero_nat @ A4 ) ) ) ).
% step_2_def
thf(fact_780_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K2: nat] :
? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( F @ K2 @ I ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F @ K4 @ I2 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_781_triangular__to__jnf__steps__dims_I3_J,axiom,
! [A3: mat_complex] :
( ( dim_row_complex @ ( jordan7871273693253786478omplex @ A3 ) )
= ( dim_row_complex @ A3 ) ) ).
% triangular_to_jnf_steps_dims(3)
thf(fact_782_triangular__to__jnf__steps__dims_I4_J,axiom,
! [A3: mat_complex] :
( ( dim_col_complex @ ( jordan7871273693253786478omplex @ A3 ) )
= ( dim_col_complex @ A3 ) ) ).
% triangular_to_jnf_steps_dims(4)
thf(fact_783_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I: nat] :
( ( ord_less_nat @ K2 @ I )
=> ( P @ I ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_784_density__collapse__carrier,axiom,
! [R: mat_complex,P: mat_complex,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ R ) )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( member_mat_complex @ ( projec3470689467825365843llapse @ R @ P ) @ ( carrier_mat_complex @ N @ N ) ) ) ) ) ).
% density_collapse_carrier
thf(fact_785_step__1__def,axiom,
( jordan2017415923357163885omplex
= ( ^ [A4: mat_complex] : ( jordan9130142659770429862omplex @ ( dim_row_complex @ A4 ) @ zero_zero_nat @ zero_zero_nat @ A4 ) ) ) ).
% step_1_def
thf(fact_786_triangular__to__jnf__steps__dims_I1_J,axiom,
! [A3: mat_complex] :
( ( dim_row_complex @ ( jordan2017415923357163885omplex @ A3 ) )
= ( dim_row_complex @ A3 ) ) ).
% triangular_to_jnf_steps_dims(1)
thf(fact_787_triangular__to__jnf__steps__dims_I2_J,axiom,
! [A3: mat_complex] :
( ( dim_col_complex @ ( jordan2017415923357163885omplex @ A3 ) )
= ( dim_col_complex @ A3 ) ) ).
% triangular_to_jnf_steps_dims(2)
thf(fact_788_mult__delta__right,axiom,
! [B: $o,X2: nat,Y2: nat] :
( ( B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y2 @ zero_zero_nat ) )
= ( times_times_nat @ X2 @ Y2 ) ) )
& ( ~ B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y2 @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_789_mult__delta__left,axiom,
! [B: $o,X2: nat,Y2: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y2 )
= ( times_times_nat @ X2 @ Y2 ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y2 )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_790_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_791_unit__vecs__last__carrier,axiom,
! [N: nat,I4: nat] : ( ord_le8044543173838861339omplex @ ( set_vec_complex2 @ ( unit_v8657589406246362837omplex @ N @ I4 ) ) @ ( carrier_vec_complex @ N ) ) ).
% unit_vecs_last_carrier
thf(fact_792_unit__vecs__carrier,axiom,
! [N: nat] : ( ord_le8044543173838861339omplex @ ( set_vec_complex2 @ ( unit_vecs_complex @ N ) ) @ ( carrier_vec_complex @ N ) ) ).
% unit_vecs_carrier
thf(fact_793_diag__block__mat__cong__hd,axiom,
! [Al: list_mat_a,Bl: list_mat_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_mat_a @ Al ) )
=> ( ( ( size_size_list_mat_a @ Al )
= ( size_size_list_mat_a @ Bl ) )
=> ( ( ( dim_row_a @ ( hd_mat_a @ Al ) )
= ( dim_row_a @ ( hd_mat_a @ Bl ) ) )
=> ( ( ( dim_col_a @ ( hd_mat_a @ Al ) )
= ( dim_col_a @ ( hd_mat_a @ Bl ) ) )
=> ( ( ( diag_block_mat_a @ Al )
= ( diag_block_mat_a @ Bl ) )
=> ( ( hd_mat_a @ Al )
= ( hd_mat_a @ Bl ) ) ) ) ) ) ) ).
% diag_block_mat_cong_hd
thf(fact_794_diag__block__mat__cong__hd,axiom,
! [Al: list_mat_complex,Bl: list_mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5969786470865220249omplex @ Al ) )
=> ( ( ( size_s5969786470865220249omplex @ Al )
= ( size_s5969786470865220249omplex @ Bl ) )
=> ( ( ( dim_row_complex @ ( hd_mat_complex @ Al ) )
= ( dim_row_complex @ ( hd_mat_complex @ Bl ) ) )
=> ( ( ( dim_col_complex @ ( hd_mat_complex @ Al ) )
= ( dim_col_complex @ ( hd_mat_complex @ Bl ) ) )
=> ( ( ( diag_b9145358668110806138omplex @ Al )
= ( diag_b9145358668110806138omplex @ Bl ) )
=> ( ( hd_mat_complex @ Al )
= ( hd_mat_complex @ Bl ) ) ) ) ) ) ) ).
% diag_block_mat_cong_hd
thf(fact_795_diag__block__mat__cong__tl,axiom,
! [Al: list_mat_a,Bl: list_mat_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_mat_a @ Al ) )
=> ( ( ( size_size_list_mat_a @ Al )
= ( size_size_list_mat_a @ Bl ) )
=> ( ( ( dim_row_a @ ( hd_mat_a @ Al ) )
= ( dim_row_a @ ( hd_mat_a @ Bl ) ) )
=> ( ( ( dim_col_a @ ( hd_mat_a @ Al ) )
= ( dim_col_a @ ( hd_mat_a @ Bl ) ) )
=> ( ( ( diag_block_mat_a @ Al )
= ( diag_block_mat_a @ Bl ) )
=> ( ( diag_block_mat_a @ ( tl_mat_a @ Al ) )
= ( diag_block_mat_a @ ( tl_mat_a @ Bl ) ) ) ) ) ) ) ) ).
% diag_block_mat_cong_tl
thf(fact_796_diag__block__mat__cong__tl,axiom,
! [Al: list_mat_complex,Bl: list_mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5969786470865220249omplex @ Al ) )
=> ( ( ( size_s5969786470865220249omplex @ Al )
= ( size_s5969786470865220249omplex @ Bl ) )
=> ( ( ( dim_row_complex @ ( hd_mat_complex @ Al ) )
= ( dim_row_complex @ ( hd_mat_complex @ Bl ) ) )
=> ( ( ( dim_col_complex @ ( hd_mat_complex @ Al ) )
= ( dim_col_complex @ ( hd_mat_complex @ Bl ) ) )
=> ( ( ( diag_b9145358668110806138omplex @ Al )
= ( diag_b9145358668110806138omplex @ Bl ) )
=> ( ( diag_b9145358668110806138omplex @ ( tl_mat_complex @ Al ) )
= ( diag_b9145358668110806138omplex @ ( tl_mat_complex @ Bl ) ) ) ) ) ) ) ) ).
% diag_block_mat_cong_tl
thf(fact_797_length__code,axiom,
( size_s3451745648224563538omplex
= ( gen_length_complex @ zero_zero_nat ) ) ).
% length_code
thf(fact_798_length__code,axiom,
( size_size_list_a
= ( gen_length_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_799_length__code,axiom,
( size_s1158823550072163597omplex
= ( gen_le4087251840394152110omplex @ zero_zero_nat ) ) ).
% length_code
thf(fact_800_length__code,axiom,
( size_size_list_vec_a
= ( gen_length_vec_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_801_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_802_length__code,axiom,
( size_size_list_mat_a
= ( gen_length_mat_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_803_length__code,axiom,
( size_s5969786470865220249omplex
= ( gen_le107826107610854458omplex @ zero_zero_nat ) ) ).
% length_code
thf(fact_804_nth__equal__first__eq,axiom,
! [X2: list_mat_complex,Xs: list_l5436439031154120755omplex,N: nat] :
( ~ ( member279434397506102358omplex @ X2 @ ( set_list_mat_complex2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s479360804472521375omplex @ Xs ) )
=> ( ( ( nth_list_mat_complex @ ( cons_l4198107141827137507omplex @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_805_nth__equal__first__eq,axiom,
! [X2: list_mat_a,Xs: list_list_mat_a,N: nat] :
( ~ ( member_list_mat_a @ X2 @ ( set_list_mat_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s6656407794899724303_mat_a @ Xs ) )
=> ( ( ( nth_list_mat_a @ ( cons_list_mat_a @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_806_nth__equal__first__eq,axiom,
! [X2: complex,Xs: list_complex,N: nat] :
( ~ ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( ( ( nth_complex @ ( cons_complex @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_807_nth__equal__first__eq,axiom,
! [X2: a,Xs: list_a,N: nat] :
( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_808_nth__equal__first__eq,axiom,
! [X2: vec_complex,Xs: list_vec_complex,N: nat] :
( ~ ( member_vec_complex @ X2 @ ( set_vec_complex2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s1158823550072163597omplex @ Xs ) )
=> ( ( ( nth_vec_complex @ ( cons_vec_complex @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_809_nth__equal__first__eq,axiom,
! [X2: vec_a,Xs: list_vec_a,N: nat] :
( ~ ( member_vec_a @ X2 @ ( set_vec_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_vec_a @ Xs ) )
=> ( ( ( nth_vec_a @ ( cons_vec_a @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_810_nth__equal__first__eq,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_811_nth__equal__first__eq,axiom,
! [X2: mat_a,Xs: list_mat_a,N: nat] :
( ~ ( member_mat_a @ X2 @ ( set_mat_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_mat_a @ Xs ) )
=> ( ( ( nth_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_812_nth__equal__first__eq,axiom,
! [X2: mat_complex,Xs: list_mat_complex,N: nat] :
( ~ ( member_mat_complex @ X2 @ ( set_mat_complex2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ( ( nth_mat_complex @ ( cons_mat_complex @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_813_list_Osel_I1_J,axiom,
! [X21: mat_complex,X22: list_mat_complex] :
( ( hd_mat_complex @ ( cons_mat_complex @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_814_list_Osel_I1_J,axiom,
! [X21: mat_a,X22: list_mat_a] :
( ( hd_mat_a @ ( cons_mat_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_815_list_Osel_I1_J,axiom,
! [X21: vec_complex,X22: list_vec_complex] :
( ( hd_vec_complex @ ( cons_vec_complex @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_816_list_Osel_I1_J,axiom,
! [X21: vec_a,X22: list_vec_a] :
( ( hd_vec_a @ ( cons_vec_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_817_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_818_impossible__Cons,axiom,
! [Xs: list_complex,Ys: list_complex,X2: complex] :
( ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ ( size_s3451745648224563538omplex @ Ys ) )
=> ( Xs
!= ( cons_complex @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_819_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_820_impossible__Cons,axiom,
! [Xs: list_vec_complex,Ys: list_vec_complex,X2: vec_complex] :
( ( ord_less_eq_nat @ ( size_s1158823550072163597omplex @ Xs ) @ ( size_s1158823550072163597omplex @ Ys ) )
=> ( Xs
!= ( cons_vec_complex @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_821_impossible__Cons,axiom,
! [Xs: list_vec_a,Ys: list_vec_a,X2: vec_a] :
( ( ord_less_eq_nat @ ( size_size_list_vec_a @ Xs ) @ ( size_size_list_vec_a @ Ys ) )
=> ( Xs
!= ( cons_vec_a @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_822_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_823_impossible__Cons,axiom,
! [Xs: list_mat_a,Ys: list_mat_a,X2: mat_a] :
( ( ord_less_eq_nat @ ( size_size_list_mat_a @ Xs ) @ ( size_size_list_mat_a @ Ys ) )
=> ( Xs
!= ( cons_mat_a @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_824_impossible__Cons,axiom,
! [Xs: list_mat_complex,Ys: list_mat_complex,X2: mat_complex] :
( ( ord_less_eq_nat @ ( size_s5969786470865220249omplex @ Xs ) @ ( size_s5969786470865220249omplex @ Ys ) )
=> ( Xs
!= ( cons_mat_complex @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_825_set__subset__Cons,axiom,
! [Xs: list_l5436439031154120755omplex,X2: list_mat_complex] : ( ord_le7594668674868021933omplex @ ( set_list_mat_complex2 @ Xs ) @ ( set_list_mat_complex2 @ ( cons_l4198107141827137507omplex @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_826_set__subset__Cons,axiom,
! [Xs: list_list_mat_a,X2: list_mat_a] : ( ord_le4771995077433322369_mat_a @ ( set_list_mat_a2 @ Xs ) @ ( set_list_mat_a2 @ ( cons_list_mat_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_827_set__subset__Cons,axiom,
! [Xs: list_mat_a,X2: mat_a] : ( ord_le3318621148231462513_mat_a @ ( set_mat_a2 @ Xs ) @ ( set_mat_a2 @ ( cons_mat_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_828_set__subset__Cons,axiom,
! [Xs: list_nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_829_set__subset__Cons,axiom,
! [Xs: list_mat_complex,X2: mat_complex] : ( ord_le3632134057777142183omplex @ ( set_mat_complex2 @ Xs ) @ ( set_mat_complex2 @ ( cons_mat_complex @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_830_set__subset__Cons,axiom,
! [Xs: list_vec_complex,X2: vec_complex] : ( ord_le8044543173838861339omplex @ ( set_vec_complex2 @ Xs ) @ ( set_vec_complex2 @ ( cons_vec_complex @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_831_set__subset__Cons,axiom,
! [Xs: list_vec_a,X2: vec_a] : ( ord_le4791951621262958845_vec_a @ ( set_vec_a2 @ Xs ) @ ( set_vec_a2 @ ( cons_vec_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_832_not__Cons__self2,axiom,
! [X2: mat_complex,Xs: list_mat_complex] :
( ( cons_mat_complex @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_833_not__Cons__self2,axiom,
! [X2: mat_a,Xs: list_mat_a] :
( ( cons_mat_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_834_not__Cons__self2,axiom,
! [X2: vec_complex,Xs: list_vec_complex] :
( ( cons_vec_complex @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_835_not__Cons__self2,axiom,
! [X2: vec_a,Xs: list_vec_a] :
( ( cons_vec_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_836_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_837_list_Oinject,axiom,
! [X21: mat_complex,X22: list_mat_complex,Y21: mat_complex,Y22: list_mat_complex] :
( ( ( cons_mat_complex @ X21 @ X22 )
= ( cons_mat_complex @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_838_list_Oinject,axiom,
! [X21: mat_a,X22: list_mat_a,Y21: mat_a,Y22: list_mat_a] :
( ( ( cons_mat_a @ X21 @ X22 )
= ( cons_mat_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_839_list_Oinject,axiom,
! [X21: vec_complex,X22: list_vec_complex,Y21: vec_complex,Y22: list_vec_complex] :
( ( ( cons_vec_complex @ X21 @ X22 )
= ( cons_vec_complex @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_840_list_Oinject,axiom,
! [X21: vec_a,X22: list_vec_a,Y21: vec_a,Y22: list_vec_a] :
( ( ( cons_vec_a @ X21 @ X22 )
= ( cons_vec_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_841_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_842_list_Osel_I3_J,axiom,
! [X21: mat_complex,X22: list_mat_complex] :
( ( tl_mat_complex @ ( cons_mat_complex @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_843_list_Osel_I3_J,axiom,
! [X21: mat_a,X22: list_mat_a] :
( ( tl_mat_a @ ( cons_mat_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_844_list_Osel_I3_J,axiom,
! [X21: vec_complex,X22: list_vec_complex] :
( ( tl_vec_complex @ ( cons_vec_complex @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_845_list_Osel_I3_J,axiom,
! [X21: vec_a,X22: list_vec_a] :
( ( tl_vec_a @ ( cons_vec_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_846_list_Osel_I3_J,axiom,
! [X21: nat,X22: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_847_list__ex__simps_I1_J,axiom,
! [P: mat_complex > $o,X2: mat_complex,Xs: list_mat_complex] :
( ( list_ex_mat_complex @ P @ ( cons_mat_complex @ X2 @ Xs ) )
= ( ( P @ X2 )
| ( list_ex_mat_complex @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_848_list__ex__simps_I1_J,axiom,
! [P: mat_a > $o,X2: mat_a,Xs: list_mat_a] :
( ( list_ex_mat_a @ P @ ( cons_mat_a @ X2 @ Xs ) )
= ( ( P @ X2 )
| ( list_ex_mat_a @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_849_list__ex__simps_I1_J,axiom,
! [P: vec_complex > $o,X2: vec_complex,Xs: list_vec_complex] :
( ( list_ex_vec_complex @ P @ ( cons_vec_complex @ X2 @ Xs ) )
= ( ( P @ X2 )
| ( list_ex_vec_complex @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_850_list__ex__simps_I1_J,axiom,
! [P: vec_a > $o,X2: vec_a,Xs: list_vec_a] :
( ( list_ex_vec_a @ P @ ( cons_vec_a @ X2 @ Xs ) )
= ( ( P @ X2 )
| ( list_ex_vec_a @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_851_list__ex__simps_I1_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( list_ex_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( ( P @ X2 )
| ( list_ex_nat @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_852_list_Oset__intros_I2_J,axiom,
! [Y2: mat_complex,X22: list_mat_complex,X21: mat_complex] :
( ( member_mat_complex @ Y2 @ ( set_mat_complex2 @ X22 ) )
=> ( member_mat_complex @ Y2 @ ( set_mat_complex2 @ ( cons_mat_complex @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_853_list_Oset__intros_I1_J,axiom,
! [X21: mat_complex,X22: list_mat_complex] : ( member_mat_complex @ X21 @ ( set_mat_complex2 @ ( cons_mat_complex @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_854_list_Oset__cases,axiom,
! [E: mat_complex,A: list_mat_complex] :
( ( member_mat_complex @ E @ ( set_mat_complex2 @ A ) )
=> ( ! [Z22: list_mat_complex] :
( A
!= ( cons_mat_complex @ E @ Z22 ) )
=> ~ ! [Z1: mat_complex,Z22: list_mat_complex] :
( ( A
= ( cons_mat_complex @ Z1 @ Z22 ) )
=> ~ ( member_mat_complex @ E @ ( set_mat_complex2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_855_set__ConsD,axiom,
! [Y2: mat_complex,X2: mat_complex,Xs: list_mat_complex] :
( ( member_mat_complex @ Y2 @ ( set_mat_complex2 @ ( cons_mat_complex @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_mat_complex @ Y2 @ ( set_mat_complex2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_856_nth__Cons__0,axiom,
! [X2: mat_a,Xs: list_mat_a] :
( ( nth_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_857_nth__Cons__0,axiom,
! [X2: mat_complex,Xs: list_mat_complex] :
( ( nth_mat_complex @ ( cons_mat_complex @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_858_mat__vec__as__mat__mat__mult,axiom,
! [A3: mat_a,Nr: nat,Nc: nat,V: vec_a] :
( ( member_mat_a @ A3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ A3 @ V )
= ( col_a @ ( times_times_mat_a @ A3 @ ( mat_of_cols_a @ Nc @ ( cons_vec_a @ V @ nil_vec_a ) ) ) @ zero_zero_nat ) ) ) ) ).
% mat_vec_as_mat_mat_mult
thf(fact_859_mat__vec__as__mat__mat__mult,axiom,
! [A3: mat_complex,Nr: nat,Nc: nat,V: vec_complex] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_vec_complex @ V @ ( carrier_vec_complex @ Nc ) )
=> ( ( mult_mat_vec_complex @ A3 @ V )
= ( col_complex @ ( times_8009071140041733218omplex @ A3 @ ( mat_of_cols_complex @ Nc @ ( cons_vec_complex @ V @ nil_vec_complex ) ) ) @ zero_zero_nat ) ) ) ) ).
% mat_vec_as_mat_mat_mult
thf(fact_860_nth__non__equal__first__eq,axiom,
! [X2: mat_a,Y2: mat_a,Xs: list_mat_a,N: nat] :
( ( X2 != Y2 )
=> ( ( ( nth_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ N )
= Y2 )
= ( ( ( nth_mat_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_861_nth__non__equal__first__eq,axiom,
! [X2: mat_complex,Y2: mat_complex,Xs: list_mat_complex,N: nat] :
( ( X2 != Y2 )
=> ( ( ( nth_mat_complex @ ( cons_mat_complex @ X2 @ Xs ) @ N )
= Y2 )
= ( ( ( nth_mat_complex @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_862_nth__Cons__pos,axiom,
! [N: nat,X2: mat_a,Xs: list_mat_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ N )
= ( nth_mat_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_863_nth__Cons__pos,axiom,
! [N: nat,X2: mat_complex,Xs: list_mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_mat_complex @ ( cons_mat_complex @ X2 @ Xs ) @ N )
= ( nth_mat_complex @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_864_sorted__list__subset_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [A5: nat,As2: list_nat,B4: nat,Bs: list_nat] :
( ( ( A5 = B4 )
=> ( P @ As2 @ ( cons_nat @ B4 @ Bs ) ) )
=> ( ( ( A5 != B4 )
=> ( ( ord_less_nat @ B4 @ A5 )
=> ( P @ ( cons_nat @ A5 @ As2 ) @ Bs ) ) )
=> ( P @ ( cons_nat @ A5 @ As2 ) @ ( cons_nat @ B4 @ Bs ) ) ) )
=> ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [A5: nat,Uv: list_nat] : ( P @ ( cons_nat @ A5 @ Uv ) @ nil_nat )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sorted_list_subset.induct
thf(fact_865_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_866_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_867_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_868_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_869_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_870_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_871_nat__distrib_I4_J,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% nat_distrib(4)
thf(fact_872_nat__distrib_I3_J,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% nat_distrib(3)
thf(fact_873_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_874_diff__commute,axiom,
! [I4: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K ) @ J ) ) ).
% diff_commute
thf(fact_875_diff__diff__cancel,axiom,
! [I4: nat,N: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I4 ) )
= I4 ) ) ).
% diff_diff_cancel
thf(fact_876_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_877_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_878_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_879_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_880_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_881_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_882_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_883_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_884_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_885_minus__nat_Osimps_I1_J,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.simps(1)
thf(fact_886_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_887_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_888_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_889_verit__minus__simplify_I2_J,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_minus_simplify(2)
thf(fact_890_verit__minus__simplify_I1_J,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% verit_minus_simplify(1)
thf(fact_891_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_892_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_893_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_894_list_Osize_I3_J,axiom,
( ( size_size_list_mat_a @ nil_mat_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_895_list_Osize_I3_J,axiom,
( ( size_s5969786470865220249omplex @ nil_mat_complex )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_896_length__0__conv,axiom,
! [Xs: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_mat_a ) ) ).
% length_0_conv
thf(fact_897_length__0__conv,axiom,
! [Xs: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs )
= zero_zero_nat )
= ( Xs = nil_mat_complex ) ) ).
% length_0_conv
thf(fact_898_list__induct4,axiom,
! [Xs: list_mat_a,Ys: list_mat_a,Zs: list_mat_a,Ws: list_mat_a,P: list_mat_a > list_mat_a > list_mat_a > list_mat_a > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( ( size_size_list_mat_a @ Ys )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( ( size_size_list_mat_a @ Zs )
= ( size_size_list_mat_a @ Ws ) )
=> ( ( P @ nil_mat_a @ nil_mat_a @ nil_mat_a @ nil_mat_a )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_a,Ys4: list_mat_a,Z2: mat_a,Zs2: list_mat_a,W2: mat_a,Ws2: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( ( size_size_list_mat_a @ Zs2 )
= ( size_size_list_mat_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_a @ Y3 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs2 ) @ ( cons_mat_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_899_list__induct4,axiom,
! [Xs: list_mat_a,Ys: list_mat_a,Zs: list_mat_a,Ws: list_mat_complex,P: list_mat_a > list_mat_a > list_mat_a > list_mat_complex > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( ( size_size_list_mat_a @ Ys )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( ( size_size_list_mat_a @ Zs )
= ( size_s5969786470865220249omplex @ Ws ) )
=> ( ( P @ nil_mat_a @ nil_mat_a @ nil_mat_a @ nil_mat_complex )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_a,Ys4: list_mat_a,Z2: mat_a,Zs2: list_mat_a,W2: mat_complex,Ws2: list_mat_complex] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( ( size_size_list_mat_a @ Zs2 )
= ( size_s5969786470865220249omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_a @ Y3 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs2 ) @ ( cons_mat_complex @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_900_list__induct4,axiom,
! [Xs: list_mat_a,Ys: list_mat_a,Zs: list_mat_complex,Ws: list_mat_a,P: list_mat_a > list_mat_a > list_mat_complex > list_mat_a > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( ( size_size_list_mat_a @ Ys )
= ( size_s5969786470865220249omplex @ Zs ) )
=> ( ( ( size_s5969786470865220249omplex @ Zs )
= ( size_size_list_mat_a @ Ws ) )
=> ( ( P @ nil_mat_a @ nil_mat_a @ nil_mat_complex @ nil_mat_a )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_a,Ys4: list_mat_a,Z2: mat_complex,Zs2: list_mat_complex,W2: mat_a,Ws2: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_s5969786470865220249omplex @ Zs2 ) )
=> ( ( ( size_s5969786470865220249omplex @ Zs2 )
= ( size_size_list_mat_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_a @ Y3 @ Ys4 ) @ ( cons_mat_complex @ Z2 @ Zs2 ) @ ( cons_mat_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_901_list__induct4,axiom,
! [Xs: list_mat_a,Ys: list_mat_a,Zs: list_mat_complex,Ws: list_mat_complex,P: list_mat_a > list_mat_a > list_mat_complex > list_mat_complex > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( ( size_size_list_mat_a @ Ys )
= ( size_s5969786470865220249omplex @ Zs ) )
=> ( ( ( size_s5969786470865220249omplex @ Zs )
= ( size_s5969786470865220249omplex @ Ws ) )
=> ( ( P @ nil_mat_a @ nil_mat_a @ nil_mat_complex @ nil_mat_complex )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_a,Ys4: list_mat_a,Z2: mat_complex,Zs2: list_mat_complex,W2: mat_complex,Ws2: list_mat_complex] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_s5969786470865220249omplex @ Zs2 ) )
=> ( ( ( size_s5969786470865220249omplex @ Zs2 )
= ( size_s5969786470865220249omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_a @ Y3 @ Ys4 ) @ ( cons_mat_complex @ Z2 @ Zs2 ) @ ( cons_mat_complex @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_902_list__induct4,axiom,
! [Xs: list_mat_a,Ys: list_mat_complex,Zs: list_mat_a,Ws: list_mat_a,P: list_mat_a > list_mat_complex > list_mat_a > list_mat_a > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( ( size_size_list_mat_a @ Zs )
= ( size_size_list_mat_a @ Ws ) )
=> ( ( P @ nil_mat_a @ nil_mat_complex @ nil_mat_a @ nil_mat_a )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_complex,Ys4: list_mat_complex,Z2: mat_a,Zs2: list_mat_a,W2: mat_a,Ws2: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_s5969786470865220249omplex @ Ys4 ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys4 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( ( size_size_list_mat_a @ Zs2 )
= ( size_size_list_mat_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_complex @ Y3 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs2 ) @ ( cons_mat_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_903_list__induct4,axiom,
! [Xs: list_mat_a,Ys: list_mat_complex,Zs: list_mat_a,Ws: list_mat_complex,P: list_mat_a > list_mat_complex > list_mat_a > list_mat_complex > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( ( size_size_list_mat_a @ Zs )
= ( size_s5969786470865220249omplex @ Ws ) )
=> ( ( P @ nil_mat_a @ nil_mat_complex @ nil_mat_a @ nil_mat_complex )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_complex,Ys4: list_mat_complex,Z2: mat_a,Zs2: list_mat_a,W2: mat_complex,Ws2: list_mat_complex] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_s5969786470865220249omplex @ Ys4 ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys4 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( ( size_size_list_mat_a @ Zs2 )
= ( size_s5969786470865220249omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_complex @ Y3 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs2 ) @ ( cons_mat_complex @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_904_list__induct4,axiom,
! [Xs: list_mat_a,Ys: list_mat_complex,Zs: list_mat_complex,Ws: list_mat_a,P: list_mat_a > list_mat_complex > list_mat_complex > list_mat_a > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys )
= ( size_s5969786470865220249omplex @ Zs ) )
=> ( ( ( size_s5969786470865220249omplex @ Zs )
= ( size_size_list_mat_a @ Ws ) )
=> ( ( P @ nil_mat_a @ nil_mat_complex @ nil_mat_complex @ nil_mat_a )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_complex,Ys4: list_mat_complex,Z2: mat_complex,Zs2: list_mat_complex,W2: mat_a,Ws2: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_s5969786470865220249omplex @ Ys4 ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys4 )
= ( size_s5969786470865220249omplex @ Zs2 ) )
=> ( ( ( size_s5969786470865220249omplex @ Zs2 )
= ( size_size_list_mat_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_complex @ Y3 @ Ys4 ) @ ( cons_mat_complex @ Z2 @ Zs2 ) @ ( cons_mat_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_905_list__induct4,axiom,
! [Xs: list_mat_a,Ys: list_mat_complex,Zs: list_mat_complex,Ws: list_mat_complex,P: list_mat_a > list_mat_complex > list_mat_complex > list_mat_complex > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys )
= ( size_s5969786470865220249omplex @ Zs ) )
=> ( ( ( size_s5969786470865220249omplex @ Zs )
= ( size_s5969786470865220249omplex @ Ws ) )
=> ( ( P @ nil_mat_a @ nil_mat_complex @ nil_mat_complex @ nil_mat_complex )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_complex,Ys4: list_mat_complex,Z2: mat_complex,Zs2: list_mat_complex,W2: mat_complex,Ws2: list_mat_complex] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_s5969786470865220249omplex @ Ys4 ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys4 )
= ( size_s5969786470865220249omplex @ Zs2 ) )
=> ( ( ( size_s5969786470865220249omplex @ Zs2 )
= ( size_s5969786470865220249omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_complex @ Y3 @ Ys4 ) @ ( cons_mat_complex @ Z2 @ Zs2 ) @ ( cons_mat_complex @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_906_list__induct4,axiom,
! [Xs: list_mat_complex,Ys: list_mat_a,Zs: list_mat_a,Ws: list_mat_a,P: list_mat_complex > list_mat_a > list_mat_a > list_mat_a > $o] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( ( size_size_list_mat_a @ Ys )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( ( size_size_list_mat_a @ Zs )
= ( size_size_list_mat_a @ Ws ) )
=> ( ( P @ nil_mat_complex @ nil_mat_a @ nil_mat_a @ nil_mat_a )
=> ( ! [X4: mat_complex,Xs3: list_mat_complex,Y3: mat_a,Ys4: list_mat_a,Z2: mat_a,Zs2: list_mat_a,W2: mat_a,Ws2: list_mat_a] :
( ( ( size_s5969786470865220249omplex @ Xs3 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( ( size_size_list_mat_a @ Zs2 )
= ( size_size_list_mat_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_mat_complex @ X4 @ Xs3 ) @ ( cons_mat_a @ Y3 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs2 ) @ ( cons_mat_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_907_list__induct4,axiom,
! [Xs: list_mat_complex,Ys: list_mat_a,Zs: list_mat_a,Ws: list_mat_complex,P: list_mat_complex > list_mat_a > list_mat_a > list_mat_complex > $o] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( ( size_size_list_mat_a @ Ys )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( ( size_size_list_mat_a @ Zs )
= ( size_s5969786470865220249omplex @ Ws ) )
=> ( ( P @ nil_mat_complex @ nil_mat_a @ nil_mat_a @ nil_mat_complex )
=> ( ! [X4: mat_complex,Xs3: list_mat_complex,Y3: mat_a,Ys4: list_mat_a,Z2: mat_a,Zs2: list_mat_a,W2: mat_complex,Ws2: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs3 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( ( size_size_list_mat_a @ Zs2 )
= ( size_s5969786470865220249omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_mat_complex @ X4 @ Xs3 ) @ ( cons_mat_a @ Y3 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs2 ) @ ( cons_mat_complex @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_908_list__induct3,axiom,
! [Xs: list_mat_a,Ys: list_mat_a,Zs: list_mat_a,P: list_mat_a > list_mat_a > list_mat_a > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( ( size_size_list_mat_a @ Ys )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( P @ nil_mat_a @ nil_mat_a @ nil_mat_a )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_a,Ys4: list_mat_a,Z2: mat_a,Zs2: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_a @ Y3 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_909_list__induct3,axiom,
! [Xs: list_mat_a,Ys: list_mat_a,Zs: list_mat_complex,P: list_mat_a > list_mat_a > list_mat_complex > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( ( size_size_list_mat_a @ Ys )
= ( size_s5969786470865220249omplex @ Zs ) )
=> ( ( P @ nil_mat_a @ nil_mat_a @ nil_mat_complex )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_a,Ys4: list_mat_a,Z2: mat_complex,Zs2: list_mat_complex] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_s5969786470865220249omplex @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_a @ Y3 @ Ys4 ) @ ( cons_mat_complex @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_910_list__induct3,axiom,
! [Xs: list_mat_a,Ys: list_mat_complex,Zs: list_mat_a,P: list_mat_a > list_mat_complex > list_mat_a > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( P @ nil_mat_a @ nil_mat_complex @ nil_mat_a )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_complex,Ys4: list_mat_complex,Z2: mat_a,Zs2: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_s5969786470865220249omplex @ Ys4 ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys4 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_complex @ Y3 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_911_list__induct3,axiom,
! [Xs: list_mat_a,Ys: list_mat_complex,Zs: list_mat_complex,P: list_mat_a > list_mat_complex > list_mat_complex > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys )
= ( size_s5969786470865220249omplex @ Zs ) )
=> ( ( P @ nil_mat_a @ nil_mat_complex @ nil_mat_complex )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_complex,Ys4: list_mat_complex,Z2: mat_complex,Zs2: list_mat_complex] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_s5969786470865220249omplex @ Ys4 ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys4 )
= ( size_s5969786470865220249omplex @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_complex @ Y3 @ Ys4 ) @ ( cons_mat_complex @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_912_list__induct3,axiom,
! [Xs: list_mat_complex,Ys: list_mat_a,Zs: list_mat_a,P: list_mat_complex > list_mat_a > list_mat_a > $o] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( ( size_size_list_mat_a @ Ys )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( P @ nil_mat_complex @ nil_mat_a @ nil_mat_a )
=> ( ! [X4: mat_complex,Xs3: list_mat_complex,Y3: mat_a,Ys4: list_mat_a,Z2: mat_a,Zs2: list_mat_a] :
( ( ( size_s5969786470865220249omplex @ Xs3 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_mat_complex @ X4 @ Xs3 ) @ ( cons_mat_a @ Y3 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_913_list__induct3,axiom,
! [Xs: list_mat_complex,Ys: list_mat_a,Zs: list_mat_complex,P: list_mat_complex > list_mat_a > list_mat_complex > $o] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( ( size_size_list_mat_a @ Ys )
= ( size_s5969786470865220249omplex @ Zs ) )
=> ( ( P @ nil_mat_complex @ nil_mat_a @ nil_mat_complex )
=> ( ! [X4: mat_complex,Xs3: list_mat_complex,Y3: mat_a,Ys4: list_mat_a,Z2: mat_complex,Zs2: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs3 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( ( size_size_list_mat_a @ Ys4 )
= ( size_s5969786470865220249omplex @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_mat_complex @ X4 @ Xs3 ) @ ( cons_mat_a @ Y3 @ Ys4 ) @ ( cons_mat_complex @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_914_list__induct3,axiom,
! [Xs: list_mat_complex,Ys: list_mat_complex,Zs: list_mat_a,P: list_mat_complex > list_mat_complex > list_mat_a > $o] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys )
= ( size_size_list_mat_a @ Zs ) )
=> ( ( P @ nil_mat_complex @ nil_mat_complex @ nil_mat_a )
=> ( ! [X4: mat_complex,Xs3: list_mat_complex,Y3: mat_complex,Ys4: list_mat_complex,Z2: mat_a,Zs2: list_mat_a] :
( ( ( size_s5969786470865220249omplex @ Xs3 )
= ( size_s5969786470865220249omplex @ Ys4 ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys4 )
= ( size_size_list_mat_a @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_mat_complex @ X4 @ Xs3 ) @ ( cons_mat_complex @ Y3 @ Ys4 ) @ ( cons_mat_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_915_list__induct3,axiom,
! [Xs: list_mat_complex,Ys: list_mat_complex,Zs: list_mat_complex,P: list_mat_complex > list_mat_complex > list_mat_complex > $o] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys )
= ( size_s5969786470865220249omplex @ Zs ) )
=> ( ( P @ nil_mat_complex @ nil_mat_complex @ nil_mat_complex )
=> ( ! [X4: mat_complex,Xs3: list_mat_complex,Y3: mat_complex,Ys4: list_mat_complex,Z2: mat_complex,Zs2: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs3 )
= ( size_s5969786470865220249omplex @ Ys4 ) )
=> ( ( ( size_s5969786470865220249omplex @ Ys4 )
= ( size_s5969786470865220249omplex @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_mat_complex @ X4 @ Xs3 ) @ ( cons_mat_complex @ Y3 @ Ys4 ) @ ( cons_mat_complex @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_916_list__induct2,axiom,
! [Xs: list_mat_a,Ys: list_mat_a,P: list_mat_a > list_mat_a > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( P @ nil_mat_a @ nil_mat_a )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_a,Ys4: list_mat_a] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_a @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_917_list__induct2,axiom,
! [Xs: list_mat_a,Ys: list_mat_complex,P: list_mat_a > list_mat_complex > $o] :
( ( ( size_size_list_mat_a @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ( P @ nil_mat_a @ nil_mat_complex )
=> ( ! [X4: mat_a,Xs3: list_mat_a,Y3: mat_complex,Ys4: list_mat_complex] :
( ( ( size_size_list_mat_a @ Xs3 )
= ( size_s5969786470865220249omplex @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_mat_a @ X4 @ Xs3 ) @ ( cons_mat_complex @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_918_list__induct2,axiom,
! [Xs: list_mat_complex,Ys: list_mat_a,P: list_mat_complex > list_mat_a > $o] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_size_list_mat_a @ Ys ) )
=> ( ( P @ nil_mat_complex @ nil_mat_a )
=> ( ! [X4: mat_complex,Xs3: list_mat_complex,Y3: mat_a,Ys4: list_mat_a] :
( ( ( size_s5969786470865220249omplex @ Xs3 )
= ( size_size_list_mat_a @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_mat_complex @ X4 @ Xs3 ) @ ( cons_mat_a @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_919_list__induct2,axiom,
! [Xs: list_mat_complex,Ys: list_mat_complex,P: list_mat_complex > list_mat_complex > $o] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ( P @ nil_mat_complex @ nil_mat_complex )
=> ( ! [X4: mat_complex,Xs3: list_mat_complex,Y3: mat_complex,Ys4: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs3 )
= ( size_s5969786470865220249omplex @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_mat_complex @ X4 @ Xs3 ) @ ( cons_mat_complex @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_920_list_Oset__sel_I1_J,axiom,
! [A: list_mat_complex] :
( ( A != nil_mat_complex )
=> ( member_mat_complex @ ( hd_mat_complex @ A ) @ ( set_mat_complex2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_921_hd__in__set,axiom,
! [Xs: list_mat_complex] :
( ( Xs != nil_mat_complex )
=> ( member_mat_complex @ ( hd_mat_complex @ Xs ) @ ( set_mat_complex2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_922_list_Oset__sel_I2_J,axiom,
! [A: list_mat_complex,X2: mat_complex] :
( ( A != nil_mat_complex )
=> ( ( member_mat_complex @ X2 @ ( set_mat_complex2 @ ( tl_mat_complex @ A ) ) )
=> ( member_mat_complex @ X2 @ ( set_mat_complex2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_923_length__greater__0__conv,axiom,
! [Xs: list_mat_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_mat_a @ Xs ) )
= ( Xs != nil_mat_a ) ) ).
% length_greater_0_conv
thf(fact_924_length__greater__0__conv,axiom,
! [Xs: list_mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5969786470865220249omplex @ Xs ) )
= ( Xs != nil_mat_complex ) ) ).
% length_greater_0_conv
thf(fact_925_length__tl,axiom,
! [Xs: list_mat_a] :
( ( size_size_list_mat_a @ ( tl_mat_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_mat_a @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_926_length__tl,axiom,
! [Xs: list_mat_complex] :
( ( size_s5969786470865220249omplex @ ( tl_mat_complex @ Xs ) )
= ( minus_minus_nat @ ( size_s5969786470865220249omplex @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_927_hd__conv__nth,axiom,
! [Xs: list_mat_a] :
( ( Xs != nil_mat_a )
=> ( ( hd_mat_a @ Xs )
= ( nth_mat_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_928_hd__conv__nth,axiom,
! [Xs: list_mat_complex] :
( ( Xs != nil_mat_complex )
=> ( ( hd_mat_complex @ Xs )
= ( nth_mat_complex @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_929_length__butlast,axiom,
! [Xs: list_mat_a] :
( ( size_size_list_mat_a @ ( butlast_mat_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_mat_a @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_930_length__butlast,axiom,
! [Xs: list_mat_complex] :
( ( size_s5969786470865220249omplex @ ( butlast_mat_complex @ Xs ) )
= ( minus_minus_nat @ ( size_s5969786470865220249omplex @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_931_nth__Cons_H,axiom,
! [N: nat,X2: mat_a,Xs: list_mat_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ N )
= ( nth_mat_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_932_nth__Cons_H,axiom,
! [N: nat,X2: mat_complex,Xs: list_mat_complex] :
( ( ( N = zero_zero_nat )
=> ( ( nth_mat_complex @ ( cons_mat_complex @ X2 @ Xs ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_mat_complex @ ( cons_mat_complex @ X2 @ Xs ) @ N )
= ( nth_mat_complex @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_933_undef__vec__def,axiom,
( undef_vec_mat_a
= ( nth_mat_a @ nil_mat_a ) ) ).
% undef_vec_def
thf(fact_934_undef__vec__def,axiom,
( undef_2495355514574404529omplex
= ( nth_mat_complex @ nil_mat_complex ) ) ).
% undef_vec_def
thf(fact_935_diag__block__mat__singleton,axiom,
! [A3: mat_a] :
( ( diag_block_mat_a @ ( cons_mat_a @ A3 @ nil_mat_a ) )
= A3 ) ).
% diag_block_mat_singleton
thf(fact_936_diag__block__mat__singleton,axiom,
! [A3: mat_complex] :
( ( diag_b9145358668110806138omplex @ ( cons_mat_complex @ A3 @ nil_mat_complex ) )
= A3 ) ).
% diag_block_mat_singleton
thf(fact_937_index__minus__mat_I3_J,axiom,
! [A3: mat_complex,B3: mat_complex] :
( ( dim_col_complex @ ( minus_2412168080157227406omplex @ A3 @ B3 ) )
= ( dim_col_complex @ B3 ) ) ).
% index_minus_mat(3)
thf(fact_938_minus__mult__distrib__mat__vec,axiom,
! [A3: mat_complex,Nr: nat,Nc: nat,B3: mat_complex,V: vec_complex] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_vec_complex @ V @ ( carrier_vec_complex @ Nc ) )
=> ( ( mult_mat_vec_complex @ ( minus_2412168080157227406omplex @ A3 @ B3 ) @ V )
= ( minus_6391593812940525058omplex @ ( mult_mat_vec_complex @ A3 @ V ) @ ( mult_mat_vec_complex @ B3 @ V ) ) ) ) ) ) ).
% minus_mult_distrib_mat_vec
thf(fact_939_index__minus__mat_I2_J,axiom,
! [A3: mat_complex,B3: mat_complex] :
( ( dim_row_complex @ ( minus_2412168080157227406omplex @ A3 @ B3 ) )
= ( dim_row_complex @ B3 ) ) ).
% index_minus_mat(2)
thf(fact_940_minus__carrier__mat,axiom,
! [B3: mat_complex,Nr: nat,Nc: nat,A3: mat_complex] :
( ( member_mat_complex @ B3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( minus_2412168080157227406omplex @ A3 @ B3 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_941_minus__mult__distrib__mat,axiom,
! [A3: mat_complex,Nr: nat,N: nat,B3: mat_complex,C2: mat_complex,Nc: nat] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B3 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( minus_2412168080157227406omplex @ A3 @ B3 ) @ C2 )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A3 @ C2 ) @ ( times_8009071140041733218omplex @ B3 @ C2 ) ) ) ) ) ) ).
% minus_mult_distrib_mat
thf(fact_942_mult__minus__distrib__mat,axiom,
! [A3: mat_complex,Nr: nat,N: nat,B3: mat_complex,Nc: nat,C2: mat_complex] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B3 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( member_mat_complex @ C2 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A3 @ ( minus_2412168080157227406omplex @ B3 @ C2 ) )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A3 @ B3 ) @ ( times_8009071140041733218omplex @ A3 @ C2 ) ) ) ) ) ) ).
% mult_minus_distrib_mat
thf(fact_943_transpose__minus,axiom,
! [A3: mat_complex,Nr: nat,Nc: nat,B3: mat_complex] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( transp3074176993011536131omplex @ ( minus_2412168080157227406omplex @ A3 @ B3 ) )
= ( minus_2412168080157227406omplex @ ( transp3074176993011536131omplex @ A3 ) @ ( transp3074176993011536131omplex @ B3 ) ) ) ) ) ).
% transpose_minus
thf(fact_944_mult__minus__distrib__mat__vec,axiom,
! [A3: mat_complex,Nr: nat,Nc: nat,V: vec_complex,W: vec_complex] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_vec_complex @ V @ ( carrier_vec_complex @ Nc ) )
=> ( ( member_vec_complex @ W @ ( carrier_vec_complex @ Nc ) )
=> ( ( mult_mat_vec_complex @ A3 @ ( minus_6391593812940525058omplex @ V @ W ) )
= ( minus_6391593812940525058omplex @ ( mult_mat_vec_complex @ A3 @ V ) @ ( mult_mat_vec_complex @ A3 @ W ) ) ) ) ) ) ).
% mult_minus_distrib_mat_vec
thf(fact_945_length__n__lists__elem,axiom,
! [Ys: list_mat_a,N: nat,Xs: list_mat_a] :
( ( member_list_mat_a @ Ys @ ( set_list_mat_a2 @ ( n_lists_mat_a @ N @ Xs ) ) )
=> ( ( size_size_list_mat_a @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_946_length__n__lists__elem,axiom,
! [Ys: list_mat_complex,N: nat,Xs: list_mat_complex] :
( ( member279434397506102358omplex @ Ys @ ( set_list_mat_complex2 @ ( n_lists_mat_complex @ N @ Xs ) ) )
=> ( ( size_s5969786470865220249omplex @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_947_sum__list__geq__tl,axiom,
! [L: list_nat] :
( ( L != nil_nat )
=> ( ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ L ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( nth_nat @ L @ J3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( tl_nat @ L ) ) @ ( groups4561878855575611511st_nat @ L ) ) ) ) ).
% sum_list_geq_tl
thf(fact_948_mat__prod__unit__vec__cong,axiom,
! [A3: mat_complex,N: nat,B3: mat_complex] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( mult_mat_vec_complex @ A3 @ ( unit_vec_complex @ N @ I2 ) )
= ( mult_mat_vec_complex @ B3 @ ( unit_vec_complex @ N @ I2 ) ) ) )
=> ( A3 = B3 ) ) ) ) ).
% mat_prod_unit_vec_cong
thf(fact_949_mat__unit__vec__col,axiom,
! [A3: mat_complex,N: nat,I4: nat] :
( ( member_mat_complex @ A3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ I4 @ N )
=> ( ( mult_mat_vec_complex @ A3 @ ( unit_vec_complex @ N @ I4 ) )
= ( col_complex @ A3 @ I4 ) ) ) ) ).
% mat_unit_vec_col
thf(fact_950_sum__list__geq__0,axiom,
! [L: list_nat] :
( ( L != nil_nat )
=> ( ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ L ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( nth_nat @ L @ J3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4561878855575611511st_nat @ L ) ) ) ) ).
% sum_list_geq_0
thf(fact_951_sum__list__mono2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).
% sum_list_mono2
thf(fact_952_elem__le__sum__list,axiom,
! [K: nat,Ns: list_nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Ns @ K ) @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).
% elem_le_sum_list
thf(fact_953_n__sum__sum__list,axiom,
! [I4: nat,L: list_nat] :
( ( ord_less_eq_nat @ I4 @ ( size_size_list_nat @ L ) )
=> ( ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ L ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( nth_nat @ L @ J3 ) ) )
=> ( ord_less_eq_nat @ ( commut2019222099004354946um_nat @ I4 @ L ) @ ( groups4561878855575611511st_nat @ L ) ) ) ) ).
% n_sum_sum_list
thf(fact_954_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_955_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_956_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_957_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_958_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_959_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_960_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_961_not__less__simps_I1_J,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_simps(1)
thf(fact_962_Nat_OlessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( ( K
!= ( suc @ I4 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_963_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_964_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_965_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_966_Suc__lessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_967_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_968_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_969_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_970_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_971_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_972_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_973_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_974_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_975_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_976_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_977_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_978_less__trans__Suc,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I4 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_979_less__Suc__induct,axiom,
! [I4: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I4 @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I2 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I4 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_980_strict__inc__induct,axiom,
! [I4: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I4 @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I4 ) ) ) ) ).
% strict_inc_induct
thf(fact_981_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_982_nat_Osimps_I1_J,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.simps(1)
thf(fact_983_old_Onat_Osimps_I1_J,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.simps(1)
thf(fact_984_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_985_Suc__n__not__n,axiom,
! [N: nat] :
( ( suc @ N )
!= N ) ).
% Suc_n_not_n
thf(fact_986_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z2: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z2 )
=> ( R @ X4 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_987_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_988_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_989_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_990_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_991_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_992_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_993_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M7: nat] :
( M6
= ( suc @ M7 ) ) ) ).
% Suc_le_D
thf(fact_994_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_995_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_996_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_997_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_998_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_999_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_1000_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1001_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1002_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1003_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M7: nat] :
( N
= ( suc @ M7 ) ) ) ).
% gr0_implies_Suc
thf(fact_1004_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1005_nat_Osimps_I3_J,axiom,
! [X23: nat] :
( ( suc @ X23 )
!= zero_zero_nat ) ).
% nat.simps(3)
thf(fact_1006_old_Onat_Osimps_I3_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.simps(3)
thf(fact_1007_old_Onat_Osimps_I2_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.simps(2)
thf(fact_1008_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1009_nat_Oinduct,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% nat.induct
thf(fact_1010_nat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [X24: nat] :
( Y2
!= ( suc @ X24 ) ) ) ).
% nat.exhaust
thf(fact_1011_unit__vecs__last_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [N2: nat] : ( P @ N2 @ zero_zero_nat )
=> ( ! [N2: nat,I2: nat] :
( ( P @ N2 @ I2 )
=> ( P @ N2 @ ( suc @ I2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% unit_vecs_last.induct
thf(fact_1012_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1013_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1014_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1015_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1016_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_not_Zero
thf(fact_1017_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1018_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M7: nat] :
( N
= ( suc @ M7 ) ) ) ).
% not0_implies_Suc
thf(fact_1019_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1020_le__simps_I3_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% le_simps(3)
thf(fact_1021_le__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% le_simps(2)
thf(fact_1022_not__less__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_simps(2)
thf(fact_1023_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1024_dec__induct,axiom,
! [I4: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( P @ I4 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I4 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1025_inc__induct,axiom,
! [I4: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I4 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% inc_induct
thf(fact_1026_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1027_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1028_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1029_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1030_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1031_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I4: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I4 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1032_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1033_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1034_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1035_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1036_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1037_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1038_nth__Cons__Suc,axiom,
! [X2: mat_a,Xs: list_mat_a,N: nat] :
( ( nth_mat_a @ ( cons_mat_a @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_mat_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1039_nth__Cons__Suc,axiom,
! [X2: mat_complex,Xs: list_mat_complex,N: nat] :
( ( nth_mat_complex @ ( cons_mat_complex @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_mat_complex @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1040_length__Suc__conv,axiom,
! [Xs: list_mat_a,N: nat] :
( ( ( size_size_list_mat_a @ Xs )
= ( suc @ N ) )
= ( ? [Y5: mat_a,Ys2: list_mat_a] :
( ( Xs
= ( cons_mat_a @ Y5 @ Ys2 ) )
& ( ( size_size_list_mat_a @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1041_length__Suc__conv,axiom,
! [Xs: list_mat_complex,N: nat] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( suc @ N ) )
= ( ? [Y5: mat_complex,Ys2: list_mat_complex] :
( ( Xs
= ( cons_mat_complex @ Y5 @ Ys2 ) )
& ( ( size_s5969786470865220249omplex @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1042_Suc__length__conv,axiom,
! [N: nat,Xs: list_mat_a] :
( ( ( suc @ N )
= ( size_size_list_mat_a @ Xs ) )
= ( ? [Y5: mat_a,Ys2: list_mat_a] :
( ( Xs
= ( cons_mat_a @ Y5 @ Ys2 ) )
& ( ( size_size_list_mat_a @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1043_Suc__length__conv,axiom,
! [N: nat,Xs: list_mat_complex] :
( ( ( suc @ N )
= ( size_s5969786470865220249omplex @ Xs ) )
= ( ? [Y5: mat_complex,Ys2: list_mat_complex] :
( ( Xs
= ( cons_mat_complex @ Y5 @ Ys2 ) )
& ( ( size_s5969786470865220249omplex @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( times_times_mat_a @ ( nth_mat_a @ al @ i ) @ ( nth_mat_a @ bl @ i ) )
= ( times_times_mat_a @ ( nth_mat_a @ bl @ i ) @ ( nth_mat_a @ al @ i ) ) ) ).
%------------------------------------------------------------------------------