TPTP Problem File: SLH0455^1.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_00652_025818__19386556_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1322 ( 459 unt; 183 typ; 0 def)
% Number of atoms : 3276 (1110 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10960 ( 234 ~; 97 |; 163 &;8986 @)
% ( 0 <=>;1480 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 7 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 443 ( 443 >; 0 *; 0 +; 0 <<)
% Number of symbols : 167 ( 164 usr; 18 con; 0-5 aty)
% Number of variables : 3132 ( 122 ^;2931 !; 79 ?;3132 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:36:50.074
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
list_P6605091754902497125omplex: $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__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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Real__Oreal_J_J,type,
set_mat_real: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Omat_Itf__a_J_J,type,
list_mat_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__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__Matrix__Omat_It__Real__Oreal_J,type,
mat_real: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $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__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (164)
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Complex__Matrix_Odensity__operator,type,
comple5220265106149225959erator: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Ohermitian_001t__Complex__Ocomplex,type,
comple8306762464034002205omplex: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Ohermitian_001tf__a,type,
complex_hermitian_a: mat_a > $o ).
thf(sy_c_Complex__Matrix_Olowner__le,type,
complex_lowner_le: mat_complex > mat_complex > $o ).
thf(sy_c_Complex__Matrix_Opartial__density__operator,type,
comple1169154605998056944erator: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Opositive,type,
complex_positive: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Otrace_001t__Complex__Ocomplex,type,
comple3184165445352484367omplex: mat_complex > complex ).
thf(sy_c_Complex__Matrix_Otrace_001t__Real__Oreal,type,
complex_trace_real: mat_real > real ).
thf(sy_c_Complex__Matrix_Otrace_001tf__a,type,
complex_trace_a: mat_a > a ).
thf(sy_c_Complex__Matrix_Ounitary_001t__Complex__Ocomplex,type,
comple6660659447773130958omplex: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Ounitary_001tf__a,type,
complex_unitary_a: mat_a > $o ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
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_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_If_001t__Complex__Ocomplex,type,
if_complex: $o > complex > complex > complex ).
thf(sy_c_If_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
if_mat_complex: $o > mat_complex > mat_complex > mat_complex ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Jordan__Normal__Form_Ojordan__matrix_001t__Complex__Ocomplex,type,
jordan5739059635872469039omplex: list_P6605091754902497125omplex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Odiff__ev_001t__Complex__Ocomplex,type,
jordan8650160714669549932omplex: mat_complex > nat > nat > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Odiff__ev_001tf__a,type,
jordan1888133435898081728f_ev_a: mat_a > nat > nat > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oev__block_001t__Complex__Ocomplex,type,
jordan8042990603089931364omplex: nat > mat_complex > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oev__block_001tf__a,type,
jordan1479931431598099656lock_a: nat > mat_a > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oev__blocks_001t__Complex__Ocomplex,type,
jordan4650062548456832493omplex: nat > mat_complex > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oev__blocks_001tf__a,type,
jordan8767189289504586111ocks_a: nat > mat_a > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oinv__all_001t__Complex__Ocomplex,type,
jordan5244935068081719878omplex: nat > ( mat_complex > nat > nat > $o ) > mat_complex > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oinv__all_001tf__a,type,
jordan7439094043700944742_all_a: nat > ( mat_a > nat > nat > $o ) > mat_a > $o ).
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_Oinv__all_H_001tf__a,type,
jordan4251489913308508029_all_a: ( mat_a > nat > nat > $o ) > mat_a > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oinv__upto_001t__Complex__Ocomplex,type,
jordan5475473882837061487omplex: nat > ( mat_complex > nat > nat > $o ) > mat_complex > nat > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oinv__upto_001tf__a,type,
jordan1574482064217505917upto_a: nat > ( mat_a > nat > nat > $o ) > mat_a > nat > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Ojb_001t__Complex__Ocomplex,type,
jordan4971026570492200526omplex: mat_complex > nat > nat > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Ojb_001tf__a,type,
jordan8102412511815959902m_jb_a: mat_a > nat > nat > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Ojnf__vector_001t__Complex__Ocomplex,type,
jordan387279176131498413omplex: mat_complex > list_P6605091754902497125omplex ).
thf(sy_c_Jordan__Normal__Form__Existence_Opartition__ev__blocks_001t__Complex__Ocomplex,type,
jordan5009815537632354121omplex: mat_complex > list_mat_complex > list_mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Opartition__ev__blocks_001tf__a,type,
jordan501837315015147299ocks_a: mat_a > list_mat_a > list_mat_a ).
thf(sy_c_Jordan__Normal__Form__Existence_Osame__diag_001t__Complex__Ocomplex,type,
jordan2620430285385836103omplex: nat > mat_complex > mat_complex > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Osame__diag_001tf__a,type,
jordan8308822787700309925diag_a: nat > mat_a > mat_a > $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_001tf__a,type,
jordan2295368353748153855ep_1_a: mat_a > mat_a ).
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__1__main_001tf__a,type,
jordan1365744739707490310main_a: nat > nat > nat > mat_a > mat_a ).
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_001tf__a,type,
jordan8731284808630253630ep_2_a: mat_a > mat_a ).
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__2__main_001tf__a,type,
jordan913024080637330373main_a: nat > nat > mat_a > mat_a ).
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_001tf__a,type,
jordan5943829226657577597ep_3_a: mat_a > mat_a ).
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__a_001tf__a,type,
jordan2309642243919034460_3_a_a: nat > nat > mat_a > mat_a ).
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_001tf__a,type,
jordan5958103116828458202_3_c_a: a > nat > nat > list_P6011104703257516679at_nat > mat_a > mat_a ).
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__c__inner__loop_001tf__a,type,
jordan8889242743715136678loop_a: a > nat > nat > nat > mat_a > mat_a ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__3__main_001t__Complex__Ocomplex,type,
jordan4702481308941288104omplex: nat > nat > mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Ostep__3__main_001tf__a,type,
jordan460303421567170436main_a: nat > nat > mat_a > mat_a ).
thf(sy_c_Jordan__Normal__Form__Existence_Oswap__cols__rows__block_001t__Complex__Ocomplex,type,
jordan8990321789093393430omplex: nat > nat > mat_complex > mat_complex ).
thf(sy_c_Jordan__Normal__Form__Existence_Oswap__cols__rows__block_001tf__a,type,
jordan7507754584721484182lock_a: nat > nat > mat_a > mat_a ).
thf(sy_c_Jordan__Normal__Form__Existence_Ouppert_001t__Complex__Ocomplex,type,
jordan3528196489273997576omplex: mat_complex > nat > nat > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Ouppert_001tf__a,type,
jordan2755030923421653284pert_a: mat_a > nat > nat > $o ).
thf(sy_c_Linear__Algebra__Complements_Ocpx__sq__mat,type,
linear7199532782703566157sq_mat: nat > nat > set_mat_complex > $o ).
thf(sy_c_Linear__Algebra__Complements_Ocpx__sq__mat__axioms,type,
linear2040860143340867312axioms: nat > nat > $o ).
thf(sy_c_Linear__Algebra__Complements_Oprojector_001t__Complex__Ocomplex,type,
linear5633924348262549461omplex: mat_complex > $o ).
thf(sy_c_Linear__Algebra__Complements_Oprojector_001tf__a,type,
linear2821214051344812439ctor_a: mat_a > $o ).
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_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_Matrix_Oappend__rows_001t__Complex__Ocomplex,type,
append_rows_complex: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Matrix_Oappend__rows_001tf__a,type,
append_rows_a: mat_a > mat_a > mat_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__Real__Oreal,type,
carrier_mat_real: nat > nat > set_mat_real ).
thf(sy_c_Matrix_Ocarrier__mat_001tf__a,type,
carrier_mat_a: nat > nat > set_mat_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_001tf__a,type,
diag_mat_a: mat_a > list_a ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Complex__Ocomplex,type,
diagonal_mat_complex: mat_complex > $o ).
thf(sy_c_Matrix_Odiagonal__mat_001tf__a,type,
diagonal_mat_a: mat_a > $o ).
thf(sy_c_Matrix_Odim__col_001t__Complex__Ocomplex,type,
dim_col_complex: mat_complex > nat ).
thf(sy_c_Matrix_Odim__col_001t__Real__Oreal,type,
dim_col_real: mat_real > 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__Real__Oreal,type,
dim_row_real: mat_real > nat ).
thf(sy_c_Matrix_Odim__row_001tf__a,type,
dim_row_a: mat_a > nat ).
thf(sy_c_Matrix_Ofour__block__mat_001t__Complex__Ocomplex,type,
four_b559179830521662709omplex: mat_complex > mat_complex > mat_complex > mat_complex > mat_complex ).
thf(sy_c_Matrix_Ofour__block__mat_001tf__a,type,
four_block_mat_a: mat_a > mat_a > mat_a > mat_a > mat_a ).
thf(sy_c_Matrix_Oinvertible__mat_001t__Complex__Ocomplex,type,
invert2568027935824841882omplex: mat_complex > $o ).
thf(sy_c_Matrix_Oinvertible__mat_001tf__a,type,
invertible_mat_a: mat_a > $o ).
thf(sy_c_Matrix_Oinverts__mat_001t__Complex__Ocomplex,type,
inverts_mat_complex: mat_complex > mat_complex > $o ).
thf(sy_c_Matrix_Oinverts__mat_001tf__a,type,
inverts_mat_a: mat_a > mat_a > $o ).
thf(sy_c_Matrix_Osimilar__mat__wit_001t__Complex__Ocomplex,type,
simila5774310414453981135omplex: mat_complex > mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_Matrix_Osimilar__mat__wit_001tf__a,type,
similar_mat_wit_a: mat_a > mat_a > mat_a > mat_a > $o ).
thf(sy_c_Matrix_Osmult__mat_001t__Complex__Ocomplex,type,
smult_mat_complex: complex > mat_complex > mat_complex ).
thf(sy_c_Matrix_Osmult__mat_001t__Nat__Onat,type,
smult_mat_nat: nat > mat_nat > mat_nat ).
thf(sy_c_Matrix_Osmult__mat_001t__Real__Oreal,type,
smult_mat_real: real > mat_real > mat_real ).
thf(sy_c_Matrix_Osmult__mat_001tf__a,type,
smult_mat_a: a > mat_a > mat_a ).
thf(sy_c_Matrix_Osquare__mat_001t__Complex__Ocomplex,type,
square_mat_complex: mat_complex > $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_001tf__a,type,
transpose_mat_a: mat_a > mat_a ).
thf(sy_c_Matrix_Oupdate__mat_001t__Complex__Ocomplex,type,
update_mat_complex: mat_complex > product_prod_nat_nat > complex > mat_complex ).
thf(sy_c_Matrix_Oupdate__mat_001tf__a,type,
update_mat_a: mat_a > product_prod_nat_nat > a > mat_a ).
thf(sy_c_Matrix_Oupper__triangular_001t__Complex__Ocomplex,type,
upper_4850907204721561915omplex: mat_complex > $o ).
thf(sy_c_Matrix_Oupper__triangular_001tf__a,type,
upper_triangular_a: mat_a > $o ).
thf(sy_c_Matrix_Ozero__mat_001t__Complex__Ocomplex,type,
zero_mat_complex: nat > nat > mat_complex ).
thf(sy_c_Matrix_Ozero__mat_001t__Real__Oreal,type,
zero_mat_real: nat > nat > mat_real ).
thf(sy_c_Matrix_Ozero__mat_001tf__a,type,
zero_mat_a: nat > nat > mat_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Complex__Ocomplex,type,
ord_less_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Complex__Ocomplex,type,
ord_less_eq_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Projective__Measurements_Odensity__collapse,type,
projec3470689467825365843llapse: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Projective__Measurements_Oeigvals_001t__Complex__Ocomplex,type,
projec6785268565095433026omplex: mat_complex > list_complex ).
thf(sy_c_Projective__Measurements_Ohermitian__decomp_001t__Complex__Ocomplex,type,
projec5943904436471448624omplex: mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_Projective__Measurements_Ohermitian__decomp_001tf__a,type,
projec1926941670171670524comp_a: mat_a > mat_a > mat_a > $o ).
thf(sy_c_Projective__Measurements_Omax__mix__density,type,
projec8360710381328234318ensity: nat > mat_complex ).
thf(sy_c_Quantum_Ocpx__mat__cnj,type,
cpx_mat_cnj: mat_complex > mat_complex ).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
real_V4546457046886955230omplex: real > complex ).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
real_V1803761363581548252l_real: real > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
dvd_dvd_complex: complex > complex > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
dvd_dvd_real: real > real > $o ).
thf(sy_c_Schur__Decomposition_Ocorthogonal__mat_001t__Complex__Ocomplex,type,
schur_549222400177443379omplex: mat_complex > $o ).
thf(sy_c_Schur__Decomposition_Omat__adjoint_001t__Complex__Ocomplex,type,
schur_5982229384592763574omplex: mat_complex > mat_complex ).
thf(sy_c_Schur__Decomposition_Omat__adjoint_001tf__a,type,
schur_mat_adjoint_a: mat_a > 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_Spectral__Theory__Complements_Omat__conj_001t__Complex__Ocomplex,type,
spectr5699176650994449695omplex: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Spectral__Theory__Complements_Omat__conj_001tf__a,type,
spectr5828033140197310157conj_a: mat_a > mat_a > mat_a ).
thf(sy_c_Spectral__Theory__Complements_Oreal__diag__decomp_001t__Complex__Ocomplex,type,
spectr5409772854192057952omplex: mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_Spectral__Theory__Complements_Oreal__diag__decomp_001tf__a,type,
spectr3403749184330357196comp_a: mat_a > mat_a > mat_a > $o ).
thf(sy_c_Spectral__Theory__Complements_Ounitarily__equiv_001t__Complex__Ocomplex,type,
spectr6340060708231679580omplex: mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_Spectral__Theory__Complements_Ounitarily__equiv_001tf__a,type,
spectr4825054497075562704quiv_a: mat_a > mat_a > mat_a > $o ).
thf(sy_c_Spectral__Theory__Complements_Ounitary__diag_001t__Complex__Ocomplex,type,
spectr532731689276696518omplex: mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_Spectral__Theory__Complements_Ounitary__diag_001tf__a,type,
spectr4894841263502123494diag_a: mat_a > mat_a > mat_a > $o ).
thf(sy_c_Tensor_Otensor__mat,type,
tensor_mat: mat_complex > mat_complex > mat_complex ).
thf(sy_c_VS__Connect_Ovec__space_Ocol__space_001t__Complex__Ocomplex,type,
vS_vec1879987866596122552omplex: nat > mat_complex > set_vec_complex ).
thf(sy_c_VS__Connect_Ovec__space_Orow__space_001t__Complex__Ocomplex,type,
vS_vec3284807721666986142omplex: nat > mat_complex > set_vec_complex ).
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__Real__Oreal_J,type,
member_mat_real: mat_real > set_mat_real > $o ).
thf(sy_c_member_001t__Matrix__Omat_Itf__a_J,type,
member_mat_a: mat_a > set_mat_a > $o ).
thf(sy_v_A1,type,
a1: mat_a ).
thf(sy_v_A2,type,
a2: mat_a ).
thf(sy_v_B1,type,
b1: mat_a ).
thf(sy_v_B2,type,
b2: mat_a ).
thf(sy_v_U1,type,
u1: mat_a ).
thf(sy_v_U2,type,
u2: mat_a ).
thf(sy_v_i____,type,
i: nat ).
% Relevant facts (1129)
thf(fact_0_assms_I4_J,axiom,
( ( dim_row_a @ a2 )
= ( dim_col_a @ a2 ) ) ).
% assms(4)
thf(fact_1_assms_I3_J,axiom,
( ( dim_row_a @ a1 )
= ( dim_col_a @ a1 ) ) ).
% assms(3)
thf(fact_2_assms_I1_J,axiom,
spectr3403749184330357196comp_a @ a1 @ b1 @ u1 ).
% assms(1)
thf(fact_3_step__2__main__dims__main,axiom,
! [N: nat,J: nat,A: mat_a] :
( ( ( dim_row_a @ ( jordan913024080637330373main_a @ N @ J @ A ) )
= ( dim_row_a @ A ) )
& ( ( dim_col_a @ ( jordan913024080637330373main_a @ N @ J @ A ) )
= ( dim_col_a @ A ) ) ) ).
% step_2_main_dims_main
thf(fact_4_step__2__main__dims__main,axiom,
! [N: nat,J: nat,A: mat_complex] :
( ( ( dim_row_complex @ ( jordan6916311984355858983omplex @ N @ J @ A ) )
= ( dim_row_complex @ A ) )
& ( ( dim_col_complex @ ( jordan6916311984355858983omplex @ N @ J @ A ) )
= ( dim_col_complex @ A ) ) ) ).
% step_2_main_dims_main
thf(fact_5_square__mat_Oelims_I3_J,axiom,
! [X: mat_a] :
( ~ ( square_mat_a @ X )
=> ( ( dim_col_a @ X )
!= ( dim_row_a @ X ) ) ) ).
% square_mat.elims(3)
thf(fact_6_square__mat_Oelims_I3_J,axiom,
! [X: mat_complex] :
( ~ ( square_mat_complex @ X )
=> ( ( dim_col_complex @ X )
!= ( dim_row_complex @ X ) ) ) ).
% square_mat.elims(3)
thf(fact_7_square__mat_Oelims_I2_J,axiom,
! [X: mat_a] :
( ( square_mat_a @ X )
=> ( ( dim_col_a @ X )
= ( dim_row_a @ X ) ) ) ).
% square_mat.elims(2)
thf(fact_8_square__mat_Oelims_I2_J,axiom,
! [X: mat_complex] :
( ( square_mat_complex @ X )
=> ( ( dim_col_complex @ X )
= ( dim_row_complex @ X ) ) ) ).
% square_mat.elims(2)
thf(fact_9_square__mat_Oelims_I1_J,axiom,
! [X: mat_a,Y: $o] :
( ( ( square_mat_a @ X )
= Y )
=> ( Y
= ( ( dim_col_a @ X )
= ( dim_row_a @ X ) ) ) ) ).
% square_mat.elims(1)
thf(fact_10_square__mat_Oelims_I1_J,axiom,
! [X: mat_complex,Y: $o] :
( ( ( square_mat_complex @ X )
= Y )
=> ( Y
= ( ( dim_col_complex @ X )
= ( dim_row_complex @ X ) ) ) ) ).
% square_mat.elims(1)
thf(fact_11_square__mat_Osimps,axiom,
( square_mat_a
= ( ^ [A2: mat_a] :
( ( dim_col_a @ A2 )
= ( dim_row_a @ A2 ) ) ) ) ).
% square_mat.simps
thf(fact_12_square__mat_Osimps,axiom,
( square_mat_complex
= ( ^ [A2: mat_complex] :
( ( dim_col_complex @ A2 )
= ( dim_row_complex @ A2 ) ) ) ) ).
% square_mat.simps
thf(fact_13_step__3__main__dims__main,axiom,
! [N: nat,K: nat,A: mat_a] :
( ( ( dim_row_a @ ( jordan460303421567170436main_a @ N @ K @ A ) )
= ( dim_row_a @ A ) )
& ( ( dim_col_a @ ( jordan460303421567170436main_a @ N @ K @ A ) )
= ( dim_col_a @ A ) ) ) ).
% step_3_main_dims_main
thf(fact_14_step__3__main__dims__main,axiom,
! [N: nat,K: nat,A: mat_complex] :
( ( ( dim_row_complex @ ( jordan4702481308941288104omplex @ N @ K @ A ) )
= ( dim_row_complex @ A ) )
& ( ( dim_col_complex @ ( jordan4702481308941288104omplex @ N @ K @ A ) )
= ( dim_col_complex @ A ) ) ) ).
% step_3_main_dims_main
thf(fact_15_step__3__c__inner__loop__dims__main,axiom,
! [Val: a,L: nat,I: nat,J: nat,A: mat_a] :
( ( ( dim_row_a @ ( jordan8889242743715136678loop_a @ Val @ L @ I @ J @ A ) )
= ( dim_row_a @ A ) )
& ( ( dim_col_a @ ( jordan8889242743715136678loop_a @ Val @ L @ I @ J @ A ) )
= ( dim_col_a @ A ) ) ) ).
% step_3_c_inner_loop_dims_main
thf(fact_16_step__3__c__inner__loop__dims__main,axiom,
! [Val: complex,L: nat,I: nat,J: nat,A: mat_complex] :
( ( ( dim_row_complex @ ( jordan7656109678144820486omplex @ Val @ L @ I @ J @ A ) )
= ( dim_row_complex @ A ) )
& ( ( dim_col_complex @ ( jordan7656109678144820486omplex @ Val @ L @ I @ J @ A ) )
= ( dim_col_complex @ A ) ) ) ).
% step_3_c_inner_loop_dims_main
thf(fact_17_step__3__a__dims__main,axiom,
! [I: nat,J: nat,A: mat_a] :
( ( ( dim_row_a @ ( jordan2309642243919034460_3_a_a @ I @ J @ A ) )
= ( dim_row_a @ A ) )
& ( ( dim_col_a @ ( jordan2309642243919034460_3_a_a @ I @ J @ A ) )
= ( dim_col_a @ A ) ) ) ).
% step_3_a_dims_main
thf(fact_18_step__3__a__dims__main,axiom,
! [I: nat,J: nat,A: mat_complex] :
( ( ( dim_row_complex @ ( jordan2858886415929732048omplex @ I @ J @ A ) )
= ( dim_row_complex @ A ) )
& ( ( dim_col_complex @ ( jordan2858886415929732048omplex @ I @ J @ A ) )
= ( dim_col_complex @ A ) ) ) ).
% step_3_a_dims_main
thf(fact_19_step__3__c__dims__main,axiom,
! [X: a,L: nat,K: nat,I: list_P6011104703257516679at_nat,A: mat_a] :
( ( ( dim_row_a @ ( jordan5958103116828458202_3_c_a @ X @ L @ K @ I @ A ) )
= ( dim_row_a @ A ) )
& ( ( dim_col_a @ ( jordan5958103116828458202_3_c_a @ X @ L @ K @ I @ A ) )
= ( dim_col_a @ A ) ) ) ).
% step_3_c_dims_main
thf(fact_20_step__3__c__dims__main,axiom,
! [X: complex,L: nat,K: nat,I: list_P6011104703257516679at_nat,A: mat_complex] :
( ( ( dim_row_complex @ ( jordan5343229918868201426omplex @ X @ L @ K @ I @ A ) )
= ( dim_row_complex @ A ) )
& ( ( dim_col_complex @ ( jordan5343229918868201426omplex @ X @ L @ K @ I @ A ) )
= ( dim_col_complex @ A ) ) ) ).
% step_3_c_dims_main
thf(fact_21_step__1__main__dims__main,axiom,
! [N: nat,I: nat,J: nat,A: mat_a] :
( ( ( dim_row_a @ ( jordan1365744739707490310main_a @ N @ I @ J @ A ) )
= ( dim_row_a @ A ) )
& ( ( dim_col_a @ ( jordan1365744739707490310main_a @ N @ I @ J @ A ) )
= ( dim_col_a @ A ) ) ) ).
% step_1_main_dims_main
thf(fact_22_step__1__main__dims__main,axiom,
! [N: nat,I: nat,J: nat,A: mat_complex] :
( ( ( dim_row_complex @ ( jordan9130142659770429862omplex @ N @ I @ J @ A ) )
= ( dim_row_complex @ A ) )
& ( ( dim_col_complex @ ( jordan9130142659770429862omplex @ N @ I @ J @ A ) )
= ( dim_col_complex @ A ) ) ) ).
% step_1_main_dims_main
thf(fact_23_swap__cols__rows__block__dims__main,axiom,
! [I: nat,J: nat,A: mat_a] :
( ( ( dim_row_a @ ( jordan7507754584721484182lock_a @ I @ J @ A ) )
= ( dim_row_a @ A ) )
& ( ( dim_col_a @ ( jordan7507754584721484182lock_a @ I @ J @ A ) )
= ( dim_col_a @ A ) ) ) ).
% swap_cols_rows_block_dims_main
thf(fact_24_swap__cols__rows__block__dims__main,axiom,
! [I: nat,J: nat,A: mat_complex] :
( ( ( dim_row_complex @ ( jordan8990321789093393430omplex @ I @ J @ A ) )
= ( dim_row_complex @ A ) )
& ( ( dim_col_complex @ ( jordan8990321789093393430omplex @ I @ J @ A ) )
= ( dim_col_complex @ A ) ) ) ).
% swap_cols_rows_block_dims_main
thf(fact_25_swap__cols__rows__block__dims_I1_J,axiom,
! [I: nat,J: nat,A: mat_a] :
( ( dim_row_a @ ( jordan7507754584721484182lock_a @ I @ J @ A ) )
= ( dim_row_a @ A ) ) ).
% swap_cols_rows_block_dims(1)
thf(fact_26_swap__cols__rows__block__dims_I1_J,axiom,
! [I: nat,J: nat,A: mat_complex] :
( ( dim_row_complex @ ( jordan8990321789093393430omplex @ I @ J @ A ) )
= ( dim_row_complex @ A ) ) ).
% swap_cols_rows_block_dims(1)
thf(fact_27_swap__cols__rows__block__dims_I2_J,axiom,
! [I: nat,J: nat,A: mat_a] :
( ( dim_col_a @ ( jordan7507754584721484182lock_a @ I @ J @ A ) )
= ( dim_col_a @ A ) ) ).
% swap_cols_rows_block_dims(2)
thf(fact_28_swap__cols__rows__block__dims_I2_J,axiom,
! [I: nat,J: nat,A: mat_complex] :
( ( dim_col_complex @ ( jordan8990321789093393430omplex @ I @ J @ A ) )
= ( dim_col_complex @ A ) ) ).
% swap_cols_rows_block_dims(2)
thf(fact_29_step__1__main__dims_I1_J,axiom,
! [N: nat,I: nat,J: nat,A: mat_a] :
( ( dim_row_a @ ( jordan1365744739707490310main_a @ N @ I @ J @ A ) )
= ( dim_row_a @ A ) ) ).
% step_1_main_dims(1)
thf(fact_30_step__1__main__dims_I1_J,axiom,
! [N: nat,I: nat,J: nat,A: mat_complex] :
( ( dim_row_complex @ ( jordan9130142659770429862omplex @ N @ I @ J @ A ) )
= ( dim_row_complex @ A ) ) ).
% step_1_main_dims(1)
thf(fact_31_step__3__c__dims_I1_J,axiom,
! [X: a,L: nat,K: nat,I: list_P6011104703257516679at_nat,A: mat_a] :
( ( dim_row_a @ ( jordan5958103116828458202_3_c_a @ X @ L @ K @ I @ A ) )
= ( dim_row_a @ A ) ) ).
% step_3_c_dims(1)
thf(fact_32_step__3__c__dims_I1_J,axiom,
! [X: complex,L: nat,K: nat,I: list_P6011104703257516679at_nat,A: mat_complex] :
( ( dim_row_complex @ ( jordan5343229918868201426omplex @ X @ L @ K @ I @ A ) )
= ( dim_row_complex @ A ) ) ).
% step_3_c_dims(1)
thf(fact_33_step__3__a__dims_I1_J,axiom,
! [I: nat,J: nat,A: mat_a] :
( ( dim_row_a @ ( jordan2309642243919034460_3_a_a @ I @ J @ A ) )
= ( dim_row_a @ A ) ) ).
% step_3_a_dims(1)
thf(fact_34_step__3__a__dims_I1_J,axiom,
! [I: nat,J: nat,A: mat_complex] :
( ( dim_row_complex @ ( jordan2858886415929732048omplex @ I @ J @ A ) )
= ( dim_row_complex @ A ) ) ).
% step_3_a_dims(1)
thf(fact_35_step__1__main__dims_I2_J,axiom,
! [N: nat,I: nat,J: nat,A: mat_a] :
( ( dim_col_a @ ( jordan1365744739707490310main_a @ N @ I @ J @ A ) )
= ( dim_col_a @ A ) ) ).
% step_1_main_dims(2)
thf(fact_36_step__1__main__dims_I2_J,axiom,
! [N: nat,I: nat,J: nat,A: mat_complex] :
( ( dim_col_complex @ ( jordan9130142659770429862omplex @ N @ I @ J @ A ) )
= ( dim_col_complex @ A ) ) ).
% step_1_main_dims(2)
thf(fact_37_step__3__c__dims_I2_J,axiom,
! [X: a,L: nat,K: nat,I: list_P6011104703257516679at_nat,A: mat_a] :
( ( dim_col_a @ ( jordan5958103116828458202_3_c_a @ X @ L @ K @ I @ A ) )
= ( dim_col_a @ A ) ) ).
% step_3_c_dims(2)
thf(fact_38_step__3__c__dims_I2_J,axiom,
! [X: complex,L: nat,K: nat,I: list_P6011104703257516679at_nat,A: mat_complex] :
( ( dim_col_complex @ ( jordan5343229918868201426omplex @ X @ L @ K @ I @ A ) )
= ( dim_col_complex @ A ) ) ).
% step_3_c_dims(2)
thf(fact_39_step__3__a__dims_I2_J,axiom,
! [I: nat,J: nat,A: mat_a] :
( ( dim_col_a @ ( jordan2309642243919034460_3_a_a @ I @ J @ A ) )
= ( dim_col_a @ A ) ) ).
% step_3_a_dims(2)
thf(fact_40_step__3__a__dims_I2_J,axiom,
! [I: nat,J: nat,A: mat_complex] :
( ( dim_col_complex @ ( jordan2858886415929732048omplex @ I @ J @ A ) )
= ( dim_col_complex @ A ) ) ).
% step_3_a_dims(2)
thf(fact_41_step__3__c__inner__loop__dims_I1_J,axiom,
! [Val: complex,L: nat,I: nat,J: nat,A: mat_complex] :
( ( dim_row_complex @ ( jordan7656109678144820486omplex @ Val @ L @ I @ J @ A ) )
= ( dim_row_complex @ A ) ) ).
% step_3_c_inner_loop_dims(1)
thf(fact_42_step__3__c__inner__loop__dims_I1_J,axiom,
! [Val: a,L: nat,I: nat,J: nat,A: mat_a] :
( ( dim_row_a @ ( jordan8889242743715136678loop_a @ Val @ L @ I @ J @ A ) )
= ( dim_row_a @ A ) ) ).
% step_3_c_inner_loop_dims(1)
thf(fact_43_step__3__main__dims_I1_J,axiom,
! [N: nat,J: nat,A: mat_complex] :
( ( dim_row_complex @ ( jordan4702481308941288104omplex @ N @ J @ A ) )
= ( dim_row_complex @ A ) ) ).
% step_3_main_dims(1)
thf(fact_44_step__3__main__dims_I1_J,axiom,
! [N: nat,J: nat,A: mat_a] :
( ( dim_row_a @ ( jordan460303421567170436main_a @ N @ J @ A ) )
= ( dim_row_a @ A ) ) ).
% step_3_main_dims(1)
thf(fact_45_step__3__c__inner__loop__dims_I2_J,axiom,
! [Val: complex,L: nat,I: nat,J: nat,A: mat_complex] :
( ( dim_col_complex @ ( jordan7656109678144820486omplex @ Val @ L @ I @ J @ A ) )
= ( dim_col_complex @ A ) ) ).
% step_3_c_inner_loop_dims(2)
thf(fact_46_step__3__c__inner__loop__dims_I2_J,axiom,
! [Val: a,L: nat,I: nat,J: nat,A: mat_a] :
( ( dim_col_a @ ( jordan8889242743715136678loop_a @ Val @ L @ I @ J @ A ) )
= ( dim_col_a @ A ) ) ).
% step_3_c_inner_loop_dims(2)
thf(fact_47_step__3__main__dims_I2_J,axiom,
! [N: nat,J: nat,A: mat_complex] :
( ( dim_col_complex @ ( jordan4702481308941288104omplex @ N @ J @ A ) )
= ( dim_col_complex @ A ) ) ).
% step_3_main_dims(2)
thf(fact_48_step__3__main__dims_I2_J,axiom,
! [N: nat,J: nat,A: mat_a] :
( ( dim_col_a @ ( jordan460303421567170436main_a @ N @ J @ A ) )
= ( dim_col_a @ A ) ) ).
% step_3_main_dims(2)
thf(fact_49_step__2__main__dims_I1_J,axiom,
! [N: nat,J: nat,A: mat_complex] :
( ( dim_row_complex @ ( jordan6916311984355858983omplex @ N @ J @ A ) )
= ( dim_row_complex @ A ) ) ).
% step_2_main_dims(1)
thf(fact_50_step__2__main__dims_I1_J,axiom,
! [N: nat,J: nat,A: mat_a] :
( ( dim_row_a @ ( jordan913024080637330373main_a @ N @ J @ A ) )
= ( dim_row_a @ A ) ) ).
% step_2_main_dims(1)
thf(fact_51_step__2__main__dims_I2_J,axiom,
! [N: nat,J: nat,A: mat_complex] :
( ( dim_col_complex @ ( jordan6916311984355858983omplex @ N @ J @ A ) )
= ( dim_col_complex @ A ) ) ).
% step_2_main_dims(2)
thf(fact_52_step__2__main__dims_I2_J,axiom,
! [N: nat,J: nat,A: mat_a] :
( ( dim_col_a @ ( jordan913024080637330373main_a @ N @ J @ A ) )
= ( dim_col_a @ A ) ) ).
% step_2_main_dims(2)
thf(fact_53_assms_I2_J,axiom,
spectr3403749184330357196comp_a @ a2 @ b2 @ u2 ).
% assms(2)
thf(fact_54__092_060open_062unitary__diag_A_Ifour__block__diag_AA1_AA2_J_A_Ifour__block__diag_AB1_AB2_J_A_Ifour__block__diag_AU1_AU2_J_092_060close_062,axiom,
spectr4894841263502123494diag_a @ ( four_block_mat_a @ a1 @ ( zero_mat_a @ ( dim_row_a @ a1 ) @ ( dim_col_a @ a2 ) ) @ ( zero_mat_a @ ( dim_row_a @ a2 ) @ ( dim_col_a @ a1 ) ) @ a2 ) @ ( four_block_mat_a @ b1 @ ( zero_mat_a @ ( dim_row_a @ b1 ) @ ( dim_col_a @ b2 ) ) @ ( zero_mat_a @ ( dim_row_a @ b2 ) @ ( dim_col_a @ b1 ) ) @ b2 ) @ ( four_block_mat_a @ u1 @ ( zero_mat_a @ ( dim_row_a @ u1 ) @ ( dim_col_a @ u2 ) ) @ ( zero_mat_a @ ( dim_row_a @ u2 ) @ ( dim_col_a @ u1 ) ) @ u2 ) ).
% \<open>unitary_diag (four_block_diag A1 A2) (four_block_diag B1 B2) (four_block_diag U1 U2)\<close>
thf(fact_55_triangular__to__jnf__steps__dims_I6_J,axiom,
! [A: mat_a] :
( ( dim_col_a @ ( jordan5943829226657577597ep_3_a @ A ) )
= ( dim_col_a @ A ) ) ).
% triangular_to_jnf_steps_dims(6)
thf(fact_56_triangular__to__jnf__steps__dims_I6_J,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( jordan4501759426295633263omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% triangular_to_jnf_steps_dims(6)
thf(fact_57_dim__update__mat_I2_J,axiom,
! [A: mat_a,Ij: product_prod_nat_nat,A3: a] :
( ( dim_col_a @ ( update_mat_a @ A @ Ij @ A3 ) )
= ( dim_col_a @ A ) ) ).
% dim_update_mat(2)
thf(fact_58_dim__update__mat_I2_J,axiom,
! [A: mat_complex,Ij: product_prod_nat_nat,A3: complex] :
( ( dim_col_complex @ ( update_mat_complex @ A @ Ij @ A3 ) )
= ( dim_col_complex @ A ) ) ).
% dim_update_mat(2)
thf(fact_59_triangular__to__jnf__steps__dims_I5_J,axiom,
! [A: mat_a] :
( ( dim_row_a @ ( jordan5943829226657577597ep_3_a @ A ) )
= ( dim_row_a @ A ) ) ).
% triangular_to_jnf_steps_dims(5)
thf(fact_60_triangular__to__jnf__steps__dims_I5_J,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( jordan4501759426295633263omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% triangular_to_jnf_steps_dims(5)
thf(fact_61_dim__update__mat_I1_J,axiom,
! [A: mat_a,Ij: product_prod_nat_nat,A3: a] :
( ( dim_row_a @ ( update_mat_a @ A @ Ij @ A3 ) )
= ( dim_row_a @ A ) ) ).
% dim_update_mat(1)
thf(fact_62_dim__update__mat_I1_J,axiom,
! [A: mat_complex,Ij: product_prod_nat_nat,A3: complex] :
( ( dim_row_complex @ ( update_mat_complex @ A @ Ij @ A3 ) )
= ( dim_row_complex @ A ) ) ).
% dim_update_mat(1)
thf(fact_63_triangular__to__jnf__steps__dims_I2_J,axiom,
! [A: mat_a] :
( ( dim_col_a @ ( jordan2295368353748153855ep_1_a @ A ) )
= ( dim_col_a @ A ) ) ).
% triangular_to_jnf_steps_dims(2)
thf(fact_64_triangular__to__jnf__steps__dims_I2_J,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( jordan2017415923357163885omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% triangular_to_jnf_steps_dims(2)
thf(fact_65_triangular__to__jnf__steps__dims_I4_J,axiom,
! [A: mat_a] :
( ( dim_col_a @ ( jordan8731284808630253630ep_2_a @ A ) )
= ( dim_col_a @ A ) ) ).
% triangular_to_jnf_steps_dims(4)
thf(fact_66_triangular__to__jnf__steps__dims_I4_J,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( jordan7871273693253786478omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% triangular_to_jnf_steps_dims(4)
thf(fact_67_triangular__to__jnf__steps__dims_I1_J,axiom,
! [A: mat_a] :
( ( dim_row_a @ ( jordan2295368353748153855ep_1_a @ A ) )
= ( dim_row_a @ A ) ) ).
% triangular_to_jnf_steps_dims(1)
thf(fact_68_triangular__to__jnf__steps__dims_I1_J,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( jordan2017415923357163885omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% triangular_to_jnf_steps_dims(1)
thf(fact_69_triangular__to__jnf__steps__dims_I3_J,axiom,
! [A: mat_a] :
( ( dim_row_a @ ( jordan8731284808630253630ep_2_a @ A ) )
= ( dim_row_a @ A ) ) ).
% triangular_to_jnf_steps_dims(3)
thf(fact_70_triangular__to__jnf__steps__dims_I3_J,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( jordan7871273693253786478omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% triangular_to_jnf_steps_dims(3)
thf(fact_71_index__transpose__mat_I2_J,axiom,
! [A: mat_a] :
( ( dim_row_a @ ( transpose_mat_a @ A ) )
= ( dim_col_a @ A ) ) ).
% index_transpose_mat(2)
thf(fact_72_index__transpose__mat_I2_J,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( transp3074176993011536131omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% index_transpose_mat(2)
thf(fact_73_index__transpose__mat_I3_J,axiom,
! [A: mat_a] :
( ( dim_col_a @ ( transpose_mat_a @ A ) )
= ( dim_row_a @ A ) ) ).
% index_transpose_mat(3)
thf(fact_74_index__transpose__mat_I3_J,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( transp3074176993011536131omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_transpose_mat(3)
thf(fact_75_transpose__mat__eq,axiom,
! [A: mat_complex,B: mat_complex] :
( ( ( transp3074176993011536131omplex @ A )
= ( transp3074176993011536131omplex @ B ) )
= ( A = B ) ) ).
% transpose_mat_eq
thf(fact_76_zero__transpose__mat,axiom,
! [N: nat,M: nat] :
( ( transpose_mat_a @ ( zero_mat_a @ N @ M ) )
= ( zero_mat_a @ M @ N ) ) ).
% zero_transpose_mat
thf(fact_77_zero__transpose__mat,axiom,
! [N: nat,M: nat] :
( ( transp3074176993011536131omplex @ ( zero_mat_complex @ N @ M ) )
= ( zero_mat_complex @ M @ N ) ) ).
% zero_transpose_mat
thf(fact_78_cong__four__block__mat,axiom,
! [A1: mat_a,B1: mat_a,A22: mat_a,B2: mat_a,A32: mat_a,B3: mat_a,A4: mat_a,B4: mat_a] :
( ( A1 = B1 )
=> ( ( A22 = B2 )
=> ( ( A32 = B3 )
=> ( ( A4 = B4 )
=> ( ( four_block_mat_a @ A1 @ A22 @ A32 @ A4 )
= ( four_block_mat_a @ B1 @ B2 @ B3 @ B4 ) ) ) ) ) ) ).
% cong_four_block_mat
thf(fact_79_Matrix_Otranspose__transpose,axiom,
! [A: mat_complex] :
( ( transp3074176993011536131omplex @ ( transp3074176993011536131omplex @ A ) )
= A ) ).
% Matrix.transpose_transpose
thf(fact_80_four__block__unitary__diag,axiom,
! [A1: mat_a,B1: mat_a,U1: mat_a,A22: mat_a,B2: mat_a,U2: mat_a] :
( ( spectr4894841263502123494diag_a @ A1 @ B1 @ U1 )
=> ( ( spectr4894841263502123494diag_a @ A22 @ B2 @ U2 )
=> ( ( ( dim_row_a @ A1 )
= ( dim_col_a @ A1 ) )
=> ( ( ( dim_row_a @ A22 )
= ( dim_col_a @ A22 ) )
=> ( spectr4894841263502123494diag_a @ ( four_block_mat_a @ A1 @ ( zero_mat_a @ ( dim_row_a @ A1 ) @ ( dim_col_a @ A22 ) ) @ ( zero_mat_a @ ( dim_row_a @ A22 ) @ ( dim_col_a @ A1 ) ) @ A22 ) @ ( four_block_mat_a @ B1 @ ( zero_mat_a @ ( dim_row_a @ B1 ) @ ( dim_col_a @ B2 ) ) @ ( zero_mat_a @ ( dim_row_a @ B2 ) @ ( dim_col_a @ B1 ) ) @ B2 ) @ ( four_block_mat_a @ U1 @ ( zero_mat_a @ ( dim_row_a @ U1 ) @ ( dim_col_a @ U2 ) ) @ ( zero_mat_a @ ( dim_row_a @ U2 ) @ ( dim_col_a @ U1 ) ) @ U2 ) ) ) ) ) ) ).
% four_block_unitary_diag
thf(fact_81_four__block__unitary__diag,axiom,
! [A1: mat_complex,B1: mat_complex,U1: mat_complex,A22: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr532731689276696518omplex @ A1 @ B1 @ U1 )
=> ( ( spectr532731689276696518omplex @ A22 @ B2 @ U2 )
=> ( ( ( dim_row_complex @ A1 )
= ( dim_col_complex @ A1 ) )
=> ( ( ( dim_row_complex @ A22 )
= ( dim_col_complex @ A22 ) )
=> ( spectr532731689276696518omplex @ ( four_b559179830521662709omplex @ A1 @ ( zero_mat_complex @ ( dim_row_complex @ A1 ) @ ( dim_col_complex @ A22 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ A22 ) @ ( dim_col_complex @ A1 ) ) @ A22 ) @ ( four_b559179830521662709omplex @ B1 @ ( zero_mat_complex @ ( dim_row_complex @ B1 ) @ ( dim_col_complex @ B2 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ B2 ) @ ( dim_col_complex @ B1 ) ) @ B2 ) @ ( four_b559179830521662709omplex @ U1 @ ( zero_mat_complex @ ( dim_row_complex @ U1 ) @ ( dim_col_complex @ U2 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ U2 ) @ ( dim_col_complex @ U1 ) ) @ U2 ) ) ) ) ) ) ).
% four_block_unitary_diag
thf(fact_82_mem__Collect__eq,axiom,
! [A3: mat_complex,P: mat_complex > $o] :
( ( member_mat_complex @ A3 @ ( collect_mat_complex @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A: set_mat_complex] :
( ( collect_mat_complex
@ ^ [X2: mat_complex] : ( member_mat_complex @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_84_four__block__diag__cong__comp,axiom,
! [A1: mat_complex,B1: mat_complex,A22: mat_complex,B2: mat_complex] :
( ( ( dim_row_complex @ A1 )
= ( dim_row_complex @ B1 ) )
=> ( ( ( dim_col_complex @ A1 )
= ( dim_col_complex @ B1 ) )
=> ( ( ( four_b559179830521662709omplex @ A1 @ ( zero_mat_complex @ ( dim_row_complex @ A1 ) @ ( dim_col_complex @ A22 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ A22 ) @ ( dim_col_complex @ A1 ) ) @ A22 )
= ( four_b559179830521662709omplex @ B1 @ ( zero_mat_complex @ ( dim_row_complex @ B1 ) @ ( dim_col_complex @ B2 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ B2 ) @ ( dim_col_complex @ B1 ) ) @ B2 ) )
=> ( A1 = B1 ) ) ) ) ).
% four_block_diag_cong_comp
thf(fact_85_four__block__diag__cong__comp,axiom,
! [A1: mat_a,B1: mat_a,A22: mat_a,B2: mat_a] :
( ( ( dim_row_a @ A1 )
= ( dim_row_a @ B1 ) )
=> ( ( ( dim_col_a @ A1 )
= ( dim_col_a @ B1 ) )
=> ( ( ( four_block_mat_a @ A1 @ ( zero_mat_a @ ( dim_row_a @ A1 ) @ ( dim_col_a @ A22 ) ) @ ( zero_mat_a @ ( dim_row_a @ A22 ) @ ( dim_col_a @ A1 ) ) @ A22 )
= ( four_block_mat_a @ B1 @ ( zero_mat_a @ ( dim_row_a @ B1 ) @ ( dim_col_a @ B2 ) ) @ ( zero_mat_a @ ( dim_row_a @ B2 ) @ ( dim_col_a @ B1 ) ) @ B2 ) )
=> ( A1 = B1 ) ) ) ) ).
% four_block_diag_cong_comp
thf(fact_86_four__block__diag__cong__comp_H,axiom,
! [A1: mat_complex,B1: mat_complex,A22: mat_complex,B2: mat_complex] :
( ( ( dim_row_complex @ A1 )
= ( dim_row_complex @ B1 ) )
=> ( ( ( dim_col_complex @ A1 )
= ( dim_col_complex @ B1 ) )
=> ( ( ( four_b559179830521662709omplex @ A1 @ ( zero_mat_complex @ ( dim_row_complex @ A1 ) @ ( dim_col_complex @ A22 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ A22 ) @ ( dim_col_complex @ A1 ) ) @ A22 )
= ( four_b559179830521662709omplex @ B1 @ ( zero_mat_complex @ ( dim_row_complex @ B1 ) @ ( dim_col_complex @ B2 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ B2 ) @ ( dim_col_complex @ B1 ) ) @ B2 ) )
=> ( A22 = B2 ) ) ) ) ).
% four_block_diag_cong_comp'
thf(fact_87_four__block__diag__cong__comp_H,axiom,
! [A1: mat_a,B1: mat_a,A22: mat_a,B2: mat_a] :
( ( ( dim_row_a @ A1 )
= ( dim_row_a @ B1 ) )
=> ( ( ( dim_col_a @ A1 )
= ( dim_col_a @ B1 ) )
=> ( ( ( four_block_mat_a @ A1 @ ( zero_mat_a @ ( dim_row_a @ A1 ) @ ( dim_col_a @ A22 ) ) @ ( zero_mat_a @ ( dim_row_a @ A22 ) @ ( dim_col_a @ A1 ) ) @ A22 )
= ( four_block_mat_a @ B1 @ ( zero_mat_a @ ( dim_row_a @ B1 ) @ ( dim_col_a @ B2 ) ) @ ( zero_mat_a @ ( dim_row_a @ B2 ) @ ( dim_col_a @ B1 ) ) @ B2 ) )
=> ( A22 = B2 ) ) ) ) ).
% four_block_diag_cong_comp'
thf(fact_88_assoc__four__block__mat,axiom,
! [A: mat_complex,B: mat_complex,C: mat_complex] :
( ( four_b559179830521662709omplex @ A @ ( zero_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ ( four_b559179830521662709omplex @ B @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ C ) ) @ ( zero_mat_complex @ ( dim_row_complex @ C ) @ ( dim_col_complex @ B ) ) @ C ) ) ) @ ( zero_mat_complex @ ( dim_row_complex @ ( four_b559179830521662709omplex @ B @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ C ) ) @ ( zero_mat_complex @ ( dim_row_complex @ C ) @ ( dim_col_complex @ B ) ) @ C ) ) @ ( dim_col_complex @ A ) ) @ ( four_b559179830521662709omplex @ B @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ C ) ) @ ( zero_mat_complex @ ( dim_row_complex @ C ) @ ( dim_col_complex @ B ) ) @ C ) )
= ( four_b559179830521662709omplex @ ( four_b559179830521662709omplex @ A @ ( zero_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ B ) ) @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ A ) ) @ B ) @ ( zero_mat_complex @ ( dim_row_complex @ ( four_b559179830521662709omplex @ A @ ( zero_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ B ) ) @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ A ) ) @ B ) ) @ ( dim_col_complex @ C ) ) @ ( zero_mat_complex @ ( dim_row_complex @ C ) @ ( dim_col_complex @ ( four_b559179830521662709omplex @ A @ ( zero_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ B ) ) @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ A ) ) @ B ) ) ) @ C ) ) ).
% assoc_four_block_mat
thf(fact_89_assoc__four__block__mat,axiom,
! [A: mat_a,B: mat_a,C: mat_a] :
( ( four_block_mat_a @ A @ ( zero_mat_a @ ( dim_row_a @ A ) @ ( dim_col_a @ ( four_block_mat_a @ B @ ( zero_mat_a @ ( dim_row_a @ B ) @ ( dim_col_a @ C ) ) @ ( zero_mat_a @ ( dim_row_a @ C ) @ ( dim_col_a @ B ) ) @ C ) ) ) @ ( zero_mat_a @ ( dim_row_a @ ( four_block_mat_a @ B @ ( zero_mat_a @ ( dim_row_a @ B ) @ ( dim_col_a @ C ) ) @ ( zero_mat_a @ ( dim_row_a @ C ) @ ( dim_col_a @ B ) ) @ C ) ) @ ( dim_col_a @ A ) ) @ ( four_block_mat_a @ B @ ( zero_mat_a @ ( dim_row_a @ B ) @ ( dim_col_a @ C ) ) @ ( zero_mat_a @ ( dim_row_a @ C ) @ ( dim_col_a @ B ) ) @ C ) )
= ( four_block_mat_a @ ( four_block_mat_a @ A @ ( zero_mat_a @ ( dim_row_a @ A ) @ ( dim_col_a @ B ) ) @ ( zero_mat_a @ ( dim_row_a @ B ) @ ( dim_col_a @ A ) ) @ B ) @ ( zero_mat_a @ ( dim_row_a @ ( four_block_mat_a @ A @ ( zero_mat_a @ ( dim_row_a @ A ) @ ( dim_col_a @ B ) ) @ ( zero_mat_a @ ( dim_row_a @ B ) @ ( dim_col_a @ A ) ) @ B ) ) @ ( dim_col_a @ C ) ) @ ( zero_mat_a @ ( dim_row_a @ C ) @ ( dim_col_a @ ( four_block_mat_a @ A @ ( zero_mat_a @ ( dim_row_a @ A ) @ ( dim_col_a @ B ) ) @ ( zero_mat_a @ ( dim_row_a @ B ) @ ( dim_col_a @ A ) ) @ B ) ) ) @ C ) ) ).
% assoc_four_block_mat
thf(fact_90_index__zero__mat_I2_J,axiom,
! [Nr: nat,Nc: nat] :
( ( dim_row_a @ ( zero_mat_a @ Nr @ Nc ) )
= Nr ) ).
% index_zero_mat(2)
thf(fact_91_index__zero__mat_I2_J,axiom,
! [Nr: nat,Nc: nat] :
( ( dim_row_complex @ ( zero_mat_complex @ Nr @ Nc ) )
= Nr ) ).
% index_zero_mat(2)
thf(fact_92_index__zero__mat_I3_J,axiom,
! [Nr: nat,Nc: nat] :
( ( dim_col_a @ ( zero_mat_a @ Nr @ Nc ) )
= Nc ) ).
% index_zero_mat(3)
thf(fact_93_index__zero__mat_I3_J,axiom,
! [Nr: nat,Nc: nat] :
( ( dim_col_complex @ ( zero_mat_complex @ Nr @ Nc ) )
= Nc ) ).
% index_zero_mat(3)
thf(fact_94__092_060open_062i_A_060_Adim__row_A_Ifour__block__diag_AB1_AB2_J_092_060close_062,axiom,
ord_less_nat @ i @ ( dim_row_a @ ( four_block_mat_a @ b1 @ ( zero_mat_a @ ( dim_row_a @ b1 ) @ ( dim_col_a @ b2 ) ) @ ( zero_mat_a @ ( dim_row_a @ b2 ) @ ( dim_col_a @ b1 ) ) @ b2 ) ) ).
% \<open>i < dim_row (four_block_diag B1 B2)\<close>
thf(fact_95_real__diag__decompD_I1_J,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr3403749184330357196comp_a @ A @ B @ U )
=> ( spectr4894841263502123494diag_a @ A @ B @ U ) ) ).
% real_diag_decompD(1)
thf(fact_96_real__diag__decompD_I1_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr5409772854192057952omplex @ A @ B @ U )
=> ( spectr532731689276696518omplex @ A @ B @ U ) ) ).
% real_diag_decompD(1)
thf(fact_97_four__block__diag__unitary,axiom,
! [U1: mat_a,U2: mat_a] :
( ( complex_unitary_a @ U1 )
=> ( ( complex_unitary_a @ U2 )
=> ( complex_unitary_a @ ( four_block_mat_a @ U1 @ ( zero_mat_a @ ( dim_row_a @ U1 ) @ ( dim_col_a @ U2 ) ) @ ( zero_mat_a @ ( dim_row_a @ U2 ) @ ( dim_col_a @ U1 ) ) @ U2 ) ) ) ) ).
% four_block_diag_unitary
thf(fact_98_four__block__diag__unitary,axiom,
! [U1: mat_complex,U2: mat_complex] :
( ( comple6660659447773130958omplex @ U1 )
=> ( ( comple6660659447773130958omplex @ U2 )
=> ( comple6660659447773130958omplex @ ( four_b559179830521662709omplex @ U1 @ ( zero_mat_complex @ ( dim_row_complex @ U1 ) @ ( dim_col_complex @ U2 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ U2 ) @ ( dim_col_complex @ U1 ) ) @ U2 ) ) ) ) ).
% four_block_diag_unitary
thf(fact_99_four__block__unitarily__equiv,axiom,
! [A1: mat_complex,B1: mat_complex,U1: mat_complex,A22: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr6340060708231679580omplex @ A1 @ B1 @ U1 )
=> ( ( spectr6340060708231679580omplex @ A22 @ B2 @ U2 )
=> ( ( ( dim_row_complex @ A1 )
= ( dim_col_complex @ A1 ) )
=> ( ( ( dim_row_complex @ A22 )
= ( dim_col_complex @ A22 ) )
=> ( spectr6340060708231679580omplex @ ( four_b559179830521662709omplex @ A1 @ ( zero_mat_complex @ ( dim_row_complex @ A1 ) @ ( dim_col_complex @ A22 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ A22 ) @ ( dim_col_complex @ A1 ) ) @ A22 ) @ ( four_b559179830521662709omplex @ B1 @ ( zero_mat_complex @ ( dim_row_complex @ B1 ) @ ( dim_col_complex @ B2 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ B2 ) @ ( dim_col_complex @ B1 ) ) @ B2 ) @ ( four_b559179830521662709omplex @ U1 @ ( zero_mat_complex @ ( dim_row_complex @ U1 ) @ ( dim_col_complex @ U2 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ U2 ) @ ( dim_col_complex @ U1 ) ) @ U2 ) ) ) ) ) ) ).
% four_block_unitarily_equiv
thf(fact_100_four__block__unitarily__equiv,axiom,
! [A1: mat_a,B1: mat_a,U1: mat_a,A22: mat_a,B2: mat_a,U2: mat_a] :
( ( spectr4825054497075562704quiv_a @ A1 @ B1 @ U1 )
=> ( ( spectr4825054497075562704quiv_a @ A22 @ B2 @ U2 )
=> ( ( ( dim_row_a @ A1 )
= ( dim_col_a @ A1 ) )
=> ( ( ( dim_row_a @ A22 )
= ( dim_col_a @ A22 ) )
=> ( spectr4825054497075562704quiv_a @ ( four_block_mat_a @ A1 @ ( zero_mat_a @ ( dim_row_a @ A1 ) @ ( dim_col_a @ A22 ) ) @ ( zero_mat_a @ ( dim_row_a @ A22 ) @ ( dim_col_a @ A1 ) ) @ A22 ) @ ( four_block_mat_a @ B1 @ ( zero_mat_a @ ( dim_row_a @ B1 ) @ ( dim_col_a @ B2 ) ) @ ( zero_mat_a @ ( dim_row_a @ B2 ) @ ( dim_col_a @ B1 ) ) @ B2 ) @ ( four_block_mat_a @ U1 @ ( zero_mat_a @ ( dim_row_a @ U1 ) @ ( dim_col_a @ U2 ) ) @ ( zero_mat_a @ ( dim_row_a @ U2 ) @ ( dim_col_a @ U1 ) ) @ U2 ) ) ) ) ) ) ).
% four_block_unitarily_equiv
thf(fact_101_four__block__diagonal,axiom,
! [B1: mat_complex,B2: mat_complex] :
( ( ( dim_row_complex @ B1 )
= ( dim_col_complex @ B1 ) )
=> ( ( ( dim_row_complex @ B2 )
= ( dim_col_complex @ B2 ) )
=> ( ( diagonal_mat_complex @ B1 )
=> ( ( diagonal_mat_complex @ B2 )
=> ( diagonal_mat_complex @ ( four_b559179830521662709omplex @ B1 @ ( zero_mat_complex @ ( dim_row_complex @ B1 ) @ ( dim_col_complex @ B2 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ B2 ) @ ( dim_col_complex @ B1 ) ) @ B2 ) ) ) ) ) ) ).
% four_block_diagonal
thf(fact_102_four__block__diagonal,axiom,
! [B1: mat_a,B2: mat_a] :
( ( ( dim_row_a @ B1 )
= ( dim_col_a @ B1 ) )
=> ( ( ( dim_row_a @ B2 )
= ( dim_col_a @ B2 ) )
=> ( ( diagonal_mat_a @ B1 )
=> ( ( diagonal_mat_a @ B2 )
=> ( diagonal_mat_a @ ( four_block_mat_a @ B1 @ ( zero_mat_a @ ( dim_row_a @ B1 ) @ ( dim_col_a @ B2 ) ) @ ( zero_mat_a @ ( dim_row_a @ B2 ) @ ( dim_col_a @ B1 ) ) @ B2 ) ) ) ) ) ) ).
% four_block_diagonal
thf(fact_103_step__3__def,axiom,
( jordan4501759426295633263omplex
= ( ^ [A2: mat_complex] : ( jordan4702481308941288104omplex @ ( dim_row_complex @ A2 ) @ one_one_nat @ A2 ) ) ) ).
% step_3_def
thf(fact_104_step__3__def,axiom,
( jordan5943829226657577597ep_3_a
= ( ^ [A2: mat_a] : ( jordan460303421567170436main_a @ ( dim_row_a @ A2 ) @ one_one_nat @ A2 ) ) ) ).
% step_3_def
thf(fact_105_step__2__def,axiom,
( jordan7871273693253786478omplex
= ( ^ [A2: mat_complex] : ( jordan6916311984355858983omplex @ ( dim_row_complex @ A2 ) @ zero_zero_nat @ A2 ) ) ) ).
% step_2_def
thf(fact_106_step__2__def,axiom,
( jordan8731284808630253630ep_2_a
= ( ^ [A2: mat_a] : ( jordan913024080637330373main_a @ ( dim_row_a @ A2 ) @ zero_zero_nat @ A2 ) ) ) ).
% step_2_def
thf(fact_107_step__1__def,axiom,
( jordan2295368353748153855ep_1_a
= ( ^ [A2: mat_a] : ( jordan1365744739707490310main_a @ ( dim_row_a @ A2 ) @ zero_zero_nat @ zero_zero_nat @ A2 ) ) ) ).
% step_1_def
thf(fact_108_step__1__def,axiom,
( jordan2017415923357163885omplex
= ( ^ [A2: mat_complex] : ( jordan9130142659770429862omplex @ ( dim_row_complex @ A2 ) @ zero_zero_nat @ zero_zero_nat @ A2 ) ) ) ).
% step_1_def
thf(fact_109_four__block__diag__adjoint,axiom,
! [A1: mat_a,A22: mat_a] :
( ( schur_mat_adjoint_a @ ( four_block_mat_a @ A1 @ ( zero_mat_a @ ( dim_row_a @ A1 ) @ ( dim_col_a @ A22 ) ) @ ( zero_mat_a @ ( dim_row_a @ A22 ) @ ( dim_col_a @ A1 ) ) @ A22 ) )
= ( four_block_mat_a @ ( schur_mat_adjoint_a @ A1 ) @ ( zero_mat_a @ ( dim_row_a @ ( schur_mat_adjoint_a @ A1 ) ) @ ( dim_col_a @ ( schur_mat_adjoint_a @ A22 ) ) ) @ ( zero_mat_a @ ( dim_row_a @ ( schur_mat_adjoint_a @ A22 ) ) @ ( dim_col_a @ ( schur_mat_adjoint_a @ A1 ) ) ) @ ( schur_mat_adjoint_a @ A22 ) ) ) ).
% four_block_diag_adjoint
thf(fact_110_four__block__diag__adjoint,axiom,
! [A1: mat_complex,A22: mat_complex] :
( ( schur_5982229384592763574omplex @ ( four_b559179830521662709omplex @ A1 @ ( zero_mat_complex @ ( dim_row_complex @ A1 ) @ ( dim_col_complex @ A22 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ A22 ) @ ( dim_col_complex @ A1 ) ) @ A22 ) )
= ( four_b559179830521662709omplex @ ( schur_5982229384592763574omplex @ A1 ) @ ( zero_mat_complex @ ( dim_row_complex @ ( schur_5982229384592763574omplex @ A1 ) ) @ ( dim_col_complex @ ( schur_5982229384592763574omplex @ A22 ) ) ) @ ( zero_mat_complex @ ( dim_row_complex @ ( schur_5982229384592763574omplex @ A22 ) ) @ ( dim_col_complex @ ( schur_5982229384592763574omplex @ A1 ) ) ) @ ( schur_5982229384592763574omplex @ A22 ) ) ) ).
% four_block_diag_adjoint
thf(fact_111_diag__mat__transpose,axiom,
! [A: mat_a] :
( ( ( dim_row_a @ A )
= ( dim_col_a @ A ) )
=> ( ( diag_mat_a @ ( transpose_mat_a @ A ) )
= ( diag_mat_a @ A ) ) ) ).
% diag_mat_transpose
thf(fact_112_diag__mat__transpose,axiom,
! [A: mat_complex] :
( ( ( dim_row_complex @ A )
= ( dim_col_complex @ A ) )
=> ( ( diag_mat_complex @ ( transp3074176993011536131omplex @ A ) )
= ( diag_mat_complex @ A ) ) ) ).
% diag_mat_transpose
thf(fact_113_left__mult__zero__mat_H,axiom,
! [A: mat_a,N: nat,Nr: nat] :
( ( ( dim_row_a @ A )
= N )
=> ( ( times_times_mat_a @ ( zero_mat_a @ Nr @ N ) @ A )
= ( zero_mat_a @ Nr @ ( dim_col_a @ A ) ) ) ) ).
% left_mult_zero_mat'
thf(fact_114_left__mult__zero__mat_H,axiom,
! [A: mat_complex,N: nat,Nr: nat] :
( ( ( dim_row_complex @ A )
= N )
=> ( ( times_8009071140041733218omplex @ ( zero_mat_complex @ Nr @ N ) @ A )
= ( zero_mat_complex @ Nr @ ( dim_col_complex @ A ) ) ) ) ).
% left_mult_zero_mat'
thf(fact_115_unitary__diag__def,axiom,
( spectr4894841263502123494diag_a
= ( ^ [A2: mat_a,B5: mat_a,U3: mat_a] :
( ( spectr4825054497075562704quiv_a @ A2 @ B5 @ U3 )
& ( diagonal_mat_a @ B5 ) ) ) ) ).
% unitary_diag_def
thf(fact_116_unitary__diag__def,axiom,
( spectr532731689276696518omplex
= ( ^ [A2: mat_complex,B5: mat_complex,U3: mat_complex] :
( ( spectr6340060708231679580omplex @ A2 @ B5 @ U3 )
& ( diagonal_mat_complex @ B5 ) ) ) ) ).
% unitary_diag_def
thf(fact_117_unitarily__equivD_I1_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( comple6660659447773130958omplex @ U ) ) ).
% unitarily_equivD(1)
thf(fact_118_unitarily__equiv__eq,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U @ B ) @ ( schur_5982229384592763574omplex @ U ) ) ) ) ).
% unitarily_equiv_eq
thf(fact_119_unitarily__equiv__adjoint,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( spectr6340060708231679580omplex @ B @ A @ ( schur_5982229384592763574omplex @ U ) ) ) ).
% unitarily_equiv_adjoint
thf(fact_120_unitarily__equiv__commute,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex,C: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( ( ( times_8009071140041733218omplex @ A @ C )
= ( times_8009071140041733218omplex @ C @ A ) )
=> ( ( times_8009071140041733218omplex @ B @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U ) @ C ) @ U ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U ) @ C ) @ U ) @ B ) ) ) ) ).
% unitarily_equiv_commute
thf(fact_121_diag__mat__diagonal__eq,axiom,
! [A: mat_a,B: mat_a] :
( ( ( diag_mat_a @ A )
= ( diag_mat_a @ B ) )
=> ( ( diagonal_mat_a @ A )
=> ( ( diagonal_mat_a @ B )
=> ( ( ( dim_col_a @ A )
= ( dim_col_a @ B ) )
=> ( A = B ) ) ) ) ) ).
% diag_mat_diagonal_eq
thf(fact_122_diag__mat__diagonal__eq,axiom,
! [A: mat_complex,B: mat_complex] :
( ( ( diag_mat_complex @ A )
= ( diag_mat_complex @ B ) )
=> ( ( diagonal_mat_complex @ A )
=> ( ( diagonal_mat_complex @ B )
=> ( ( ( dim_col_complex @ A )
= ( dim_col_complex @ B ) )
=> ( A = B ) ) ) ) ) ).
% diag_mat_diagonal_eq
thf(fact_123_zero__adjoint,axiom,
! [N: nat,M: nat] :
( ( schur_mat_adjoint_a @ ( zero_mat_a @ N @ M ) )
= ( zero_mat_a @ M @ N ) ) ).
% zero_adjoint
thf(fact_124_zero__adjoint,axiom,
! [N: nat,M: nat] :
( ( schur_5982229384592763574omplex @ ( zero_mat_complex @ N @ M ) )
= ( zero_mat_complex @ M @ N ) ) ).
% zero_adjoint
thf(fact_125_unitary__diagD_I2_J,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4894841263502123494diag_a @ A @ B @ U )
=> ( diagonal_mat_a @ B ) ) ).
% unitary_diagD(2)
thf(fact_126_unitary__diagD_I2_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( diagonal_mat_complex @ B ) ) ).
% unitary_diagD(2)
thf(fact_127_unitary__diag__imp__unitarily__equiv,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4894841263502123494diag_a @ A @ B @ U )
=> ( spectr4825054497075562704quiv_a @ A @ B @ U ) ) ).
% unitary_diag_imp_unitarily_equiv
thf(fact_128_unitary__diag__imp__unitarily__equiv,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( spectr6340060708231679580omplex @ A @ B @ U ) ) ).
% unitary_diag_imp_unitarily_equiv
thf(fact_129_unitary__diagD_I3_J,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4894841263502123494diag_a @ A @ B @ U )
=> ( complex_unitary_a @ U ) ) ).
% unitary_diagD(3)
thf(fact_130_unitary__diagD_I3_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( comple6660659447773130958omplex @ U ) ) ).
% unitary_diagD(3)
thf(fact_131_index__mult__mat_I2_J,axiom,
! [A: mat_a,B: mat_a] :
( ( dim_row_a @ ( times_times_mat_a @ A @ B ) )
= ( dim_row_a @ A ) ) ).
% index_mult_mat(2)
thf(fact_132_index__mult__mat_I2_J,axiom,
! [A: mat_complex,B: mat_complex] :
( ( dim_row_complex @ ( times_8009071140041733218omplex @ A @ B ) )
= ( dim_row_complex @ A ) ) ).
% index_mult_mat(2)
thf(fact_133_index__mult__mat_I3_J,axiom,
! [A: mat_a,B: mat_a] :
( ( dim_col_a @ ( times_times_mat_a @ A @ B ) )
= ( dim_col_a @ B ) ) ).
% index_mult_mat(3)
thf(fact_134_index__mult__mat_I3_J,axiom,
! [A: mat_complex,B: mat_complex] :
( ( dim_col_complex @ ( times_8009071140041733218omplex @ A @ B ) )
= ( dim_col_complex @ B ) ) ).
% index_mult_mat(3)
thf(fact_135_step__3__c__inner__loop_Osimps_I1_J,axiom,
! [Val: complex,L: nat,I: nat,A: mat_complex] :
( ( jordan7656109678144820486omplex @ Val @ L @ I @ zero_zero_nat @ A )
= A ) ).
% step_3_c_inner_loop.simps(1)
thf(fact_136_step__3__c__inner__loop_Osimps_I1_J,axiom,
! [Val: a,L: nat,I: nat,A: mat_a] :
( ( jordan8889242743715136678loop_a @ Val @ L @ I @ zero_zero_nat @ A )
= A ) ).
% step_3_c_inner_loop.simps(1)
thf(fact_137_right__mult__zero__mat_H,axiom,
! [A: mat_a,N: nat,Nc: nat] :
( ( ( dim_col_a @ A )
= N )
=> ( ( times_times_mat_a @ A @ ( zero_mat_a @ N @ Nc ) )
= ( zero_mat_a @ ( dim_row_a @ A ) @ Nc ) ) ) ).
% right_mult_zero_mat'
thf(fact_138_right__mult__zero__mat_H,axiom,
! [A: mat_complex,N: nat,Nc: nat] :
( ( ( dim_col_complex @ A )
= N )
=> ( ( times_8009071140041733218omplex @ A @ ( zero_mat_complex @ N @ Nc ) )
= ( zero_mat_complex @ ( dim_row_complex @ A ) @ Nc ) ) ) ).
% right_mult_zero_mat'
thf(fact_139_adjoint__dim__row,axiom,
! [A: mat_a] :
( ( dim_row_a @ ( schur_mat_adjoint_a @ A ) )
= ( dim_col_a @ A ) ) ).
% adjoint_dim_row
thf(fact_140_adjoint__dim__row,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( schur_5982229384592763574omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% adjoint_dim_row
thf(fact_141_adjoint__dim__col,axiom,
! [A: mat_a] :
( ( dim_col_a @ ( schur_mat_adjoint_a @ A ) )
= ( dim_row_a @ A ) ) ).
% adjoint_dim_col
thf(fact_142_adjoint__dim__col,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( schur_5982229384592763574omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% adjoint_dim_col
thf(fact_143_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_144_append__rows__def,axiom,
( append_rows_complex
= ( ^ [A2: mat_complex,B5: mat_complex] : ( four_b559179830521662709omplex @ A2 @ ( zero_mat_complex @ ( dim_row_complex @ A2 ) @ zero_zero_nat ) @ B5 @ ( zero_mat_complex @ ( dim_row_complex @ B5 ) @ zero_zero_nat ) ) ) ) ).
% append_rows_def
thf(fact_145_append__rows__def,axiom,
( append_rows_a
= ( ^ [A2: mat_a,B5: mat_a] : ( four_block_mat_a @ A2 @ ( zero_mat_a @ ( dim_row_a @ A2 ) @ zero_zero_nat ) @ B5 @ ( zero_mat_a @ ( dim_row_a @ B5 ) @ zero_zero_nat ) ) ) ) ).
% append_rows_def
thf(fact_146_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_147_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_148_rel__simps_I69_J,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% rel_simps(69)
thf(fact_149_rel__simps_I69_J,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% rel_simps(69)
thf(fact_150_rel__simps_I68_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% rel_simps(68)
thf(fact_151_rel__simps_I68_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% rel_simps(68)
thf(fact_152_verit__comp__simplify_I28_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% verit_comp_simplify(28)
thf(fact_153_verit__comp__simplify_I28_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% verit_comp_simplify(28)
thf(fact_154_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_155_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_156_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_157_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_158_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_159_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_160_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_161_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_162_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_163_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_164_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_165_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_166_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_167_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_168_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_169_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_170_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_171_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_172_more__arith__simps_I11_J,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B6 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B6 @ C2 ) ) ) ).
% more_arith_simps(11)
thf(fact_173_more__arith__simps_I11_J,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( times_times_complex @ ( times_times_complex @ A3 @ B6 ) @ C2 )
= ( times_times_complex @ A3 @ ( times_times_complex @ B6 @ C2 ) ) ) ).
% more_arith_simps(11)
thf(fact_174_more__arith__simps_I11_J,axiom,
! [A3: real,B6: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B6 ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B6 @ C2 ) ) ) ).
% more_arith_simps(11)
thf(fact_175_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_176_verit__comp__simplify1_I1_J,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_177_verit__comp__simplify1_I1_J,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_178_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_179_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_180_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_181_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_182_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_183_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_184_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_185_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_186_Complex__Matrix_Oadjoint__adjoint,axiom,
! [A: mat_complex] :
( ( schur_5982229384592763574omplex @ ( schur_5982229384592763574omplex @ A ) )
= A ) ).
% Complex_Matrix.adjoint_adjoint
thf(fact_187_mult__right__cancel,axiom,
! [C2: nat,A3: nat,B6: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B6 @ C2 ) )
= ( A3 = B6 ) ) ) ).
% mult_right_cancel
thf(fact_188_mult__right__cancel,axiom,
! [C2: complex,A3: complex,B6: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B6 @ C2 ) )
= ( A3 = B6 ) ) ) ).
% mult_right_cancel
thf(fact_189_mult__right__cancel,axiom,
! [C2: real,A3: real,B6: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B6 @ C2 ) )
= ( A3 = B6 ) ) ) ).
% mult_right_cancel
thf(fact_190_mult__cancel__right,axiom,
! [A3: nat,C2: nat,B6: nat] :
( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B6 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B6 ) ) ) ).
% mult_cancel_right
thf(fact_191_mult__cancel__right,axiom,
! [A3: complex,C2: complex,B6: complex] :
( ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B6 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B6 ) ) ) ).
% mult_cancel_right
thf(fact_192_mult__cancel__right,axiom,
! [A3: real,C2: real,B6: real] :
( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B6 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B6 ) ) ) ).
% mult_cancel_right
thf(fact_193_mult__left__cancel,axiom,
! [C2: nat,A3: nat,B6: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B6 ) )
= ( A3 = B6 ) ) ) ).
% mult_left_cancel
thf(fact_194_mult__left__cancel,axiom,
! [C2: complex,A3: complex,B6: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ C2 @ A3 )
= ( times_times_complex @ C2 @ B6 ) )
= ( A3 = B6 ) ) ) ).
% mult_left_cancel
thf(fact_195_mult__left__cancel,axiom,
! [C2: real,A3: real,B6: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B6 ) )
= ( A3 = B6 ) ) ) ).
% mult_left_cancel
thf(fact_196_mult__cancel__left,axiom,
! [C2: nat,A3: nat,B6: nat] :
( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B6 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B6 ) ) ) ).
% mult_cancel_left
thf(fact_197_mult__cancel__left,axiom,
! [C2: complex,A3: complex,B6: complex] :
( ( ( times_times_complex @ C2 @ A3 )
= ( times_times_complex @ C2 @ B6 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B6 ) ) ) ).
% mult_cancel_left
thf(fact_198_mult__cancel__left,axiom,
! [C2: real,A3: real,B6: real] :
( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B6 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B6 ) ) ) ).
% mult_cancel_left
thf(fact_199_no__zero__divisors,axiom,
! [A3: nat,B6: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B6 != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B6 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_200_no__zero__divisors,axiom,
! [A3: complex,B6: complex] :
( ( A3 != zero_zero_complex )
=> ( ( B6 != zero_zero_complex )
=> ( ( times_times_complex @ A3 @ B6 )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_201_no__zero__divisors,axiom,
! [A3: real,B6: real] :
( ( A3 != zero_zero_real )
=> ( ( B6 != zero_zero_real )
=> ( ( times_times_real @ A3 @ B6 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_202_mult__eq__0__iff,axiom,
! [A3: nat,B6: nat] :
( ( ( times_times_nat @ A3 @ B6 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B6 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_203_mult__eq__0__iff,axiom,
! [A3: complex,B6: complex] :
( ( ( times_times_complex @ A3 @ B6 )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( B6 = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_204_mult__eq__0__iff,axiom,
! [A3: real,B6: real] :
( ( ( times_times_real @ A3 @ B6 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B6 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_205_divisors__zero,axiom,
! [A3: nat,B6: nat] :
( ( ( times_times_nat @ A3 @ B6 )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B6 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_206_divisors__zero,axiom,
! [A3: complex,B6: complex] :
( ( ( times_times_complex @ A3 @ B6 )
= zero_zero_complex )
=> ( ( A3 = zero_zero_complex )
| ( B6 = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_207_divisors__zero,axiom,
! [A3: real,B6: real] :
( ( ( times_times_real @ A3 @ B6 )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B6 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_208_mult__zero__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_209_mult__zero__right,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_210_mult__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_211_mult__zero__left,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_212_mult__zero__left,axiom,
! [A3: complex] :
( ( times_times_complex @ zero_zero_complex @ A3 )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_213_mult__zero__left,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_214_mult__not__zero,axiom,
! [A3: nat,B6: nat] :
( ( ( times_times_nat @ A3 @ B6 )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B6 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_215_mult__not__zero,axiom,
! [A3: complex,B6: complex] :
( ( ( times_times_complex @ A3 @ B6 )
!= zero_zero_complex )
=> ( ( A3 != zero_zero_complex )
& ( B6 != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_216_mult__not__zero,axiom,
! [A3: real,B6: real] :
( ( ( times_times_real @ A3 @ B6 )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B6 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_217_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_218_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_219_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_220_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_221_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_222_verit__prod__simplify_I1_J,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% verit_prod_simplify(1)
thf(fact_223_verit__prod__simplify_I1_J,axiom,
! [A3: complex] :
( ( times_times_complex @ one_one_complex @ A3 )
= A3 ) ).
% verit_prod_simplify(1)
thf(fact_224_verit__prod__simplify_I1_J,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% verit_prod_simplify(1)
thf(fact_225_verit__prod__simplify_I2_J,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% verit_prod_simplify(2)
thf(fact_226_verit__prod__simplify_I2_J,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% verit_prod_simplify(2)
thf(fact_227_verit__prod__simplify_I2_J,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% verit_prod_simplify(2)
thf(fact_228_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_229_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_230_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_231_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_232_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_233_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_234_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_235_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_236_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_237_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_238_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_239_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_240_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B6 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_241_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_242_mult__less__cancel__right__disj,axiom,
! [A3: real,C2: real,B6: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A3 @ B6 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B6 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_243_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B6 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_244_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_245_mult__strict__right__mono__neg,axiom,
! [B6: real,A3: real,C2: real] :
( ( ord_less_real @ B6 @ A3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_246_mult__less__cancel__left__disj,axiom,
! [C2: real,A3: real,B6: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A3 @ B6 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B6 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_247_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B6 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_248_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_249_mult__strict__left__mono__neg,axiom,
! [B6: real,A3: real,C2: real] :
( ( ord_less_real @ B6 @ A3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_250_mult__less__cancel__left__pos,axiom,
! [C2: real,A3: real,B6: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) )
= ( ord_less_real @ A3 @ B6 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_251_mult__less__cancel__left__neg,axiom,
! [C2: real,A3: real,B6: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) )
= ( ord_less_real @ B6 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_252_zero__less__mult__pos2,axiom,
! [B6: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B6 @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B6 ) ) ) ).
% zero_less_mult_pos2
thf(fact_253_zero__less__mult__pos2,axiom,
! [B6: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B6 @ A3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B6 ) ) ) ).
% zero_less_mult_pos2
thf(fact_254_zero__less__mult__pos,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B6 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B6 ) ) ) ).
% zero_less_mult_pos
thf(fact_255_zero__less__mult__pos,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B6 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B6 ) ) ) ).
% zero_less_mult_pos
thf(fact_256_zero__less__mult__iff,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B6 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ zero_zero_real @ B6 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ B6 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_257_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B6 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B6 @ A3 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_258_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B6 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B6 @ A3 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_259_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B6 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B6 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_260_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B6 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B6 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_261_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B6 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B6 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_262_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B6 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ B6 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_263_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B6 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B6 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_264_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B6 )
=> ( ord_less_real @ ( times_times_real @ A3 @ B6 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_265_mult__less__0__iff,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ B6 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ B6 @ zero_zero_real ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B6 ) ) ) ) ).
% mult_less_0_iff
thf(fact_266_not__square__less__zero,axiom,
! [A3: real] :
~ ( ord_less_real @ ( times_times_real @ A3 @ A3 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_267_mult__neg__neg,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B6 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B6 ) ) ) ) ).
% mult_neg_neg
thf(fact_268_mult__cancel__right2,axiom,
! [A3: complex,C2: complex] :
( ( ( times_times_complex @ A3 @ C2 )
= C2 )
= ( ( C2 = zero_zero_complex )
| ( A3 = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_269_mult__cancel__right2,axiom,
! [A3: real,C2: real] :
( ( ( times_times_real @ A3 @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_270_mult__cancel__right1,axiom,
! [C2: complex,B6: complex] :
( ( C2
= ( times_times_complex @ B6 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( B6 = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_271_mult__cancel__right1,axiom,
! [C2: real,B6: real] :
( ( C2
= ( times_times_real @ B6 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B6 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_272_mult__cancel__left2,axiom,
! [C2: complex,A3: complex] :
( ( ( times_times_complex @ C2 @ A3 )
= C2 )
= ( ( C2 = zero_zero_complex )
| ( A3 = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_273_mult__cancel__left2,axiom,
! [C2: real,A3: real] :
( ( ( times_times_real @ C2 @ A3 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_274_mult__cancel__left1,axiom,
! [C2: complex,B6: complex] :
( ( C2
= ( times_times_complex @ C2 @ B6 ) )
= ( ( C2 = zero_zero_complex )
| ( B6 = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_275_mult__cancel__left1,axiom,
! [C2: real,B6: real] :
( ( C2
= ( times_times_real @ C2 @ B6 ) )
= ( ( C2 = zero_zero_real )
| ( B6 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_276_mult__if__delta,axiom,
! [P: $o,Q: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_277_mult__if__delta,axiom,
! [P: $o,Q: complex] :
( ( P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q )
= zero_zero_complex ) ) ) ).
% mult_if_delta
thf(fact_278_mult__if__delta,axiom,
! [P: $o,Q: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_279_mult__less__iff1,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_280_inv__all_H__def,axiom,
( jordan4251489913308508029_all_a
= ( ^ [P2: mat_a > nat > nat > $o,A2: mat_a] :
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J2 @ ( dim_row_a @ A2 ) )
=> ( P2 @ A2 @ I2 @ J2 ) ) ) ) ) ).
% inv_all'_def
thf(fact_281_inv__all_H__def,axiom,
( jordan5032732407113867375omplex
= ( ^ [P2: mat_complex > nat > nat > $o,A2: mat_complex] :
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_complex @ A2 ) )
=> ( ( ord_less_nat @ J2 @ ( dim_row_complex @ A2 ) )
=> ( P2 @ A2 @ I2 @ J2 ) ) ) ) ) ).
% inv_all'_def
thf(fact_282_four__block__diag__similar,axiom,
! [A1: mat_a,B1: mat_a,U1: mat_a,A22: mat_a,B2: mat_a,U2: mat_a] :
( ( spectr4825054497075562704quiv_a @ A1 @ B1 @ U1 )
=> ( ( spectr4825054497075562704quiv_a @ A22 @ B2 @ U2 )
=> ( ( ( dim_row_a @ A1 )
= ( dim_col_a @ A1 ) )
=> ( ( ( dim_row_a @ A22 )
= ( dim_col_a @ A22 ) )
=> ( similar_mat_wit_a @ ( four_block_mat_a @ A1 @ ( zero_mat_a @ ( dim_row_a @ A1 ) @ ( dim_col_a @ A22 ) ) @ ( zero_mat_a @ ( dim_row_a @ A22 ) @ ( dim_col_a @ A1 ) ) @ A22 ) @ ( four_block_mat_a @ B1 @ ( zero_mat_a @ ( dim_row_a @ B1 ) @ ( dim_col_a @ B2 ) ) @ ( zero_mat_a @ ( dim_row_a @ B2 ) @ ( dim_col_a @ B1 ) ) @ B2 ) @ ( four_block_mat_a @ U1 @ ( zero_mat_a @ ( dim_row_a @ U1 ) @ ( dim_col_a @ U2 ) ) @ ( zero_mat_a @ ( dim_row_a @ U2 ) @ ( dim_col_a @ U1 ) ) @ U2 ) @ ( schur_mat_adjoint_a @ ( four_block_mat_a @ U1 @ ( zero_mat_a @ ( dim_row_a @ U1 ) @ ( dim_col_a @ U2 ) ) @ ( zero_mat_a @ ( dim_row_a @ U2 ) @ ( dim_col_a @ U1 ) ) @ U2 ) ) ) ) ) ) ) ).
% four_block_diag_similar
thf(fact_283_four__block__diag__similar,axiom,
! [A1: mat_complex,B1: mat_complex,U1: mat_complex,A22: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr6340060708231679580omplex @ A1 @ B1 @ U1 )
=> ( ( spectr6340060708231679580omplex @ A22 @ B2 @ U2 )
=> ( ( ( dim_row_complex @ A1 )
= ( dim_col_complex @ A1 ) )
=> ( ( ( dim_row_complex @ A22 )
= ( dim_col_complex @ A22 ) )
=> ( simila5774310414453981135omplex @ ( four_b559179830521662709omplex @ A1 @ ( zero_mat_complex @ ( dim_row_complex @ A1 ) @ ( dim_col_complex @ A22 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ A22 ) @ ( dim_col_complex @ A1 ) ) @ A22 ) @ ( four_b559179830521662709omplex @ B1 @ ( zero_mat_complex @ ( dim_row_complex @ B1 ) @ ( dim_col_complex @ B2 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ B2 ) @ ( dim_col_complex @ B1 ) ) @ B2 ) @ ( four_b559179830521662709omplex @ U1 @ ( zero_mat_complex @ ( dim_row_complex @ U1 ) @ ( dim_col_complex @ U2 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ U2 ) @ ( dim_col_complex @ U1 ) ) @ U2 ) @ ( schur_5982229384592763574omplex @ ( four_b559179830521662709omplex @ U1 @ ( zero_mat_complex @ ( dim_row_complex @ U1 ) @ ( dim_col_complex @ U2 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ U2 ) @ ( dim_col_complex @ U1 ) ) @ U2 ) ) ) ) ) ) ) ).
% four_block_diag_similar
thf(fact_284_vector__space__over__itself_Ovector__space__assms_I4_J,axiom,
! [X: complex] :
( ( times_times_complex @ one_one_complex @ X )
= X ) ).
% vector_space_over_itself.vector_space_assms(4)
thf(fact_285_vector__space__over__itself_Ovector__space__assms_I4_J,axiom,
! [X: real] :
( ( times_times_real @ one_one_real @ X )
= X ) ).
% vector_space_over_itself.vector_space_assms(4)
thf(fact_286_comm__monoid__mult__class_Omult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_287_comm__monoid__mult__class_Omult__1,axiom,
! [A3: complex] :
( ( times_times_complex @ one_one_complex @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_288_comm__monoid__mult__class_Omult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_289_similar__mat__wit__sym,axiom,
! [A: mat_complex,B: mat_complex,P: mat_complex,Q2: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B @ P @ Q2 )
=> ( simila5774310414453981135omplex @ B @ A @ Q2 @ P ) ) ).
% similar_mat_wit_sym
thf(fact_290_similar__mat__wit__trans,axiom,
! [A: mat_complex,B: mat_complex,P: mat_complex,Q2: mat_complex,C: mat_complex,P3: mat_complex,Q3: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B @ P @ Q2 )
=> ( ( simila5774310414453981135omplex @ B @ C @ P3 @ Q3 )
=> ( simila5774310414453981135omplex @ A @ C @ ( times_8009071140041733218omplex @ P @ P3 ) @ ( times_8009071140041733218omplex @ Q3 @ Q2 ) ) ) ) ).
% similar_mat_wit_trans
thf(fact_291_similar__mat__witD_I3_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q2: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q2 )
=> ( A
= ( times_times_mat_a @ ( times_times_mat_a @ P @ B ) @ Q2 ) ) ) ) ).
% similar_mat_witD(3)
thf(fact_292_similar__mat__witD_I3_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P: mat_complex,Q2: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q2 )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ B ) @ Q2 ) ) ) ) ).
% similar_mat_witD(3)
thf(fact_293_unitarily__equivD_I2_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( simila5774310414453981135omplex @ A @ B @ U @ ( schur_5982229384592763574omplex @ U ) ) ) ).
% unitarily_equivD(2)
thf(fact_294_unitary__diagD_I1_J,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4894841263502123494diag_a @ A @ B @ U )
=> ( similar_mat_wit_a @ A @ B @ U @ ( schur_mat_adjoint_a @ U ) ) ) ).
% unitary_diagD(1)
thf(fact_295_unitary__diagD_I1_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( simila5774310414453981135omplex @ A @ B @ U @ ( schur_5982229384592763574omplex @ U ) ) ) ).
% unitary_diagD(1)
thf(fact_296_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_297_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_298_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_299_vector__space__over__itself_Oscale__left__commute,axiom,
! [A3: complex,B6: complex,X: complex] :
( ( times_times_complex @ A3 @ ( times_times_complex @ B6 @ X ) )
= ( times_times_complex @ B6 @ ( times_times_complex @ A3 @ X ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_300_vector__space__over__itself_Oscale__left__commute,axiom,
! [A3: real,B6: real,X: real] :
( ( times_times_real @ A3 @ ( times_times_real @ B6 @ X ) )
= ( times_times_real @ B6 @ ( times_times_real @ A3 @ X ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_301_vector__space__over__itself_Oscale__scale,axiom,
! [A3: complex,B6: complex,X: complex] :
( ( times_times_complex @ A3 @ ( times_times_complex @ B6 @ X ) )
= ( times_times_complex @ ( times_times_complex @ A3 @ B6 ) @ X ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_302_vector__space__over__itself_Oscale__scale,axiom,
! [A3: real,B6: real,X: real] :
( ( times_times_real @ A3 @ ( times_times_real @ B6 @ X ) )
= ( times_times_real @ ( times_times_real @ A3 @ B6 ) @ X ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_303_mult_Oleft__commute,axiom,
! [B6: nat,A3: nat,C2: nat] :
( ( times_times_nat @ B6 @ ( times_times_nat @ A3 @ C2 ) )
= ( times_times_nat @ A3 @ ( times_times_nat @ B6 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_304_mult_Oleft__commute,axiom,
! [B6: complex,A3: complex,C2: complex] :
( ( times_times_complex @ B6 @ ( times_times_complex @ A3 @ C2 ) )
= ( times_times_complex @ A3 @ ( times_times_complex @ B6 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_305_mult_Oleft__commute,axiom,
! [B6: real,A3: real,C2: real] :
( ( times_times_real @ B6 @ ( times_times_real @ A3 @ C2 ) )
= ( times_times_real @ A3 @ ( times_times_real @ B6 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_306_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B7: nat] : ( times_times_nat @ B7 @ A5 ) ) ) ).
% mult.commute
thf(fact_307_mult_Ocommute,axiom,
( times_times_complex
= ( ^ [A5: complex,B7: complex] : ( times_times_complex @ B7 @ A5 ) ) ) ).
% mult.commute
thf(fact_308_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A5: real,B7: real] : ( times_times_real @ B7 @ A5 ) ) ) ).
% mult.commute
thf(fact_309_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B6 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B6 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_310_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( times_times_complex @ ( times_times_complex @ A3 @ B6 ) @ C2 )
= ( times_times_complex @ A3 @ ( times_times_complex @ B6 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_311_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: real,B6: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B6 ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B6 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_312_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_313_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_314_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_315_unitarily__equiv__def,axiom,
( spectr6340060708231679580omplex
= ( ^ [A2: mat_complex,B5: mat_complex,U3: mat_complex] :
( ( comple6660659447773130958omplex @ U3 )
& ( simila5774310414453981135omplex @ A2 @ B5 @ U3 @ ( schur_5982229384592763574omplex @ U3 ) ) ) ) ) ).
% unitarily_equiv_def
thf(fact_316_unitarily__equivI,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B @ U @ ( schur_5982229384592763574omplex @ U ) )
=> ( ( comple6660659447773130958omplex @ U )
=> ( spectr6340060708231679580omplex @ A @ B @ U ) ) ) ).
% unitarily_equivI
thf(fact_317_unitary__diagI,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( similar_mat_wit_a @ A @ B @ U @ ( schur_mat_adjoint_a @ U ) )
=> ( ( diagonal_mat_a @ B )
=> ( ( complex_unitary_a @ U )
=> ( spectr4894841263502123494diag_a @ A @ B @ U ) ) ) ) ).
% unitary_diagI
thf(fact_318_unitary__diagI,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B @ U @ ( schur_5982229384592763574omplex @ U ) )
=> ( ( diagonal_mat_complex @ B )
=> ( ( comple6660659447773130958omplex @ U )
=> ( spectr532731689276696518omplex @ A @ B @ U ) ) ) ) ).
% unitary_diagI
thf(fact_319_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: complex,A3: complex,B6: complex] :
( ( X != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ B6 @ X ) )
=> ( A3 = B6 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_320_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A3: real,B6: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ B6 @ X ) )
=> ( A3 = B6 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_321_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: complex,X: complex,B6: complex] :
( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ B6 @ X ) )
= ( ( A3 = B6 )
| ( X = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_322_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: real,X: real,B6: real] :
( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ B6 @ X ) )
= ( ( A3 = B6 )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_323_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: complex,X: complex,Y: complex] :
( ( A3 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ A3 @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_324_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: real,X: real,Y: real] :
( ( A3 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ A3 @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_325_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: complex,X: complex,Y: complex] :
( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ A3 @ Y ) )
= ( ( X = Y )
| ( A3 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_326_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: real,X: real,Y: real] :
( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ A3 @ Y ) )
= ( ( X = Y )
| ( A3 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_327_vector__space__over__itself_Oscale__zero__right,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ zero_zero_complex )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_328_vector__space__over__itself_Oscale__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_329_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: complex] :
( ( times_times_complex @ zero_zero_complex @ X )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_330_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_331_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: complex,X: complex] :
( ( ( times_times_complex @ A3 @ X )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( X = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_332_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: real,X: real] :
( ( ( times_times_real @ A3 @ X )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_333_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_334_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_335_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_336_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_337_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_338_mult_Ocomm__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.comm_neutral
thf(fact_339_mult_Ocomm__neutral,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% mult.comm_neutral
thf(fact_340_mult_Ocomm__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.comm_neutral
thf(fact_341_similar__mat__wit__dim__row,axiom,
! [A: mat_a,B: mat_a,Q2: mat_a,R: mat_a] :
( ( similar_mat_wit_a @ A @ B @ Q2 @ R )
=> ( ( dim_row_a @ B )
= ( dim_row_a @ A ) ) ) ).
% similar_mat_wit_dim_row
thf(fact_342_similar__mat__wit__dim__row,axiom,
! [A: mat_complex,B: mat_complex,Q2: mat_complex,R: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B @ Q2 @ R )
=> ( ( dim_row_complex @ B )
= ( dim_row_complex @ A ) ) ) ).
% similar_mat_wit_dim_row
thf(fact_343_class__field_Ozero__not__one,axiom,
zero_zero_complex != one_one_complex ).
% class_field.zero_not_one
thf(fact_344_class__field_Ozero__not__one,axiom,
zero_zero_real != one_one_real ).
% class_field.zero_not_one
thf(fact_345_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_346_mult__delta__right,axiom,
! [B6: $o,X: nat,Y: nat] :
( ( B6
=> ( ( times_times_nat @ X @ ( if_nat @ B6 @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B6
=> ( ( times_times_nat @ X @ ( if_nat @ B6 @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_347_mult__delta__right,axiom,
! [B6: $o,X: complex,Y: complex] :
( ( B6
=> ( ( times_times_complex @ X @ ( if_complex @ B6 @ Y @ zero_zero_complex ) )
= ( times_times_complex @ X @ Y ) ) )
& ( ~ B6
=> ( ( times_times_complex @ X @ ( if_complex @ B6 @ Y @ zero_zero_complex ) )
= zero_zero_complex ) ) ) ).
% mult_delta_right
thf(fact_348_mult__delta__right,axiom,
! [B6: $o,X: real,Y: real] :
( ( B6
=> ( ( times_times_real @ X @ ( if_real @ B6 @ Y @ zero_zero_real ) )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B6
=> ( ( times_times_real @ X @ ( if_real @ B6 @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_349_mult__delta__left,axiom,
! [B6: $o,X: nat,Y: nat] :
( ( B6
=> ( ( times_times_nat @ ( if_nat @ B6 @ X @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B6
=> ( ( times_times_nat @ ( if_nat @ B6 @ X @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_350_mult__delta__left,axiom,
! [B6: $o,X: complex,Y: complex] :
( ( B6
=> ( ( times_times_complex @ ( if_complex @ B6 @ X @ zero_zero_complex ) @ Y )
= ( times_times_complex @ X @ Y ) ) )
& ( ~ B6
=> ( ( times_times_complex @ ( if_complex @ B6 @ X @ zero_zero_complex ) @ Y )
= zero_zero_complex ) ) ) ).
% mult_delta_left
thf(fact_351_mult__delta__left,axiom,
! [B6: $o,X: real,Y: real] :
( ( B6
=> ( ( times_times_real @ ( if_real @ B6 @ X @ zero_zero_real ) @ Y )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B6
=> ( ( times_times_real @ ( if_real @ B6 @ X @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_352_mult__hom_Ohom__zero,axiom,
! [C2: nat] :
( ( times_times_nat @ C2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_353_mult__hom_Ohom__zero,axiom,
! [C2: complex] :
( ( times_times_complex @ C2 @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_hom.hom_zero
thf(fact_354_mult__hom_Ohom__zero,axiom,
! [C2: real] :
( ( times_times_real @ C2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_355_class__cring_Ofactors__equal,axiom,
! [A3: complex,B6: complex,C2: complex,D: complex] :
( ( A3 = B6 )
=> ( ( C2 = D )
=> ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B6 @ D ) ) ) ) ).
% class_cring.factors_equal
thf(fact_356_class__cring_Ofactors__equal,axiom,
! [A3: real,B6: real,C2: real,D: real] :
( ( A3 = B6 )
=> ( ( C2 = D )
=> ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B6 @ D ) ) ) ) ).
% class_cring.factors_equal
thf(fact_357_mult__four__block__diag,axiom,
! [A1: mat_a,Nr1: nat,N1: nat,D12: mat_a,Nr2: nat,N22: nat,A22: mat_a,Nc1: nat,D22: mat_a,Nc2: nat] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ Nr1 @ N1 ) )
=> ( ( member_mat_a @ D12 @ ( carrier_mat_a @ Nr2 @ N22 ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ N1 @ Nc1 ) )
=> ( ( member_mat_a @ D22 @ ( carrier_mat_a @ N22 @ Nc2 ) )
=> ( ( times_times_mat_a @ ( four_block_mat_a @ A1 @ ( zero_mat_a @ ( dim_row_a @ A1 ) @ ( dim_col_a @ D12 ) ) @ ( zero_mat_a @ ( dim_row_a @ D12 ) @ ( dim_col_a @ A1 ) ) @ D12 ) @ ( four_block_mat_a @ A22 @ ( zero_mat_a @ ( dim_row_a @ A22 ) @ ( dim_col_a @ D22 ) ) @ ( zero_mat_a @ ( dim_row_a @ D22 ) @ ( dim_col_a @ A22 ) ) @ D22 ) )
= ( four_block_mat_a @ ( times_times_mat_a @ A1 @ A22 ) @ ( zero_mat_a @ ( dim_row_a @ ( times_times_mat_a @ A1 @ A22 ) ) @ ( dim_col_a @ ( times_times_mat_a @ D12 @ D22 ) ) ) @ ( zero_mat_a @ ( dim_row_a @ ( times_times_mat_a @ D12 @ D22 ) ) @ ( dim_col_a @ ( times_times_mat_a @ A1 @ A22 ) ) ) @ ( times_times_mat_a @ D12 @ D22 ) ) ) ) ) ) ) ).
% mult_four_block_diag
thf(fact_358_mult__four__block__diag,axiom,
! [A1: mat_complex,Nr1: nat,N1: nat,D12: mat_complex,Nr2: nat,N22: nat,A22: mat_complex,Nc1: nat,D22: mat_complex,Nc2: nat] :
( ( member_mat_complex @ A1 @ ( carrier_mat_complex @ Nr1 @ N1 ) )
=> ( ( member_mat_complex @ D12 @ ( carrier_mat_complex @ Nr2 @ N22 ) )
=> ( ( member_mat_complex @ A22 @ ( carrier_mat_complex @ N1 @ Nc1 ) )
=> ( ( member_mat_complex @ D22 @ ( carrier_mat_complex @ N22 @ Nc2 ) )
=> ( ( times_8009071140041733218omplex @ ( four_b559179830521662709omplex @ A1 @ ( zero_mat_complex @ ( dim_row_complex @ A1 ) @ ( dim_col_complex @ D12 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ D12 ) @ ( dim_col_complex @ A1 ) ) @ D12 ) @ ( four_b559179830521662709omplex @ A22 @ ( zero_mat_complex @ ( dim_row_complex @ A22 ) @ ( dim_col_complex @ D22 ) ) @ ( zero_mat_complex @ ( dim_row_complex @ D22 ) @ ( dim_col_complex @ A22 ) ) @ D22 ) )
= ( four_b559179830521662709omplex @ ( times_8009071140041733218omplex @ A1 @ A22 ) @ ( zero_mat_complex @ ( dim_row_complex @ ( times_8009071140041733218omplex @ A1 @ A22 ) ) @ ( dim_col_complex @ ( times_8009071140041733218omplex @ D12 @ D22 ) ) ) @ ( zero_mat_complex @ ( dim_row_complex @ ( times_8009071140041733218omplex @ D12 @ D22 ) ) @ ( dim_col_complex @ ( times_8009071140041733218omplex @ A1 @ A22 ) ) ) @ ( times_8009071140041733218omplex @ D12 @ D22 ) ) ) ) ) ) ) ).
% mult_four_block_diag
thf(fact_359_four__block__diag__zero_H,axiom,
! [B: mat_complex,A: mat_complex] :
( ( member_mat_complex @ B @ ( carrier_mat_complex @ zero_zero_nat @ zero_zero_nat ) )
=> ( ( four_b559179830521662709omplex @ B @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ A ) ) @ ( zero_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ B ) ) @ A )
= A ) ) ).
% four_block_diag_zero'
thf(fact_360_four__block__diag__zero_H,axiom,
! [B: mat_a,A: mat_a] :
( ( member_mat_a @ B @ ( carrier_mat_a @ zero_zero_nat @ zero_zero_nat ) )
=> ( ( four_block_mat_a @ B @ ( zero_mat_a @ ( dim_row_a @ B ) @ ( dim_col_a @ A ) ) @ ( zero_mat_a @ ( dim_row_a @ A ) @ ( dim_col_a @ B ) ) @ A )
= A ) ) ).
% four_block_diag_zero'
thf(fact_361_four__block__diag__zero,axiom,
! [B: mat_complex,A: mat_complex] :
( ( member_mat_complex @ B @ ( carrier_mat_complex @ zero_zero_nat @ zero_zero_nat ) )
=> ( ( four_b559179830521662709omplex @ A @ ( zero_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ B ) ) @ ( zero_mat_complex @ ( dim_row_complex @ B ) @ ( dim_col_complex @ A ) ) @ B )
= A ) ) ).
% four_block_diag_zero
thf(fact_362_four__block__diag__zero,axiom,
! [B: mat_a,A: mat_a] :
( ( member_mat_a @ B @ ( carrier_mat_a @ zero_zero_nat @ zero_zero_nat ) )
=> ( ( four_block_mat_a @ A @ ( zero_mat_a @ ( dim_row_a @ A ) @ ( dim_col_a @ B ) ) @ ( zero_mat_a @ ( dim_row_a @ B ) @ ( dim_col_a @ A ) ) @ B )
= A ) ) ).
% four_block_diag_zero
thf(fact_363_hermitian__decomp__sim,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U )
=> ( simila5774310414453981135omplex @ A @ B @ U @ ( schur_5982229384592763574omplex @ U ) ) ) ).
% hermitian_decomp_sim
thf(fact_364_conjugate__eq__unitarily__equiv,axiom,
! [A: mat_complex,N: nat,V: mat_complex,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ V @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( ( comple6660659447773130958omplex @ V )
=> ( ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ V @ B ) @ ( schur_5982229384592763574omplex @ V ) )
= B )
=> ( spectr6340060708231679580omplex @ A @ B @ ( times_8009071140041733218omplex @ U @ V ) ) ) ) ) ) ) ).
% conjugate_eq_unitarily_equiv
thf(fact_365_similar__mat__wit__four__block,axiom,
! [A1: mat_a,B1: mat_a,P1: mat_a,Q1: mat_a,A22: mat_a,B2: mat_a,P22: mat_a,Q22: mat_a,URA: mat_a,UR: mat_a,LLA: mat_a,LL: mat_a,N: nat,M: nat] :
( ( similar_mat_wit_a @ A1 @ B1 @ P1 @ Q1 )
=> ( ( similar_mat_wit_a @ A22 @ B2 @ P22 @ Q22 )
=> ( ( URA
= ( times_times_mat_a @ ( times_times_mat_a @ P1 @ UR ) @ Q22 ) )
=> ( ( LLA
= ( times_times_mat_a @ ( times_times_mat_a @ P22 @ LL ) @ Q1 ) )
=> ( ( member_mat_a @ A1 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ M @ M ) )
=> ( ( member_mat_a @ LL @ ( carrier_mat_a @ M @ N ) )
=> ( ( member_mat_a @ UR @ ( carrier_mat_a @ N @ M ) )
=> ( similar_mat_wit_a @ ( four_block_mat_a @ A1 @ URA @ LLA @ A22 ) @ ( four_block_mat_a @ B1 @ UR @ LL @ B2 ) @ ( four_block_mat_a @ P1 @ ( zero_mat_a @ N @ M ) @ ( zero_mat_a @ M @ N ) @ P22 ) @ ( four_block_mat_a @ Q1 @ ( zero_mat_a @ N @ M ) @ ( zero_mat_a @ M @ N ) @ Q22 ) ) ) ) ) ) ) ) ) ) ).
% similar_mat_wit_four_block
thf(fact_366_similar__mat__wit__four__block,axiom,
! [A1: mat_complex,B1: mat_complex,P1: mat_complex,Q1: mat_complex,A22: mat_complex,B2: mat_complex,P22: mat_complex,Q22: mat_complex,URA: mat_complex,UR: mat_complex,LLA: mat_complex,LL: mat_complex,N: nat,M: nat] :
( ( simila5774310414453981135omplex @ A1 @ B1 @ P1 @ Q1 )
=> ( ( simila5774310414453981135omplex @ A22 @ B2 @ P22 @ Q22 )
=> ( ( URA
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P1 @ UR ) @ Q22 ) )
=> ( ( LLA
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P22 @ LL ) @ Q1 ) )
=> ( ( member_mat_complex @ A1 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ A22 @ ( carrier_mat_complex @ M @ M ) )
=> ( ( member_mat_complex @ LL @ ( carrier_mat_complex @ M @ N ) )
=> ( ( member_mat_complex @ UR @ ( carrier_mat_complex @ N @ M ) )
=> ( simila5774310414453981135omplex @ ( four_b559179830521662709omplex @ A1 @ URA @ LLA @ A22 ) @ ( four_b559179830521662709omplex @ B1 @ UR @ LL @ B2 ) @ ( four_b559179830521662709omplex @ P1 @ ( zero_mat_complex @ N @ M ) @ ( zero_mat_complex @ M @ N ) @ P22 ) @ ( four_b559179830521662709omplex @ Q1 @ ( zero_mat_complex @ N @ M ) @ ( zero_mat_complex @ M @ N ) @ Q22 ) ) ) ) ) ) ) ) ) ) ).
% similar_mat_wit_four_block
thf(fact_367_smult__zero,axiom,
! [A: mat_a] :
( ( smult_mat_a @ zero_zero_a @ A )
= ( zero_mat_a @ ( dim_row_a @ A ) @ ( dim_col_a @ A ) ) ) ).
% smult_zero
thf(fact_368_smult__zero,axiom,
! [A: mat_complex] :
( ( smult_mat_complex @ zero_zero_complex @ A )
= ( zero_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ A ) ) ) ).
% smult_zero
thf(fact_369_smult__zero,axiom,
! [A: mat_real] :
( ( smult_mat_real @ zero_zero_real @ A )
= ( zero_mat_real @ ( dim_row_real @ A ) @ ( dim_col_real @ A ) ) ) ).
% smult_zero
thf(fact_370_mult__smult__distrib,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat,K: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A @ ( smult_mat_complex @ K @ B ) )
= ( smult_mat_complex @ K @ ( times_8009071140041733218omplex @ A @ B ) ) ) ) ) ).
% mult_smult_distrib
thf(fact_371_mult__smult__assoc__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat,K: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( smult_mat_complex @ K @ A ) @ B )
= ( smult_mat_complex @ K @ ( times_8009071140041733218omplex @ A @ B ) ) ) ) ) ).
% mult_smult_assoc_mat
thf(fact_372_smult__four__block__mat,axiom,
! [A: mat_complex,Nr1: nat,Nc1: nat,B: mat_complex,Nc2: nat,C: mat_complex,Nr2: nat,D3: mat_complex,A3: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr1 @ Nc1 ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr1 @ Nc2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ Nr2 @ Nc1 ) )
=> ( ( member_mat_complex @ D3 @ ( carrier_mat_complex @ Nr2 @ Nc2 ) )
=> ( ( smult_mat_complex @ A3 @ ( four_b559179830521662709omplex @ A @ B @ C @ D3 ) )
= ( four_b559179830521662709omplex @ ( smult_mat_complex @ A3 @ A ) @ ( smult_mat_complex @ A3 @ B ) @ ( smult_mat_complex @ A3 @ C ) @ ( smult_mat_complex @ A3 @ D3 ) ) ) ) ) ) ) ).
% smult_four_block_mat
thf(fact_373_smult__four__block__mat,axiom,
! [A: mat_a,Nr1: nat,Nc1: nat,B: mat_a,Nc2: nat,C: mat_a,Nr2: nat,D3: mat_a,A3: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D3 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( smult_mat_a @ A3 @ ( four_block_mat_a @ A @ B @ C @ D3 ) )
= ( four_block_mat_a @ ( smult_mat_a @ A3 @ A ) @ ( smult_mat_a @ A3 @ B ) @ ( smult_mat_a @ A3 @ C ) @ ( smult_mat_a @ A3 @ D3 ) ) ) ) ) ) ) ).
% smult_four_block_mat
thf(fact_374_unitarily__equiv__smult,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex,X: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( spectr6340060708231679580omplex @ ( smult_mat_complex @ X @ A ) @ ( smult_mat_complex @ X @ B ) @ U ) ) ) ).
% unitarily_equiv_smult
thf(fact_375_smult__carrier__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat,K: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( smult_mat_complex @ K @ A ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% smult_carrier_mat
thf(fact_376_mat__conj__smult,axiom,
! [A: mat_complex,N: nat,U: mat_complex,B: mat_complex,X: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U @ B ) @ ( schur_5982229384592763574omplex @ U ) ) )
=> ( ( smult_mat_complex @ X @ A )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U @ ( smult_mat_complex @ X @ B ) ) @ ( schur_5982229384592763574omplex @ U ) ) ) ) ) ) ) ).
% mat_conj_smult
thf(fact_377_carrier__matD_I1_J,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( dim_row_a @ A )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_378_carrier__matD_I1_J,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( dim_row_complex @ A )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_379_carrier__matD_I2_J,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( dim_col_a @ A )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_380_carrier__matD_I2_J,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( dim_col_complex @ A )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_381_assoc__mult__mat,axiom,
! [A: mat_complex,N_1: nat,N_2: nat,B: mat_complex,N_3: nat,C: mat_complex,N_4: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N_1 @ N_2 ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N_2 @ N_3 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N_3 @ N_4 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ C )
= ( times_8009071140041733218omplex @ A @ ( times_8009071140041733218omplex @ B @ C ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_382_mult__carrier__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A @ B ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_383_index__smult__mat_I2_J,axiom,
! [A3: a,A: mat_a] :
( ( dim_row_a @ ( smult_mat_a @ A3 @ A ) )
= ( dim_row_a @ A ) ) ).
% index_smult_mat(2)
thf(fact_384_index__smult__mat_I2_J,axiom,
! [A3: complex,A: mat_complex] :
( ( dim_row_complex @ ( smult_mat_complex @ A3 @ A ) )
= ( dim_row_complex @ A ) ) ).
% index_smult_mat(2)
thf(fact_385_index__smult__mat_I3_J,axiom,
! [A3: a,A: mat_a] :
( ( dim_col_a @ ( smult_mat_a @ A3 @ A ) )
= ( dim_col_a @ A ) ) ).
% index_smult_mat(3)
thf(fact_386_index__smult__mat_I3_J,axiom,
! [A3: complex,A: mat_complex] :
( ( dim_col_complex @ ( smult_mat_complex @ A3 @ A ) )
= ( dim_col_complex @ A ) ) ).
% index_smult_mat(3)
thf(fact_387_zero__carrier__mat,axiom,
! [Nr: nat,Nc: nat] : ( member_mat_a @ ( zero_mat_a @ Nr @ Nc ) @ ( carrier_mat_a @ Nr @ Nc ) ) ).
% zero_carrier_mat
thf(fact_388_zero__carrier__mat,axiom,
! [Nr: nat,Nc: nat] : ( member_mat_complex @ ( zero_mat_complex @ Nr @ Nc ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ).
% zero_carrier_mat
thf(fact_389_adjoint__dim,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( member_mat_complex @ ( schur_5982229384592763574omplex @ A ) @ ( carrier_mat_complex @ N @ N ) ) ) ).
% adjoint_dim
thf(fact_390_adjoint__dim_H,axiom,
! [A: mat_complex,N: nat,M: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( member_mat_complex @ ( schur_5982229384592763574omplex @ A ) @ ( carrier_mat_complex @ M @ N ) ) ) ).
% adjoint_dim'
thf(fact_391_smult__zero__mat,axiom,
! [K: a,Nr: nat,Nc: nat] :
( ( smult_mat_a @ K @ ( zero_mat_a @ Nr @ Nc ) )
= ( zero_mat_a @ Nr @ Nc ) ) ).
% smult_zero_mat
thf(fact_392_smult__zero__mat,axiom,
! [K: complex,Nr: nat,Nc: nat] :
( ( smult_mat_complex @ K @ ( zero_mat_complex @ Nr @ Nc ) )
= ( zero_mat_complex @ Nr @ Nc ) ) ).
% smult_zero_mat
thf(fact_393_transpose__carrier__mat,axiom,
! [A: mat_complex,Nc: nat,Nr: nat] :
( ( member_mat_complex @ ( transp3074176993011536131omplex @ A ) @ ( carrier_mat_complex @ Nc @ Nr ) )
= ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_394_similar__mat__witD2_I7_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P: mat_complex,Q2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q2 )
=> ( member_mat_complex @ Q2 @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD2(7)
thf(fact_395_similar__mat__witD2_I6_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P: mat_complex,Q2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q2 )
=> ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD2(6)
thf(fact_396_similar__mat__witD2_I5_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P: mat_complex,Q2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q2 )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD2(5)
thf(fact_397_similar__mat__witD2_I4_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P: mat_complex,Q2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q2 )
=> ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD2(4)
thf(fact_398_diagonal__mat__smult,axiom,
! [A: mat_complex,X: complex] :
( ( diagonal_mat_complex @ A )
=> ( diagonal_mat_complex @ ( smult_mat_complex @ X @ A ) ) ) ).
% diagonal_mat_smult
thf(fact_399_unitarily__equiv__carrier_I1_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% unitarily_equiv_carrier(1)
thf(fact_400_unitarily__equiv__carrier_I2_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% unitarily_equiv_carrier(2)
thf(fact_401_similar__mat__wit__smult,axiom,
! [A: mat_complex,B: mat_complex,P: mat_complex,Q2: mat_complex,K: complex] :
( ( simila5774310414453981135omplex @ A @ B @ P @ Q2 )
=> ( simila5774310414453981135omplex @ ( smult_mat_complex @ K @ A ) @ ( smult_mat_complex @ K @ B ) @ P @ Q2 ) ) ).
% similar_mat_wit_smult
thf(fact_402_hermitian__decomp__dim__carrier,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( projec1926941670171670524comp_a @ A @ B @ U )
=> ( member_mat_a @ B @ ( carrier_mat_a @ ( dim_row_a @ A ) @ ( dim_col_a @ A ) ) ) ) ).
% hermitian_decomp_dim_carrier
thf(fact_403_hermitian__decomp__dim__carrier,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ A ) ) ) ) ).
% hermitian_decomp_dim_carrier
thf(fact_404_swap__cols__rows__block__dims_I3_J,axiom,
! [A: mat_complex,N: nat,I: nat,J: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( member_mat_complex @ ( jordan8990321789093393430omplex @ I @ J @ A ) @ ( carrier_mat_complex @ N @ N ) ) ) ).
% swap_cols_rows_block_dims(3)
thf(fact_405_unitary__diag__carrier_I1_J,axiom,
! [A: mat_a,N: nat,B: mat_a,U: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( spectr4894841263502123494diag_a @ A @ B @ U )
=> ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% unitary_diag_carrier(1)
thf(fact_406_unitary__diag__carrier_I1_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% unitary_diag_carrier(1)
thf(fact_407_unitary__diag__carrier_I2_J,axiom,
! [A: mat_a,N: nat,B: mat_a,U: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( spectr4894841263502123494diag_a @ A @ B @ U )
=> ( member_mat_a @ U @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% unitary_diag_carrier(2)
thf(fact_408_unitary__diag__carrier_I2_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% unitary_diag_carrier(2)
thf(fact_409_carrier__matI,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( ( dim_row_a @ A )
= Nr )
=> ( ( ( dim_col_a @ A )
= Nc )
=> ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_410_carrier__matI,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( ( dim_row_complex @ A )
= Nr )
=> ( ( ( dim_col_complex @ A )
= Nc )
=> ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_411_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_412_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_413_left__mult__zero__mat,axiom,
! [A: mat_a,N: nat,Nc: nat,Nr: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( zero_mat_a @ Nr @ N ) @ A )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% left_mult_zero_mat
thf(fact_414_left__mult__zero__mat,axiom,
! [A: mat_complex,N: nat,Nc: nat,Nr: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( zero_mat_complex @ Nr @ N ) @ A )
= ( zero_mat_complex @ Nr @ Nc ) ) ) ).
% left_mult_zero_mat
thf(fact_415_right__mult__zero__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( times_times_mat_a @ A @ ( zero_mat_a @ N @ Nc ) )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% right_mult_zero_mat
thf(fact_416_right__mult__zero__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( times_8009071140041733218omplex @ A @ ( zero_mat_complex @ N @ Nc ) )
= ( zero_mat_complex @ Nr @ Nc ) ) ) ).
% right_mult_zero_mat
thf(fact_417_unitary__zero,axiom,
! [A: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ zero_zero_nat @ zero_zero_nat ) )
=> ( comple6660659447773130958omplex @ A ) ) ).
% unitary_zero
thf(fact_418_adjoint__mult,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,L: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ M @ L ) )
=> ( ( schur_5982229384592763574omplex @ ( times_8009071140041733218omplex @ A @ B ) )
= ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ B ) @ ( schur_5982229384592763574omplex @ A ) ) ) ) ) ).
% adjoint_mult
thf(fact_419_unitary__times__unitary,axiom,
! [P: mat_complex,N: nat,Q2: mat_complex] :
( ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ Q2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ P )
=> ( ( comple6660659447773130958omplex @ Q2 )
=> ( comple6660659447773130958omplex @ ( times_8009071140041733218omplex @ P @ Q2 ) ) ) ) ) ) ).
% unitary_times_unitary
thf(fact_420_transpose__mult,axiom,
! [A: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( transp3074176993011536131omplex @ ( times_8009071140041733218omplex @ A @ B ) )
= ( times_8009071140041733218omplex @ ( transp3074176993011536131omplex @ B ) @ ( transp3074176993011536131omplex @ A ) ) ) ) ) ).
% transpose_mult
thf(fact_421_diagonal__mat__times__diag,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( diagonal_mat_complex @ A )
=> ( ( diagonal_mat_complex @ B )
=> ( diagonal_mat_complex @ ( times_8009071140041733218omplex @ A @ B ) ) ) ) ) ) ).
% diagonal_mat_times_diag
thf(fact_422_diagonal__mat__sq__diag,axiom,
! [B: mat_complex,N: nat] :
( ( diagonal_mat_complex @ B )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( diagonal_mat_complex @ ( times_8009071140041733218omplex @ B @ B ) ) ) ) ).
% diagonal_mat_sq_diag
thf(fact_423_diagonal__mat__commute,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( diagonal_mat_complex @ A )
=> ( ( diagonal_mat_complex @ B )
=> ( ( times_8009071140041733218omplex @ A @ B )
= ( times_8009071140041733218omplex @ B @ A ) ) ) ) ) ) ).
% diagonal_mat_commute
thf(fact_424_similar__mat__witD_I7_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q2: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q2 )
=> ( member_mat_a @ Q2 @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD(7)
thf(fact_425_similar__mat__witD_I7_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P: mat_complex,Q2: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q2 )
=> ( member_mat_complex @ Q2 @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD(7)
thf(fact_426_similar__mat__witD_I6_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q2: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q2 )
=> ( member_mat_a @ P @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD(6)
thf(fact_427_similar__mat__witD_I6_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P: mat_complex,Q2: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q2 )
=> ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD(6)
thf(fact_428_similar__mat__witD_I5_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q2: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q2 )
=> ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD(5)
thf(fact_429_similar__mat__witD_I5_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P: mat_complex,Q2: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q2 )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD(5)
thf(fact_430_similar__mat__witD_I4_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q2: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q2 )
=> ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD(4)
thf(fact_431_similar__mat__witD_I4_J,axiom,
! [N: nat,A: mat_complex,B: mat_complex,P: mat_complex,Q2: mat_complex] :
( ( N
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q2 )
=> ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) ) ) ) ).
% similar_mat_witD(4)
thf(fact_432_four__block__mat__adjoint,axiom,
! [A: mat_a,Nr1: nat,Nc1: nat,B: mat_a,Nc2: nat,C: mat_a,Nr2: nat,D3: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D3 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( schur_mat_adjoint_a @ ( four_block_mat_a @ A @ B @ C @ D3 ) )
= ( four_block_mat_a @ ( schur_mat_adjoint_a @ A ) @ ( schur_mat_adjoint_a @ C ) @ ( schur_mat_adjoint_a @ B ) @ ( schur_mat_adjoint_a @ D3 ) ) ) ) ) ) ) ).
% four_block_mat_adjoint
thf(fact_433_four__block__mat__adjoint,axiom,
! [A: mat_complex,Nr1: nat,Nc1: nat,B: mat_complex,Nc2: nat,C: mat_complex,Nr2: nat,D3: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr1 @ Nc1 ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr1 @ Nc2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ Nr2 @ Nc1 ) )
=> ( ( member_mat_complex @ D3 @ ( carrier_mat_complex @ Nr2 @ Nc2 ) )
=> ( ( schur_5982229384592763574omplex @ ( four_b559179830521662709omplex @ A @ B @ C @ D3 ) )
= ( four_b559179830521662709omplex @ ( schur_5982229384592763574omplex @ A ) @ ( schur_5982229384592763574omplex @ C ) @ ( schur_5982229384592763574omplex @ B ) @ ( schur_5982229384592763574omplex @ D3 ) ) ) ) ) ) ) ).
% four_block_mat_adjoint
thf(fact_434_similar__mat__witD2_I3_J,axiom,
! [A: mat_complex,N: nat,M: nat,B: mat_complex,P: mat_complex,Q2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B @ P @ Q2 )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ B ) @ Q2 ) ) ) ) ).
% similar_mat_witD2(3)
thf(fact_435_unitarily__equiv__carrier_H_I3_J,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( member_mat_a @ U @ ( carrier_mat_a @ ( dim_row_a @ A ) @ ( dim_row_a @ A ) ) ) ) ).
% unitarily_equiv_carrier'(3)
thf(fact_436_unitarily__equiv__carrier_H_I3_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( member_mat_complex @ U @ ( carrier_mat_complex @ ( dim_row_complex @ A ) @ ( dim_row_complex @ A ) ) ) ) ).
% unitarily_equiv_carrier'(3)
thf(fact_437_unitarily__equiv__carrier_H_I2_J,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( member_mat_a @ B @ ( carrier_mat_a @ ( dim_row_a @ A ) @ ( dim_row_a @ A ) ) ) ) ).
% unitarily_equiv_carrier'(2)
thf(fact_438_unitarily__equiv__carrier_H_I2_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( member_mat_complex @ B @ ( carrier_mat_complex @ ( dim_row_complex @ A ) @ ( dim_row_complex @ A ) ) ) ) ).
% unitarily_equiv_carrier'(2)
thf(fact_439_unitarily__equiv__carrier_H_I1_J,axiom,
! [A: mat_a,B: mat_a,U: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U )
=> ( member_mat_a @ A @ ( carrier_mat_a @ ( dim_row_a @ A ) @ ( dim_row_a @ A ) ) ) ) ).
% unitarily_equiv_carrier'(1)
thf(fact_440_unitarily__equiv__carrier_H_I1_J,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( member_mat_complex @ A @ ( carrier_mat_complex @ ( dim_row_complex @ A ) @ ( dim_row_complex @ A ) ) ) ) ).
% unitarily_equiv_carrier'(1)
thf(fact_441_unitarily__equiv__square,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( spectr6340060708231679580omplex @ ( times_8009071140041733218omplex @ A @ A ) @ ( times_8009071140041733218omplex @ B @ B ) @ U ) ) ) ).
% unitarily_equiv_square
thf(fact_442_unitary__adjoint,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ A )
=> ( comple6660659447773130958omplex @ ( schur_5982229384592763574omplex @ A ) ) ) ) ).
% unitary_adjoint
thf(fact_443_transpose__four__block__mat,axiom,
! [A: mat_complex,Nr1: nat,Nc1: nat,B: mat_complex,Nc2: nat,C: mat_complex,Nr2: nat,D3: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr1 @ Nc1 ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr1 @ Nc2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ Nr2 @ Nc1 ) )
=> ( ( member_mat_complex @ D3 @ ( carrier_mat_complex @ Nr2 @ Nc2 ) )
=> ( ( transp3074176993011536131omplex @ ( four_b559179830521662709omplex @ A @ B @ C @ D3 ) )
= ( four_b559179830521662709omplex @ ( transp3074176993011536131omplex @ A ) @ ( transp3074176993011536131omplex @ C ) @ ( transp3074176993011536131omplex @ B ) @ ( transp3074176993011536131omplex @ D3 ) ) ) ) ) ) ) ).
% transpose_four_block_mat
thf(fact_444_transpose__four__block__mat,axiom,
! [A: mat_a,Nr1: nat,Nc1: nat,B: mat_a,Nc2: nat,C: mat_a,Nr2: nat,D3: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D3 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( transpose_mat_a @ ( four_block_mat_a @ A @ B @ C @ D3 ) )
= ( four_block_mat_a @ ( transpose_mat_a @ A ) @ ( transpose_mat_a @ C ) @ ( transpose_mat_a @ B ) @ ( transpose_mat_a @ D3 ) ) ) ) ) ) ) ).
% transpose_four_block_mat
thf(fact_445_hermitian__decomp__unitary,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U )
=> ( comple6660659447773130958omplex @ U ) ) ).
% hermitian_decomp_unitary
thf(fact_446_hermitian__decomp__diag__mat,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U )
=> ( diagonal_mat_complex @ B ) ) ).
% hermitian_decomp_diag_mat
thf(fact_447_unitary__elim,axiom,
! [A: mat_complex,N: nat,B: mat_complex,P: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ P )
=> ( ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ A ) @ ( schur_5982229384592763574omplex @ P ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ B ) @ ( schur_5982229384592763574omplex @ P ) ) )
=> ( A = B ) ) ) ) ) ) ).
% unitary_elim
thf(fact_448_unitary__conjugate__real__diag__decomp,axiom,
! [A: mat_a,N: nat,Us: mat_a,B: mat_a,U: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ Us @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ Us )
=> ( ( spectr3403749184330357196comp_a @ ( spectr5828033140197310157conj_a @ ( schur_mat_adjoint_a @ Us ) @ A ) @ B @ U )
=> ( spectr3403749184330357196comp_a @ A @ B @ ( times_times_mat_a @ Us @ U ) ) ) ) ) ) ).
% unitary_conjugate_real_diag_decomp
thf(fact_449_unitary__conjugate__real__diag__decomp,axiom,
! [A: mat_complex,N: nat,Us: mat_complex,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ Us @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ Us )
=> ( ( spectr5409772854192057952omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ Us ) @ A ) @ B @ U )
=> ( spectr5409772854192057952omplex @ A @ B @ ( times_8009071140041733218omplex @ Us @ U ) ) ) ) ) ) ).
% unitary_conjugate_real_diag_decomp
thf(fact_450_unitarily__equiv__conjugate,axiom,
! [A: mat_complex,N: nat,V: mat_complex,U: mat_complex,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ V @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ V ) @ A ) @ B @ U )
=> ( ( comple6660659447773130958omplex @ V )
=> ( spectr6340060708231679580omplex @ A @ B @ ( times_8009071140041733218omplex @ V @ U ) ) ) ) ) ) ) ) ).
% unitarily_equiv_conjugate
thf(fact_451_Complex__Matrix_Ounitary__def,axiom,
( complex_unitary_a
= ( ^ [A2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ ( dim_row_a @ A2 ) @ ( dim_row_a @ A2 ) ) )
& ( inverts_mat_a @ A2 @ ( schur_mat_adjoint_a @ A2 ) ) ) ) ) ).
% Complex_Matrix.unitary_def
thf(fact_452_Complex__Matrix_Ounitary__def,axiom,
( comple6660659447773130958omplex
= ( ^ [A2: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ ( dim_row_complex @ A2 ) @ ( dim_row_complex @ A2 ) ) )
& ( inverts_mat_complex @ A2 @ ( schur_5982229384592763574omplex @ A2 ) ) ) ) ) ).
% Complex_Matrix.unitary_def
thf(fact_453_hermitian__decomp__eigenvalues,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U )
=> ( ( diag_mat_complex @ B )
= ( projec6785268565095433026omplex @ A ) ) ) ).
% hermitian_decomp_eigenvalues
thf(fact_454_Linear__Algebra__Complements_Osmult__one,axiom,
! [A: mat_nat] :
( ( smult_mat_nat @ one_one_nat @ A )
= A ) ).
% Linear_Algebra_Complements.smult_one
thf(fact_455_Linear__Algebra__Complements_Osmult__one,axiom,
! [A: mat_real] :
( ( smult_mat_real @ one_one_real @ A )
= A ) ).
% Linear_Algebra_Complements.smult_one
thf(fact_456_Linear__Algebra__Complements_Osmult__one,axiom,
! [A: mat_complex] :
( ( smult_mat_complex @ one_one_complex @ A )
= A ) ).
% Linear_Algebra_Complements.smult_one
thf(fact_457_smult__smult__times,axiom,
! [A3: nat,K: nat,A: mat_nat] :
( ( smult_mat_nat @ A3 @ ( smult_mat_nat @ K @ A ) )
= ( smult_mat_nat @ ( times_times_nat @ A3 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_458_smult__smult__times,axiom,
! [A3: complex,K: complex,A: mat_complex] :
( ( smult_mat_complex @ A3 @ ( smult_mat_complex @ K @ A ) )
= ( smult_mat_complex @ ( times_times_complex @ A3 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_459_smult__smult__times,axiom,
! [A3: real,K: real,A: mat_real] :
( ( smult_mat_real @ A3 @ ( smult_mat_real @ K @ A ) )
= ( smult_mat_real @ ( times_times_real @ A3 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_460_inverts__mat__unique,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( inverts_mat_complex @ A @ B )
=> ( ( inverts_mat_complex @ A @ C )
=> ( B = C ) ) ) ) ) ) ).
% inverts_mat_unique
thf(fact_461_inverts__mat__symm,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( inverts_mat_complex @ A @ B )
=> ( inverts_mat_complex @ B @ A ) ) ) ) ).
% inverts_mat_symm
thf(fact_462_mat__conj__def,axiom,
( spectr5699176650994449695omplex
= ( ^ [U3: mat_complex,V2: mat_complex] : ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U3 @ V2 ) @ ( schur_5982229384592763574omplex @ U3 ) ) ) ) ).
% mat_conj_def
thf(fact_463_mat__conj__adjoint,axiom,
! [U: mat_complex,V: mat_complex] :
( ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U ) @ V )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U ) @ V ) @ U ) ) ).
% mat_conj_adjoint
thf(fact_464_mat__conj__unit__commute,axiom,
! [U: mat_complex,A: mat_complex,N: nat] :
( ( comple6660659447773130958omplex @ U )
=> ( ( ( times_8009071140041733218omplex @ U @ A )
= ( times_8009071140041733218omplex @ A @ U ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr5699176650994449695omplex @ U @ A )
= A ) ) ) ) ) ).
% mat_conj_unit_commute
thf(fact_465_unitarily__equivI_H,axiom,
! [A: mat_complex,U: mat_complex,B: mat_complex,N: nat] :
( ( A
= ( spectr5699176650994449695omplex @ U @ B ) )
=> ( ( comple6660659447773130958omplex @ U )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( spectr6340060708231679580omplex @ A @ B @ U ) ) ) ) ) ).
% unitarily_equivI'
thf(fact_466_unitaryD2,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ A )
=> ( inverts_mat_complex @ ( schur_5982229384592763574omplex @ A ) @ A ) ) ) ).
% unitaryD2
thf(fact_467_mat__conj__commute,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ U )
=> ( ( ( times_8009071140041733218omplex @ A @ B )
= ( times_8009071140041733218omplex @ B @ A ) )
=> ( ( times_8009071140041733218omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U ) @ A ) @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U ) @ B ) )
= ( times_8009071140041733218omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U ) @ B ) @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U ) @ A ) ) ) ) ) ) ) ) ).
% mat_conj_commute
thf(fact_468_unitary__mult__conjugate,axiom,
! [A: mat_complex,N: nat,V: mat_complex,U: mat_complex,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ V @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ V )
=> ( ( ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ V ) @ A )
= ( spectr5699176650994449695omplex @ U @ B ) )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ V @ U ) @ B ) @ ( schur_5982229384592763574omplex @ ( times_8009071140041733218omplex @ V @ U ) ) ) ) ) ) ) ) ) ) ).
% unitary_mult_conjugate
thf(fact_469_unitary__diagI_H,axiom,
! [A: mat_a,N: nat,B: mat_a,U: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( diagonal_mat_a @ B )
=> ( ( complex_unitary_a @ U )
=> ( ( A
= ( spectr5828033140197310157conj_a @ U @ B ) )
=> ( spectr4894841263502123494diag_a @ A @ B @ U ) ) ) ) ) ) ).
% unitary_diagI'
thf(fact_470_unitary__diagI_H,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( diagonal_mat_complex @ B )
=> ( ( comple6660659447773130958omplex @ U )
=> ( ( A
= ( spectr5699176650994449695omplex @ U @ B ) )
=> ( spectr532731689276696518omplex @ A @ B @ U ) ) ) ) ) ) ).
% unitary_diagI'
thf(fact_471_invertible__mat__def,axiom,
( invert2568027935824841882omplex
= ( ^ [A2: mat_complex] :
( ( square_mat_complex @ A2 )
& ? [B5: mat_complex] :
( ( inverts_mat_complex @ A2 @ B5 )
& ( inverts_mat_complex @ B5 @ A2 ) ) ) ) ) ).
% invertible_mat_def
thf(fact_472_invertible__mat__def,axiom,
( invertible_mat_a
= ( ^ [A2: mat_a] :
( ( square_mat_a @ A2 )
& ? [B5: mat_a] :
( ( inverts_mat_a @ A2 @ B5 )
& ( inverts_mat_a @ B5 @ A2 ) ) ) ) ) ).
% invertible_mat_def
thf(fact_473_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_474_normal__upper__triangular__matrix__is__diagonal,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( upper_4850907204721561915omplex @ A )
=> ( ( ( times_8009071140041733218omplex @ A @ ( schur_5982229384592763574omplex @ A ) )
= ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ A ) @ A ) )
=> ( diagonal_mat_complex @ A ) ) ) ) ).
% normal_upper_triangular_matrix_is_diagonal
thf(fact_475_unitary__is__corthogonal,axiom,
! [U: mat_complex,N: nat] :
( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ U )
=> ( schur_549222400177443379omplex @ U ) ) ) ).
% unitary_is_corthogonal
thf(fact_476_cpx__sq__mat__axioms_Ointro,axiom,
! [DimR: nat,DimC: nat] :
( ( DimR = DimC )
=> ( ( ord_less_nat @ zero_zero_nat @ DimR )
=> ( linear2040860143340867312axioms @ DimR @ DimC ) ) ) ).
% cpx_sq_mat_axioms.intro
thf(fact_477_cpx__sq__mat__axioms__def,axiom,
( linear2040860143340867312axioms
= ( ^ [DimR2: nat,DimC2: nat] :
( ( DimR2 = DimC2 )
& ( ord_less_nat @ zero_zero_nat @ DimR2 ) ) ) ) ).
% cpx_sq_mat_axioms_def
thf(fact_478_smult__smult__mat,axiom,
! [A: mat_complex,Nr: nat,N: nat,K: complex,L: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( smult_mat_complex @ K @ ( smult_mat_complex @ L @ A ) )
= ( smult_mat_complex @ ( times_times_complex @ K @ L ) @ A ) ) ) ).
% smult_smult_mat
thf(fact_479_mat__assoc__test_I1_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D3: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ ( times_8009071140041733218omplex @ C @ D3 ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ C ) @ D3 ) ) ) ) ) ) ).
% mat_assoc_test(1)
thf(fact_480_mat__assoc__test_I2_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D3: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ ( times_8009071140041733218omplex @ A @ ( schur_5982229384592763574omplex @ B ) ) ) @ C )
= ( times_8009071140041733218omplex @ B @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ A ) @ C ) ) ) ) ) ) ) ).
% mat_assoc_test(2)
thf(fact_481_upper__triangular__zero,axiom,
! [N: nat] : ( upper_triangular_a @ ( zero_mat_a @ N @ N ) ) ).
% upper_triangular_zero
thf(fact_482_upper__triangular__zero,axiom,
! [N: nat] : ( upper_4850907204721561915omplex @ ( zero_mat_complex @ N @ N ) ) ).
% upper_triangular_zero
thf(fact_483_diagonal__imp__upper__triangular,axiom,
! [A: mat_complex,N: nat] :
( ( diagonal_mat_complex @ A )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( upper_4850907204721561915omplex @ A ) ) ) ).
% diagonal_imp_upper_triangular
thf(fact_484_upper__triangular__four__block,axiom,
! [A: mat_complex,N: nat,D3: mat_complex,M: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D3 @ ( carrier_mat_complex @ M @ M ) )
=> ( ( upper_4850907204721561915omplex @ A )
=> ( ( upper_4850907204721561915omplex @ D3 )
=> ( upper_4850907204721561915omplex @ ( four_b559179830521662709omplex @ A @ B @ ( zero_mat_complex @ M @ N ) @ D3 ) ) ) ) ) ) ).
% upper_triangular_four_block
thf(fact_485_upper__triangular__four__block,axiom,
! [A: mat_a,N: nat,D3: mat_a,M: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ D3 @ ( carrier_mat_a @ M @ M ) )
=> ( ( upper_triangular_a @ A )
=> ( ( upper_triangular_a @ D3 )
=> ( upper_triangular_a @ ( four_block_mat_a @ A @ B @ ( zero_mat_a @ M @ N ) @ D3 ) ) ) ) ) ) ).
% upper_triangular_four_block
thf(fact_486_step__1__2__inv_I3_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( upper_4850907204721561915omplex @ A )
=> ( ( B
= ( jordan7871273693253786478omplex @ ( jordan2017415923357163885omplex @ A ) ) )
=> ( jordan4650062548456832493omplex @ N @ B ) ) ) ) ).
% step_1_2_inv(3)
thf(fact_487_vec__space_Orow__space__is__preserved,axiom,
! [P: mat_complex,M: nat,A: mat_complex,N: nat] :
( ( invert2568027935824841882omplex @ P )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ M @ M ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ M @ N ) )
=> ( ( vS_vec3284807721666986142omplex @ N @ ( times_8009071140041733218omplex @ P @ A ) )
= ( vS_vec3284807721666986142omplex @ N @ A ) ) ) ) ) ).
% vec_space.row_space_is_preserved
thf(fact_488_step__1__2__inv_I2_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( upper_4850907204721561915omplex @ A )
=> ( ( B
= ( jordan7871273693253786478omplex @ ( jordan2017415923357163885omplex @ A ) ) )
=> ( jordan5244935068081719878omplex @ N @ jordan8650160714669549932omplex @ B ) ) ) ) ).
% step_1_2_inv(2)
thf(fact_489_density__collapse__operator,axiom,
! [P: mat_complex,R: mat_complex,N: nat] :
( ( linear5633924348262549461omplex @ P )
=> ( ( comple5220265106149225959erator @ R )
=> ( ( 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 ) )
=> ( comple5220265106149225959erator @ ( projec3470689467825365843llapse @ R @ P ) ) ) ) ) ) ) ).
% density_collapse_operator
thf(fact_490_hermitian__mat__conj_H,axiom,
! [A: mat_complex,N: nat,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U ) @ A ) ) ) ) ) ).
% hermitian_mat_conj'
thf(fact_491_projector__def,axiom,
( linear5633924348262549461omplex
= ( ^ [M3: mat_complex] :
( ( comple8306762464034002205omplex @ M3 )
& ( ( times_8009071140041733218omplex @ M3 @ M3 )
= M3 ) ) ) ) ).
% projector_def
thf(fact_492_hermitian__square__similar__mat__wit,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( comple8306762464034002205omplex @ A )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr532731689276696518omplex @ A @ B @ U )
=> ( simila5774310414453981135omplex @ ( times_8009071140041733218omplex @ A @ A ) @ ( times_8009071140041733218omplex @ B @ B ) @ U @ ( schur_5982229384592763574omplex @ U ) ) ) ) ) ).
% hermitian_square_similar_mat_wit
thf(fact_493_hermitian__square__hermitian,axiom,
! [A: mat_complex] :
( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( times_8009071140041733218omplex @ A @ A ) ) ) ).
% hermitian_square_hermitian
thf(fact_494_zero__hermitian,axiom,
! [N: nat] : ( complex_hermitian_a @ ( zero_mat_a @ N @ N ) ) ).
% zero_hermitian
thf(fact_495_zero__hermitian,axiom,
! [N: nat] : ( comple8306762464034002205omplex @ ( zero_mat_complex @ N @ N ) ) ).
% zero_hermitian
thf(fact_496_hermitian__def,axiom,
( comple8306762464034002205omplex
= ( ^ [A2: mat_complex] :
( ( schur_5982229384592763574omplex @ A2 )
= A2 ) ) ) ).
% hermitian_def
thf(fact_497_unitary__density,axiom,
! [R: mat_complex,U: mat_complex,N: nat] :
( ( comple5220265106149225959erator @ R )
=> ( ( comple6660659447773130958omplex @ U )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( comple5220265106149225959erator @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U @ R ) @ ( schur_5982229384592763574omplex @ U ) ) ) ) ) ) ) ).
% unitary_density
thf(fact_498_hermitian__square,axiom,
! [M4: mat_a] :
( ( complex_hermitian_a @ M4 )
=> ( member_mat_a @ M4 @ ( carrier_mat_a @ ( dim_row_a @ M4 ) @ ( dim_row_a @ M4 ) ) ) ) ).
% hermitian_square
thf(fact_499_hermitian__square,axiom,
! [M4: mat_complex] :
( ( comple8306762464034002205omplex @ M4 )
=> ( member_mat_complex @ M4 @ ( carrier_mat_complex @ ( dim_row_complex @ M4 ) @ ( dim_row_complex @ M4 ) ) ) ) ).
% hermitian_square
thf(fact_500_hermitian__real__diag__decomp,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( comple8306762464034002205omplex @ A )
=> ~ ! [B8: mat_complex,U4: mat_complex] :
~ ( spectr5409772854192057952omplex @ A @ B8 @ U4 ) ) ) ) ).
% hermitian_real_diag_decomp
thf(fact_501_hermitian__is__normal,axiom,
! [A: mat_complex] :
( ( comple8306762464034002205omplex @ A )
=> ( ( times_8009071140041733218omplex @ A @ ( schur_5982229384592763574omplex @ A ) )
= ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ A ) @ A ) ) ) ).
% hermitian_is_normal
thf(fact_502_hermitian__mat__conj,axiom,
! [A: mat_complex,N: nat,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( spectr5699176650994449695omplex @ U @ A ) ) ) ) ) ).
% hermitian_mat_conj
thf(fact_503_projector__square__eq,axiom,
! [M4: mat_complex] :
( ( linear5633924348262549461omplex @ M4 )
=> ( ( times_8009071140041733218omplex @ M4 @ M4 )
= M4 ) ) ).
% projector_square_eq
thf(fact_504_zero__projector,axiom,
! [N: nat] : ( linear2821214051344812439ctor_a @ ( zero_mat_a @ N @ N ) ) ).
% zero_projector
thf(fact_505_zero__projector,axiom,
! [N: nat] : ( linear5633924348262549461omplex @ ( zero_mat_complex @ N @ N ) ) ).
% zero_projector
thf(fact_506_mult__adjoint__hermitian,axiom,
! [A: mat_complex,N: nat,M: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ M ) )
=> ( comple8306762464034002205omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ A ) @ A ) ) ) ).
% mult_adjoint_hermitian
thf(fact_507_step__1__2__inv_I1_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( upper_4850907204721561915omplex @ A )
=> ( ( B
= ( jordan7871273693253786478omplex @ ( jordan2017415923357163885omplex @ A ) ) )
=> ( jordan5244935068081719878omplex @ N @ jordan3528196489273997576omplex @ B ) ) ) ) ).
% step_1_2_inv(1)
thf(fact_508_max__mix__is__density,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( comple5220265106149225959erator @ ( projec8360710381328234318ensity @ N ) ) ) ).
% max_mix_is_density
thf(fact_509_transpose__of__prod,axiom,
! [M4: mat_complex,N3: mat_complex] :
( ( ( dim_col_complex @ M4 )
= ( dim_row_complex @ N3 ) )
=> ( ( transp3074176993011536131omplex @ ( times_8009071140041733218omplex @ M4 @ N3 ) )
= ( times_8009071140041733218omplex @ ( transp3074176993011536131omplex @ N3 ) @ ( transp3074176993011536131omplex @ M4 ) ) ) ) ).
% transpose_of_prod
thf(fact_510_vec__space_Orow__space__eq__col__space__transpose,axiom,
( vS_vec3284807721666986142omplex
= ( ^ [N4: nat,A2: mat_complex] : ( vS_vec1879987866596122552omplex @ N4 @ ( transp3074176993011536131omplex @ A2 ) ) ) ) ).
% vec_space.row_space_eq_col_space_transpose
thf(fact_511_real__diag__decomp__hermitian,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr5409772854192057952omplex @ A @ B @ U )
=> ( comple8306762464034002205omplex @ A ) ) ).
% real_diag_decomp_hermitian
thf(fact_512_hermitian__decomp__decomp_H,axiom,
! [A: mat_complex,B: mat_complex,U: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B @ U )
=> ( spectr5409772854192057952omplex @ A @ B @ U ) ) ).
% hermitian_decomp_decomp'
thf(fact_513_vec__space_Ocol__space__eq__row__space__transpose,axiom,
( vS_vec1879987866596122552omplex
= ( ^ [N4: nat,A2: mat_complex] : ( vS_vec3284807721666986142omplex @ N4 @ ( transp3074176993011536131omplex @ A2 ) ) ) ) ).
% vec_space.col_space_eq_row_space_transpose
thf(fact_514_uppert__to__jb,axiom,
! [N: nat,A: mat_complex] :
( ( jordan5244935068081719878omplex @ N @ jordan3528196489273997576omplex @ A )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( jordan5475473882837061487omplex @ N @ jordan4971026570492200526omplex @ A @ one_one_nat ) ) ) ).
% uppert_to_jb
thf(fact_515_cpx__mat__cnj__prod,axiom,
! [M4: mat_complex,N3: mat_complex] :
( ( ( dim_col_complex @ M4 )
= ( dim_row_complex @ N3 ) )
=> ( ( cpx_mat_cnj @ ( times_8009071140041733218omplex @ M4 @ N3 ) )
= ( times_8009071140041733218omplex @ ( cpx_mat_cnj @ M4 ) @ ( cpx_mat_cnj @ N3 ) ) ) ) ).
% cpx_mat_cnj_prod
thf(fact_516_tensor__mat__carrier,axiom,
! [U: mat_complex,V: mat_complex] : ( member_mat_complex @ ( tensor_mat @ U @ V ) @ ( carrier_mat_complex @ ( times_times_nat @ ( dim_row_complex @ U ) @ ( dim_row_complex @ V ) ) @ ( times_times_nat @ ( dim_col_complex @ U ) @ ( dim_col_complex @ V ) ) ) ) ).
% tensor_mat_carrier
thf(fact_517_tensor__mat__unitary,axiom,
! [U: mat_complex,V: mat_complex] :
( ( comple6660659447773130958omplex @ U )
=> ( ( comple6660659447773130958omplex @ V )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ U ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ V ) )
=> ( comple6660659447773130958omplex @ ( tensor_mat @ U @ V ) ) ) ) ) ) ).
% tensor_mat_unitary
thf(fact_518_tensor__mat__adjoint,axiom,
! [M1: mat_complex,R1: nat,C1: nat,M22: mat_complex,R2: nat,C22: nat] :
( ( member_mat_complex @ M1 @ ( carrier_mat_complex @ R1 @ C1 ) )
=> ( ( member_mat_complex @ M22 @ ( carrier_mat_complex @ R2 @ C22 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C1 )
=> ( ( ord_less_nat @ zero_zero_nat @ C22 )
=> ( ( ord_less_nat @ zero_zero_nat @ R1 )
=> ( ( ord_less_nat @ zero_zero_nat @ R2 )
=> ( ( schur_5982229384592763574omplex @ ( tensor_mat @ M1 @ M22 ) )
= ( tensor_mat @ ( schur_5982229384592763574omplex @ M1 ) @ ( schur_5982229384592763574omplex @ M22 ) ) ) ) ) ) ) ) ) ).
% tensor_mat_adjoint
thf(fact_519_dim__col__of__cjn__prod,axiom,
! [M4: mat_complex,N3: mat_complex] :
( ( dim_col_complex @ ( times_8009071140041733218omplex @ ( cpx_mat_cnj @ M4 ) @ ( cpx_mat_cnj @ N3 ) ) )
= ( dim_col_complex @ N3 ) ) ).
% dim_col_of_cjn_prod
thf(fact_520_dim__row__of__cjn__prod,axiom,
! [M4: mat_complex,N3: mat_complex] :
( ( dim_row_complex @ ( times_8009071140041733218omplex @ ( cpx_mat_cnj @ M4 ) @ ( cpx_mat_cnj @ N3 ) ) )
= ( dim_row_complex @ M4 ) ) ).
% dim_row_of_cjn_prod
thf(fact_521_tensor__mat__hermitian,axiom,
! [A: mat_complex,N: nat,B: mat_complex,N5: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N5 @ N5 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ( comple8306762464034002205omplex @ A )
=> ( ( comple8306762464034002205omplex @ B )
=> ( comple8306762464034002205omplex @ ( tensor_mat @ A @ B ) ) ) ) ) ) ) ) ).
% tensor_mat_hermitian
thf(fact_522_mult__distr__tensor,axiom,
! [A: mat_complex,B: mat_complex,C: mat_complex,D3: mat_complex] :
( ( ( dim_col_complex @ A )
= ( dim_row_complex @ B ) )
=> ( ( ( dim_col_complex @ C )
= ( dim_row_complex @ D3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ D3 ) )
=> ( ( tensor_mat @ ( times_8009071140041733218omplex @ A @ B ) @ ( times_8009071140041733218omplex @ C @ D3 ) )
= ( times_8009071140041733218omplex @ ( tensor_mat @ A @ C ) @ ( tensor_mat @ B @ D3 ) ) ) ) ) ) ) ) ) ).
% mult_distr_tensor
thf(fact_523_step__3__main__inv,axiom,
! [A: mat_complex,N: nat,K: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( jordan5244935068081719878omplex @ N @ jordan3528196489273997576omplex @ A )
=> ( ( jordan8042990603089931364omplex @ N @ A )
=> ( ( jordan5475473882837061487omplex @ N @ jordan4971026570492200526omplex @ A @ K )
=> ( ( jordan5244935068081719878omplex @ N @ jordan4971026570492200526omplex @ ( jordan4702481308941288104omplex @ N @ K @ A ) )
& ( jordan2620430285385836103omplex @ N @ A @ ( jordan4702481308941288104omplex @ N @ K @ A ) ) ) ) ) ) ) ) ).
% step_3_main_inv
thf(fact_524_step__3__main__inv,axiom,
! [A: mat_a,N: nat,K: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( jordan7439094043700944742_all_a @ N @ jordan2755030923421653284pert_a @ A )
=> ( ( jordan1479931431598099656lock_a @ N @ A )
=> ( ( jordan1574482064217505917upto_a @ N @ jordan8102412511815959902m_jb_a @ A @ K )
=> ( ( jordan7439094043700944742_all_a @ N @ jordan8102412511815959902m_jb_a @ ( jordan460303421567170436main_a @ N @ K @ A ) )
& ( jordan8308822787700309925diag_a @ N @ A @ ( jordan460303421567170436main_a @ N @ K @ A ) ) ) ) ) ) ) ) ).
% step_3_main_inv
thf(fact_525_dim__row__tensor__mat,axiom,
! [A: mat_complex,B: mat_complex] :
( ( dim_row_complex @ ( tensor_mat @ A @ B ) )
= ( times_times_nat @ ( dim_row_complex @ A ) @ ( dim_row_complex @ B ) ) ) ).
% dim_row_tensor_mat
thf(fact_526_dim__col__tensor__mat,axiom,
! [A: mat_complex,B: mat_complex] :
( ( dim_col_complex @ ( tensor_mat @ A @ B ) )
= ( times_times_nat @ ( dim_col_complex @ A ) @ ( dim_col_complex @ B ) ) ) ).
% dim_col_tensor_mat
thf(fact_527_unitary__operator__keep__trace,axiom,
! [U: mat_complex,N: nat,A: mat_complex] :
( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ U )
=> ( ( comple3184165445352484367omplex @ A )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U ) @ A ) @ U ) ) ) ) ) ) ).
% unitary_operator_keep_trace
thf(fact_528_mat__assoc__test_I10_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D3: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ C ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ B @ C ) @ A ) ) ) ) ) ) ) ).
% mat_assoc_test(10)
thf(fact_529_mat__assoc__test_I11_J,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex,D3: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D3 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B ) @ C ) @ D3 ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ C @ D3 ) @ A ) @ B ) ) ) ) ) ) ) ).
% mat_assoc_test(11)
thf(fact_530_trace__zero,axiom,
! [N: nat] :
( ( complex_trace_a @ ( zero_mat_a @ N @ N ) )
= zero_zero_a ) ).
% trace_zero
thf(fact_531_trace__zero,axiom,
! [N: nat] :
( ( comple3184165445352484367omplex @ ( zero_mat_complex @ N @ N ) )
= zero_zero_complex ) ).
% trace_zero
thf(fact_532_trace__zero,axiom,
! [N: nat] :
( ( complex_trace_real @ ( zero_mat_real @ N @ N ) )
= zero_zero_real ) ).
% trace_zero
thf(fact_533_trace__comm,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ B ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B @ A ) ) ) ) ) ).
% trace_comm
thf(fact_534_unitarily__equiv__trace,axiom,
! [A: mat_complex,N: nat,B: mat_complex,U: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( spectr6340060708231679580omplex @ A @ B @ U )
=> ( ( comple3184165445352484367omplex @ A )
= ( comple3184165445352484367omplex @ B ) ) ) ) ).
% unitarily_equiv_trace
thf(fact_535_projector__collapse__trace,axiom,
! [P: mat_complex,N: nat,R: mat_complex] :
( ( linear5633924348262549461omplex @ P )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ R ) @ P ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ P ) ) ) ) ) ) ).
% projector_collapse_trace
thf(fact_536_trace__smult,axiom,
! [A: mat_complex,N: nat,C2: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple3184165445352484367omplex @ ( smult_mat_complex @ C2 @ A ) )
= ( times_times_complex @ C2 @ ( comple3184165445352484367omplex @ A ) ) ) ) ).
% trace_smult
thf(fact_537_trace__smult,axiom,
! [A: mat_real,N: nat,C2: real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N @ N ) )
=> ( ( complex_trace_real @ ( smult_mat_real @ C2 @ A ) )
= ( times_times_real @ C2 @ ( complex_trace_real @ A ) ) ) ) ).
% trace_smult
thf(fact_538_trace__pdo__eq__imp__eq,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ! [Rho: mat_complex] :
( ( member_mat_complex @ Rho @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple1169154605998056944erator @ Rho )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ Rho ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B @ Rho ) ) ) ) )
=> ( A = B ) ) ) ) ).
% trace_pdo_eq_imp_eq
thf(fact_539_partition__jb_I2_J,axiom,
! [A: mat_a,N: nat,Bs: list_mat_a,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( jordan7439094043700944742_all_a @ N @ jordan2755030923421653284pert_a @ A )
=> ( ( jordan7439094043700944742_all_a @ N @ jordan1888133435898081728f_ev_a @ A )
=> ( ( jordan8767189289504586111ocks_a @ N @ A )
=> ( ( ( jordan501837315015147299ocks_a @ A @ nil_mat_a )
= Bs )
=> ( ( member_mat_a @ B @ ( set_mat_a2 @ Bs ) )
=> ( ( jordan4251489913308508029_all_a @ jordan2755030923421653284pert_a @ B )
& ( jordan1479931431598099656lock_a @ ( dim_col_a @ B ) @ B )
& ( ( dim_row_a @ B )
= ( dim_col_a @ B ) ) ) ) ) ) ) ) ) ).
% partition_jb(2)
thf(fact_540_partition__jb_I2_J,axiom,
! [A: mat_complex,N: nat,Bs: list_mat_complex,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( jordan5244935068081719878omplex @ N @ jordan3528196489273997576omplex @ A )
=> ( ( jordan5244935068081719878omplex @ N @ jordan8650160714669549932omplex @ A )
=> ( ( jordan4650062548456832493omplex @ N @ A )
=> ( ( ( jordan5009815537632354121omplex @ A @ nil_mat_complex )
= Bs )
=> ( ( member_mat_complex @ B @ ( set_mat_complex2 @ Bs ) )
=> ( ( jordan5032732407113867375omplex @ jordan3528196489273997576omplex @ B )
& ( jordan8042990603089931364omplex @ ( dim_col_complex @ B ) @ B )
& ( ( dim_row_complex @ B )
= ( dim_col_complex @ B ) ) ) ) ) ) ) ) ) ).
% partition_jb(2)
thf(fact_541_jnf__vector_I1_J,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( ord_less_nat @ J3 @ N )
=> ( jordan4971026570492200526omplex @ A @ I3 @ J3 ) ) )
=> ( ( jordan8042990603089931364omplex @ N @ A )
=> ( ( jordan5739059635872469039omplex @ ( jordan387279176131498413omplex @ A ) )
= A ) ) ) ) ).
% jnf_vector(1)
thf(fact_542_lowner__le__keep__under__measurement,axiom,
! [M4: mat_complex,N: nat,A: mat_complex,B: mat_complex] :
( ( member_mat_complex @ M4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ A @ B )
=> ( complex_lowner_le @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ M4 ) @ A ) @ M4 ) @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ M4 ) @ B ) @ M4 ) ) ) ) ) ) ).
% lowner_le_keep_under_measurement
thf(fact_543_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_544_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_545_dvd__0__left__iff,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
= ( A3 = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_546_dvd__0__left__iff,axiom,
! [A3: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A3 )
= ( A3 = zero_zero_complex ) ) ).
% dvd_0_left_iff
thf(fact_547_dvd__0__left__iff,axiom,
! [A3: real] :
( ( dvd_dvd_real @ zero_zero_real @ A3 )
= ( A3 = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_548_dvd__0__right,axiom,
! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_549_dvd__0__right,axiom,
! [A3: complex] : ( dvd_dvd_complex @ A3 @ zero_zero_complex ) ).
% dvd_0_right
thf(fact_550_dvd__0__right,axiom,
! [A3: real] : ( dvd_dvd_real @ A3 @ zero_zero_real ) ).
% dvd_0_right
thf(fact_551_dvd__0__left,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
=> ( A3 = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_552_dvd__0__left,axiom,
! [A3: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A3 )
=> ( A3 = zero_zero_complex ) ) ).
% dvd_0_left
thf(fact_553_dvd__0__left,axiom,
! [A3: real] :
( ( dvd_dvd_real @ zero_zero_real @ A3 )
=> ( A3 = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_554_dvd__triv__right,axiom,
! [A3: nat,B6: nat] : ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B6 @ A3 ) ) ).
% dvd_triv_right
thf(fact_555_dvd__triv__right,axiom,
! [A3: complex,B6: complex] : ( dvd_dvd_complex @ A3 @ ( times_times_complex @ B6 @ A3 ) ) ).
% dvd_triv_right
thf(fact_556_dvd__triv__right,axiom,
! [A3: real,B6: real] : ( dvd_dvd_real @ A3 @ ( times_times_real @ B6 @ A3 ) ) ).
% dvd_triv_right
thf(fact_557_dvd__mult__right,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B6 ) @ C2 )
=> ( dvd_dvd_nat @ B6 @ C2 ) ) ).
% dvd_mult_right
thf(fact_558_dvd__mult__right,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A3 @ B6 ) @ C2 )
=> ( dvd_dvd_complex @ B6 @ C2 ) ) ).
% dvd_mult_right
thf(fact_559_dvd__mult__right,axiom,
! [A3: real,B6: real,C2: real] :
( ( dvd_dvd_real @ ( times_times_real @ A3 @ B6 ) @ C2 )
=> ( dvd_dvd_real @ B6 @ C2 ) ) ).
% dvd_mult_right
thf(fact_560_mult__dvd__mono,axiom,
! [A3: nat,B6: nat,C2: nat,D: nat] :
( ( dvd_dvd_nat @ A3 @ B6 )
=> ( ( dvd_dvd_nat @ C2 @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B6 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_561_mult__dvd__mono,axiom,
! [A3: complex,B6: complex,C2: complex,D: complex] :
( ( dvd_dvd_complex @ A3 @ B6 )
=> ( ( dvd_dvd_complex @ C2 @ D )
=> ( dvd_dvd_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B6 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_562_mult__dvd__mono,axiom,
! [A3: real,B6: real,C2: real,D: real] :
( ( dvd_dvd_real @ A3 @ B6 )
=> ( ( dvd_dvd_real @ C2 @ D )
=> ( dvd_dvd_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_563_dvd__triv__left,axiom,
! [A3: nat,B6: nat] : ( dvd_dvd_nat @ A3 @ ( times_times_nat @ A3 @ B6 ) ) ).
% dvd_triv_left
thf(fact_564_dvd__triv__left,axiom,
! [A3: complex,B6: complex] : ( dvd_dvd_complex @ A3 @ ( times_times_complex @ A3 @ B6 ) ) ).
% dvd_triv_left
thf(fact_565_dvd__triv__left,axiom,
! [A3: real,B6: real] : ( dvd_dvd_real @ A3 @ ( times_times_real @ A3 @ B6 ) ) ).
% dvd_triv_left
thf(fact_566_dvd__mult__left,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B6 ) @ C2 )
=> ( dvd_dvd_nat @ A3 @ C2 ) ) ).
% dvd_mult_left
thf(fact_567_dvd__mult__left,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A3 @ B6 ) @ C2 )
=> ( dvd_dvd_complex @ A3 @ C2 ) ) ).
% dvd_mult_left
thf(fact_568_dvd__mult__left,axiom,
! [A3: real,B6: real,C2: real] :
( ( dvd_dvd_real @ ( times_times_real @ A3 @ B6 ) @ C2 )
=> ( dvd_dvd_real @ A3 @ C2 ) ) ).
% dvd_mult_left
thf(fact_569_dvd__mult2,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( dvd_dvd_nat @ A3 @ B6 )
=> ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B6 @ C2 ) ) ) ).
% dvd_mult2
thf(fact_570_dvd__mult2,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( dvd_dvd_complex @ A3 @ B6 )
=> ( dvd_dvd_complex @ A3 @ ( times_times_complex @ B6 @ C2 ) ) ) ).
% dvd_mult2
thf(fact_571_dvd__mult2,axiom,
! [A3: real,B6: real,C2: real] :
( ( dvd_dvd_real @ A3 @ B6 )
=> ( dvd_dvd_real @ A3 @ ( times_times_real @ B6 @ C2 ) ) ) ).
% dvd_mult2
thf(fact_572_dvd__mult,axiom,
! [A3: nat,C2: nat,B6: nat] :
( ( dvd_dvd_nat @ A3 @ C2 )
=> ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B6 @ C2 ) ) ) ).
% dvd_mult
thf(fact_573_dvd__mult,axiom,
! [A3: complex,C2: complex,B6: complex] :
( ( dvd_dvd_complex @ A3 @ C2 )
=> ( dvd_dvd_complex @ A3 @ ( times_times_complex @ B6 @ C2 ) ) ) ).
% dvd_mult
thf(fact_574_dvd__mult,axiom,
! [A3: real,C2: real,B6: real] :
( ( dvd_dvd_real @ A3 @ C2 )
=> ( dvd_dvd_real @ A3 @ ( times_times_real @ B6 @ C2 ) ) ) ).
% dvd_mult
thf(fact_575_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B7: nat,A5: nat] :
? [K2: nat] :
( A5
= ( times_times_nat @ B7 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_576_dvd__def,axiom,
( dvd_dvd_complex
= ( ^ [B7: complex,A5: complex] :
? [K2: complex] :
( A5
= ( times_times_complex @ B7 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_577_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B7: real,A5: real] :
? [K2: real] :
( A5
= ( times_times_real @ B7 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_578_dvdI,axiom,
! [A3: nat,B6: nat,K: nat] :
( ( A3
= ( times_times_nat @ B6 @ K ) )
=> ( dvd_dvd_nat @ B6 @ A3 ) ) ).
% dvdI
thf(fact_579_dvdI,axiom,
! [A3: complex,B6: complex,K: complex] :
( ( A3
= ( times_times_complex @ B6 @ K ) )
=> ( dvd_dvd_complex @ B6 @ A3 ) ) ).
% dvdI
thf(fact_580_dvdI,axiom,
! [A3: real,B6: real,K: real] :
( ( A3
= ( times_times_real @ B6 @ K ) )
=> ( dvd_dvd_real @ B6 @ A3 ) ) ).
% dvdI
thf(fact_581_dvdE,axiom,
! [B6: nat,A3: nat] :
( ( dvd_dvd_nat @ B6 @ A3 )
=> ~ ! [K3: nat] :
( A3
!= ( times_times_nat @ B6 @ K3 ) ) ) ).
% dvdE
thf(fact_582_dvdE,axiom,
! [B6: complex,A3: complex] :
( ( dvd_dvd_complex @ B6 @ A3 )
=> ~ ! [K3: complex] :
( A3
!= ( times_times_complex @ B6 @ K3 ) ) ) ).
% dvdE
thf(fact_583_dvdE,axiom,
! [B6: real,A3: real] :
( ( dvd_dvd_real @ B6 @ A3 )
=> ~ ! [K3: real] :
( A3
!= ( times_times_real @ B6 @ K3 ) ) ) ).
% dvdE
thf(fact_584_dvd__unit__imp__unit,axiom,
! [A3: nat,B6: nat] :
( ( dvd_dvd_nat @ A3 @ B6 )
=> ( ( dvd_dvd_nat @ B6 @ one_one_nat )
=> ( dvd_dvd_nat @ A3 @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_585_algebraic__semidom__class_Ounit__imp__dvd,axiom,
! [B6: nat,A3: nat] :
( ( dvd_dvd_nat @ B6 @ one_one_nat )
=> ( dvd_dvd_nat @ B6 @ A3 ) ) ).
% algebraic_semidom_class.unit_imp_dvd
thf(fact_586_one__dvd,axiom,
! [A3: nat] : ( dvd_dvd_nat @ one_one_nat @ A3 ) ).
% one_dvd
thf(fact_587_one__dvd,axiom,
! [A3: real] : ( dvd_dvd_real @ one_one_real @ A3 ) ).
% one_dvd
thf(fact_588_one__dvd,axiom,
! [A3: complex] : ( dvd_dvd_complex @ one_one_complex @ A3 ) ).
% one_dvd
thf(fact_589_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_590_dvd__trans,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( dvd_dvd_nat @ A3 @ B6 )
=> ( ( dvd_dvd_nat @ B6 @ C2 )
=> ( dvd_dvd_nat @ A3 @ C2 ) ) ) ).
% dvd_trans
thf(fact_591_dvd__refl,axiom,
! [A3: nat] : ( dvd_dvd_nat @ A3 @ A3 ) ).
% dvd_refl
thf(fact_592_lowner__le__refl,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( complex_lowner_le @ A @ A ) ) ).
% lowner_le_refl
thf(fact_593_lowner__le__trans,axiom,
! [A: mat_complex,N: nat,B: mat_complex,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ A @ B )
=> ( ( complex_lowner_le @ B @ C )
=> ( complex_lowner_le @ A @ C ) ) ) ) ) ) ).
% lowner_le_trans
thf(fact_594_lowner__le__antisym,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ A @ B )
=> ( ( complex_lowner_le @ B @ A )
=> ( A = B ) ) ) ) ) ).
% lowner_le_antisym
thf(fact_595_dvd__mult__cancel__left,axiom,
! [C2: complex,A3: complex,B6: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B6 ) )
= ( ( C2 = zero_zero_complex )
| ( dvd_dvd_complex @ A3 @ B6 ) ) ) ).
% dvd_mult_cancel_left
thf(fact_596_dvd__mult__cancel__left,axiom,
! [C2: real,A3: real,B6: real] :
( ( dvd_dvd_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) )
= ( ( C2 = zero_zero_real )
| ( dvd_dvd_real @ A3 @ B6 ) ) ) ).
% dvd_mult_cancel_left
thf(fact_597_dvd__mult__cancel__right,axiom,
! [A3: complex,C2: complex,B6: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B6 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( dvd_dvd_complex @ A3 @ B6 ) ) ) ).
% dvd_mult_cancel_right
thf(fact_598_dvd__mult__cancel__right,axiom,
! [A3: real,C2: real,B6: real] :
( ( dvd_dvd_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( dvd_dvd_real @ A3 @ B6 ) ) ) ).
% dvd_mult_cancel_right
thf(fact_599_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( A3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B6 ) @ ( times_times_nat @ A3 @ C2 ) )
= ( dvd_dvd_nat @ B6 @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_600_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( A3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B6 @ A3 ) @ ( times_times_nat @ C2 @ A3 ) )
= ( dvd_dvd_nat @ B6 @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_601_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_602_unit__dvd__iff_I4_J,axiom,
! [B6: nat,A3: nat,C2: nat] :
( ( dvd_dvd_nat @ B6 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B6 @ C2 ) )
= ( dvd_dvd_nat @ A3 @ C2 ) ) ) ).
% unit_dvd_iff(4)
thf(fact_603_unit__dvd__iff_I3_J,axiom,
! [B6: nat,A3: nat,C2: nat] :
( ( dvd_dvd_nat @ B6 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A3 @ ( times_times_nat @ C2 @ B6 ) )
= ( dvd_dvd_nat @ A3 @ C2 ) ) ) ).
% unit_dvd_iff(3)
thf(fact_604_unit__dvd__iff_I2_J,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B6 ) @ C2 )
= ( dvd_dvd_nat @ B6 @ C2 ) ) ) ).
% unit_dvd_iff(2)
thf(fact_605_unit__dvd__iff_I1_J,axiom,
! [B6: nat,A3: nat,C2: nat] :
( ( dvd_dvd_nat @ B6 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B6 ) @ C2 )
= ( dvd_dvd_nat @ A3 @ C2 ) ) ) ).
% unit_dvd_iff(1)
thf(fact_606_algebraic__semidom__class_Ounit__prod,axiom,
! [A3: nat,B6: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B6 @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B6 ) @ one_one_nat ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_607_algebraic__semidom__class_Ois__unit__mult__iff,axiom,
! [A3: nat,B6: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B6 ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A3 @ one_one_nat )
& ( dvd_dvd_nat @ B6 @ one_one_nat ) ) ) ).
% algebraic_semidom_class.is_unit_mult_iff
thf(fact_608_algebraic__semidom__class_Ounit__mult__left__cancel,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( ( times_times_nat @ A3 @ B6 )
= ( times_times_nat @ A3 @ C2 ) )
= ( B6 = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_left_cancel
thf(fact_609_algebraic__semidom__class_Ounit__mult__right__cancel,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( ( times_times_nat @ B6 @ A3 )
= ( times_times_nat @ C2 @ A3 ) )
= ( B6 = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_right_cancel
thf(fact_610_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_611_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_612_unit__dvdE,axiom,
! [A3: nat,B6: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ~ ( ( A3 != zero_zero_nat )
=> ! [C3: nat] :
( B6
!= ( times_times_nat @ A3 @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_613_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_614_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_615_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_616_partition__jb_I1_J,axiom,
! [A: mat_complex,N: nat,Bs: list_mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( jordan5244935068081719878omplex @ N @ jordan3528196489273997576omplex @ A )
=> ( ( jordan5244935068081719878omplex @ N @ jordan8650160714669549932omplex @ A )
=> ( ( jordan4650062548456832493omplex @ N @ A )
=> ( ( ( jordan5009815537632354121omplex @ A @ nil_mat_complex )
= Bs )
=> ( A
= ( diag_b9145358668110806138omplex @ Bs ) ) ) ) ) ) ) ).
% partition_jb(1)
thf(fact_617_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_618_idom__class_Odvd__mult__unit__iff,axiom,
! [B6: complex,A3: complex,C2: complex] :
( ( dvd_dvd_complex @ B6 @ one_one_complex )
=> ( ( dvd_dvd_complex @ A3 @ ( times_times_complex @ C2 @ B6 ) )
= ( dvd_dvd_complex @ A3 @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff
thf(fact_619_idom__class_Odvd__mult__unit__iff,axiom,
! [B6: real,A3: real,C2: real] :
( ( dvd_dvd_real @ B6 @ one_one_real )
=> ( ( dvd_dvd_real @ A3 @ ( times_times_real @ C2 @ B6 ) )
= ( dvd_dvd_real @ A3 @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff
thf(fact_620_idom__class_Odvd__mult__unit__iff_H,axiom,
! [B6: complex,A3: complex,C2: complex] :
( ( dvd_dvd_complex @ B6 @ one_one_complex )
=> ( ( dvd_dvd_complex @ A3 @ ( times_times_complex @ B6 @ C2 ) )
= ( dvd_dvd_complex @ A3 @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff'
thf(fact_621_idom__class_Odvd__mult__unit__iff_H,axiom,
! [B6: real,A3: real,C2: real] :
( ( dvd_dvd_real @ B6 @ one_one_real )
=> ( ( dvd_dvd_real @ A3 @ ( times_times_real @ B6 @ C2 ) )
= ( dvd_dvd_real @ A3 @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff'
thf(fact_622_comm__monoid__mult__class_Ounit__prod,axiom,
! [A3: nat,B6: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B6 @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B6 ) @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_623_comm__monoid__mult__class_Ounit__prod,axiom,
! [A3: complex,B6: complex] :
( ( dvd_dvd_complex @ A3 @ one_one_complex )
=> ( ( dvd_dvd_complex @ B6 @ one_one_complex )
=> ( dvd_dvd_complex @ ( times_times_complex @ A3 @ B6 ) @ one_one_complex ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_624_comm__monoid__mult__class_Ounit__prod,axiom,
! [A3: real,B6: real] :
( ( dvd_dvd_real @ A3 @ one_one_real )
=> ( ( dvd_dvd_real @ B6 @ one_one_real )
=> ( dvd_dvd_real @ ( times_times_real @ A3 @ B6 ) @ one_one_real ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_625_diag__block__mat_Osimps_I1_J,axiom,
( ( diag_block_mat_a @ nil_mat_a )
= ( zero_mat_a @ zero_zero_nat @ zero_zero_nat ) ) ).
% diag_block_mat.simps(1)
thf(fact_626_diag__block__mat_Osimps_I1_J,axiom,
( ( diag_b9145358668110806138omplex @ nil_mat_complex )
= ( zero_mat_complex @ zero_zero_nat @ zero_zero_nat ) ) ).
% diag_block_mat.simps(1)
thf(fact_627_division__decomp,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B6 @ C2 ) )
=> ? [B9: nat,C4: nat] :
( ( A3
= ( times_times_nat @ B9 @ C4 ) )
& ( dvd_dvd_nat @ B9 @ B6 )
& ( dvd_dvd_nat @ C4 @ C2 ) ) ) ).
% division_decomp
thf(fact_628_dvd__productE,axiom,
! [P4: nat,A3: nat,B6: nat] :
( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A3 @ B6 ) )
=> ~ ! [X3: nat,Y2: nat] :
( ( P4
= ( times_times_nat @ X3 @ Y2 ) )
=> ( ( dvd_dvd_nat @ X3 @ A3 )
=> ~ ( dvd_dvd_nat @ Y2 @ B6 ) ) ) ) ).
% dvd_productE
thf(fact_629_idom__class_Ounit__imp__dvd,axiom,
! [B6: real,A3: real] :
( ( dvd_dvd_real @ B6 @ one_one_real )
=> ( dvd_dvd_real @ B6 @ A3 ) ) ).
% idom_class.unit_imp_dvd
thf(fact_630_idom__class_Ounit__imp__dvd,axiom,
! [B6: complex,A3: complex] :
( ( dvd_dvd_complex @ B6 @ one_one_complex )
=> ( dvd_dvd_complex @ B6 @ A3 ) ) ).
% idom_class.unit_imp_dvd
thf(fact_631_gcd__nat_Oextremum,axiom,
! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_632_gcd__nat_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
& ( zero_zero_nat != A3 ) ) ).
% gcd_nat.extremum_strict
thf(fact_633_gcd__nat_Oextremum__unique,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
= ( A3 = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_634_gcd__nat_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ( dvd_dvd_nat @ A3 @ zero_zero_nat )
& ( A3 != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_635_gcd__nat_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
=> ( A3 = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_636_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( A3 != zero_zero_complex )
=> ( ( dvd_dvd_complex @ ( times_times_complex @ B6 @ A3 ) @ ( times_times_complex @ C2 @ A3 ) )
= ( dvd_dvd_complex @ B6 @ C2 ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_637_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A3: real,B6: real,C2: real] :
( ( A3 != zero_zero_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ B6 @ A3 ) @ ( times_times_real @ C2 @ A3 ) )
= ( dvd_dvd_real @ B6 @ C2 ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_638_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( A3 != zero_zero_complex )
=> ( ( dvd_dvd_complex @ ( times_times_complex @ A3 @ B6 ) @ ( times_times_complex @ A3 @ C2 ) )
= ( dvd_dvd_complex @ B6 @ C2 ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_639_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A3: real,B6: real,C2: real] :
( ( A3 != zero_zero_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A3 @ B6 ) @ ( times_times_real @ A3 @ C2 ) )
= ( dvd_dvd_real @ B6 @ C2 ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_640_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B6 ) @ C2 )
= ( dvd_dvd_nat @ B6 @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_641_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( dvd_dvd_complex @ A3 @ one_one_complex )
=> ( ( dvd_dvd_complex @ ( times_times_complex @ A3 @ B6 ) @ C2 )
= ( dvd_dvd_complex @ B6 @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_642_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A3: real,B6: real,C2: real] :
( ( dvd_dvd_real @ A3 @ one_one_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A3 @ B6 ) @ C2 )
= ( dvd_dvd_real @ B6 @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_643_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B6: nat,A3: nat,C2: nat] :
( ( dvd_dvd_nat @ B6 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B6 ) @ C2 )
= ( dvd_dvd_nat @ A3 @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_644_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B6: complex,A3: complex,C2: complex] :
( ( dvd_dvd_complex @ B6 @ one_one_complex )
=> ( ( dvd_dvd_complex @ ( times_times_complex @ A3 @ B6 ) @ C2 )
= ( dvd_dvd_complex @ A3 @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_645_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B6: real,A3: real,C2: real] :
( ( dvd_dvd_real @ B6 @ one_one_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A3 @ B6 ) @ C2 )
= ( dvd_dvd_real @ A3 @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_646_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A3: nat,B6: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B6 ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A3 @ one_one_nat )
& ( dvd_dvd_nat @ B6 @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_647_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A3: complex,B6: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A3 @ B6 ) @ one_one_complex )
= ( ( dvd_dvd_complex @ A3 @ one_one_complex )
& ( dvd_dvd_complex @ B6 @ one_one_complex ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_648_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A3: real,B6: real] :
( ( dvd_dvd_real @ ( times_times_real @ A3 @ B6 ) @ one_one_real )
= ( ( dvd_dvd_real @ A3 @ one_one_real )
& ( dvd_dvd_real @ B6 @ one_one_real ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_649_idom__class_Ounit__mult__right__cancel,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( dvd_dvd_complex @ A3 @ one_one_complex )
=> ( ( ( times_times_complex @ B6 @ A3 )
= ( times_times_complex @ C2 @ A3 ) )
= ( B6 = C2 ) ) ) ).
% idom_class.unit_mult_right_cancel
thf(fact_650_idom__class_Ounit__mult__right__cancel,axiom,
! [A3: real,B6: real,C2: real] :
( ( dvd_dvd_real @ A3 @ one_one_real )
=> ( ( ( times_times_real @ B6 @ A3 )
= ( times_times_real @ C2 @ A3 ) )
= ( B6 = C2 ) ) ) ).
% idom_class.unit_mult_right_cancel
thf(fact_651_idom__class_Ounit__mult__left__cancel,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( dvd_dvd_complex @ A3 @ one_one_complex )
=> ( ( ( times_times_complex @ A3 @ B6 )
= ( times_times_complex @ A3 @ C2 ) )
= ( B6 = C2 ) ) ) ).
% idom_class.unit_mult_left_cancel
thf(fact_652_idom__class_Ounit__mult__left__cancel,axiom,
! [A3: real,B6: real,C2: real] :
( ( dvd_dvd_real @ A3 @ one_one_real )
=> ( ( ( times_times_real @ A3 @ B6 )
= ( times_times_real @ A3 @ C2 ) )
= ( B6 = C2 ) ) ) ).
% idom_class.unit_mult_left_cancel
thf(fact_653_dvd__field__iff,axiom,
( dvd_dvd_complex
= ( ^ [A5: complex,B7: complex] :
( ( A5 = zero_zero_complex )
=> ( B7 = zero_zero_complex ) ) ) ) ).
% dvd_field_iff
thf(fact_654_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A5: real,B7: real] :
( ( A5 = zero_zero_real )
=> ( B7 = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_655_lowner__le__traceI,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ! [Rho: mat_complex] :
( ( member_mat_complex @ Rho @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple1169154605998056944erator @ Rho )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ Rho ) ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B @ Rho ) ) ) ) )
=> ( complex_lowner_le @ A @ B ) ) ) ) ).
% lowner_le_traceI
thf(fact_656_lowner__le__traceD,axiom,
! [A: mat_complex,N: nat,B: mat_complex,Rho2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ Rho2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ A @ B )
=> ( ( comple1169154605998056944erator @ Rho2 )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ Rho2 ) ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B @ Rho2 ) ) ) ) ) ) ) ) ).
% lowner_le_traceD
thf(fact_657_lowner__le__trace,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ A @ B )
= ( ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple1169154605998056944erator @ X2 )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ X2 ) ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B @ X2 ) ) ) ) ) ) ) ) ) ).
% lowner_le_trace
thf(fact_658_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_659_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_660_verit__la__disequality,axiom,
! [A3: nat,B6: nat] :
( ( A3 = B6 )
| ~ ( ord_less_eq_nat @ A3 @ B6 )
| ~ ( ord_less_eq_nat @ B6 @ A3 ) ) ).
% verit_la_disequality
thf(fact_661_verit__la__disequality,axiom,
! [A3: real,B6: real] :
( ( A3 = B6 )
| ~ ( ord_less_eq_real @ A3 @ B6 )
| ~ ( ord_less_eq_real @ B6 @ A3 ) ) ).
% verit_la_disequality
thf(fact_662_verit__comp__simplify1_I2_J,axiom,
! [A3: complex] : ( ord_less_eq_complex @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_663_verit__comp__simplify1_I2_J,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_664_verit__comp__simplify1_I2_J,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_665_verit__eq__simplify_I6_J,axiom,
! [X: complex,Y: complex] :
( ( X = Y )
=> ( ord_less_eq_complex @ X @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_666_verit__eq__simplify_I6_J,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_667_verit__eq__simplify_I6_J,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_668_verit__comp__simplify1_I3_J,axiom,
! [B10: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B10 @ A6 ) )
= ( ord_less_nat @ A6 @ B10 ) ) ).
% verit_comp_simplify1(3)
thf(fact_669_verit__comp__simplify1_I3_J,axiom,
! [B10: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B10 @ A6 ) )
= ( ord_less_real @ A6 @ B10 ) ) ).
% verit_comp_simplify1(3)
thf(fact_670_zero__order_I2_J,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% zero_order(2)
thf(fact_671_zero__order_I1_J,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_order(1)
thf(fact_672_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_673_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_674_zero__compare__simps_I8_J,axiom,
! [A3: real,B6: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ B6 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B6 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B6 ) ) ) ) ).
% zero_compare_simps(8)
thf(fact_675_zero__compare__simps_I4_J,axiom,
! [A3: real,B6: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B6 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B6 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B6 @ zero_zero_real ) ) ) ) ).
% zero_compare_simps(4)
thf(fact_676_mult__sign__intros_I4_J,axiom,
! [A3: complex,B6: complex] :
( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B6 @ zero_zero_complex )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B6 ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_677_mult__sign__intros_I4_J,axiom,
! [A3: real,B6: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B6 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B6 ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_678_mult__sign__intros_I3_J,axiom,
! [A3: complex,B6: complex] :
( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B6 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B6 ) @ zero_zero_complex ) ) ) ).
% mult_sign_intros(3)
thf(fact_679_mult__sign__intros_I3_J,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B6 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B6 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(3)
thf(fact_680_mult__sign__intros_I3_J,axiom,
! [A3: real,B6: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B6 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B6 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(3)
thf(fact_681_mult__sign__intros_I2_J,axiom,
! [A3: complex,B6: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ B6 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B6 ) @ zero_zero_complex ) ) ) ).
% mult_sign_intros(2)
thf(fact_682_mult__sign__intros_I2_J,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B6 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B6 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(2)
thf(fact_683_mult__sign__intros_I2_J,axiom,
! [A3: real,B6: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B6 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B6 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(2)
thf(fact_684_mult__sign__intros_I1_J,axiom,
! [A3: complex,B6: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B6 )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B6 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_685_mult__sign__intros_I1_J,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B6 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B6 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_686_mult__sign__intros_I1_J,axiom,
! [A3: real,B6: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B6 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B6 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_687_mult__mono,axiom,
! [A3: complex,B6: complex,C2: complex,D: complex] :
( ( ord_less_eq_complex @ A3 @ B6 )
=> ( ( ord_less_eq_complex @ C2 @ D )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B6 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B6 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_688_mult__mono,axiom,
! [A3: nat,B6: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B6 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_689_mult__mono,axiom,
! [A3: real,B6: real,C2: real,D: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_690_mult__mono_H,axiom,
! [A3: complex,B6: complex,C2: complex,D: complex] :
( ( ord_less_eq_complex @ A3 @ B6 )
=> ( ( ord_less_eq_complex @ C2 @ D )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B6 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_691_mult__mono_H,axiom,
! [A3: nat,B6: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B6 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_692_mult__mono_H,axiom,
! [A3: real,B6: real,C2: real,D: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_693_zero__le__square,axiom,
! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_694_split__mult__pos__le,axiom,
! [A3: complex,B6: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
& ( ord_less_eq_complex @ zero_zero_complex @ B6 ) )
| ( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
& ( ord_less_eq_complex @ B6 @ zero_zero_complex ) ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B6 ) ) ) ).
% split_mult_pos_le
thf(fact_695_split__mult__pos__le,axiom,
! [A3: real,B6: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B6 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B6 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B6 ) ) ) ).
% split_mult_pos_le
thf(fact_696_mult__left__mono__neg,axiom,
! [B6: complex,A3: complex,C2: complex] :
( ( ord_less_eq_complex @ B6 @ A3 )
=> ( ( ord_less_eq_complex @ C2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B6 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_697_mult__left__mono__neg,axiom,
! [B6: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B6 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_698_mult__left__mono,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B6 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B6 ) ) ) ) ).
% mult_left_mono
thf(fact_699_mult__left__mono,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B6 ) ) ) ) ).
% mult_left_mono
thf(fact_700_mult__left__mono,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) ) ) ) ).
% mult_left_mono
thf(fact_701_mult__right__mono__neg,axiom,
! [B6: complex,A3: complex,C2: complex] :
( ( ord_less_eq_complex @ B6 @ A3 )
=> ( ( ord_less_eq_complex @ C2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B6 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_702_mult__right__mono__neg,axiom,
! [B6: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B6 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_703_mult__right__mono,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B6 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B6 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_704_mult__right__mono,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B6 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_705_mult__right__mono,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_706_split__mult__neg__le,axiom,
! [A3: complex,B6: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
& ( ord_less_eq_complex @ B6 @ zero_zero_complex ) )
| ( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
& ( ord_less_eq_complex @ zero_zero_complex @ B6 ) ) )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B6 ) @ zero_zero_complex ) ) ).
% split_mult_neg_le
thf(fact_707_split__mult__neg__le,axiom,
! [A3: nat,B6: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
& ( ord_less_eq_nat @ B6 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B6 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B6 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_708_split__mult__neg__le,axiom,
! [A3: real,B6: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B6 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B6 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B6 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_709_mult__nonneg__nonpos2,axiom,
! [A3: complex,B6: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ B6 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ B6 @ A3 ) @ zero_zero_complex ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_710_mult__nonneg__nonpos2,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B6 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B6 @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_711_mult__nonneg__nonpos2,axiom,
! [A3: real,B6: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B6 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B6 @ A3 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_712_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B6 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B6 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_713_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B6 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_714_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_715_rel__simps_I45_J,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% rel_simps(45)
thf(fact_716_rel__simps_I45_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% rel_simps(45)
thf(fact_717_rel__simps_I44_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% rel_simps(44)
thf(fact_718_rel__simps_I44_J,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% rel_simps(44)
thf(fact_719_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_720_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_721_field__le__mult__one__interval,axiom,
! [X: real,Y: real] :
( ! [Z2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ Z2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ Y ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_722_mult__le__cancel__iff1,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_723_mult__le__cancel__iff2,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_724_mult__le__cancel__left,axiom,
! [C2: real,A3: real,B6: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B6 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B6 @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_725_mult__le__cancel__right,axiom,
! [A3: real,C2: real,B6: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B6 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B6 @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_726_mult__left__less__imp__less,axiom,
! [C2: nat,A3: nat,B6: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B6 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A3 @ B6 ) ) ) ).
% mult_left_less_imp_less
thf(fact_727_mult__left__less__imp__less,axiom,
! [C2: real,A3: real,B6: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B6 ) ) ) ).
% mult_left_less_imp_less
thf(fact_728_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: nat,B6: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B6 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_729_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: real,B6: real,C2: real,D: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ B6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_730_mult__less__cancel__left,axiom,
! [C2: real,A3: real,B6: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B6 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B6 @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_731_mult__right__less__imp__less,axiom,
! [A3: nat,C2: nat,B6: nat] :
( ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B6 @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A3 @ B6 ) ) ) ).
% mult_right_less_imp_less
thf(fact_732_mult__right__less__imp__less,axiom,
! [A3: real,C2: real,B6: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ C2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B6 ) ) ) ).
% mult_right_less_imp_less
thf(fact_733_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: nat,B6: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B6 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_734_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: real,B6: real,C2: real,D: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_735_mult__less__cancel__right,axiom,
! [A3: real,C2: real,B6: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B6 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B6 @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_736_mult__le__cancel__left__neg,axiom,
! [C2: real,A3: real,B6: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) )
= ( ord_less_eq_real @ B6 @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_737_mult__le__cancel__left__pos,axiom,
! [C2: real,A3: real,B6: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) )
= ( ord_less_eq_real @ A3 @ B6 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_738_mult__left__le__imp__le,axiom,
! [C2: nat,A3: nat,B6: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B6 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A3 @ B6 ) ) ) ).
% mult_left_le_imp_le
thf(fact_739_mult__left__le__imp__le,axiom,
! [C2: real,A3: real,B6: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B6 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B6 ) ) ) ).
% mult_left_le_imp_le
thf(fact_740_mult__right__le__imp__le,axiom,
! [A3: nat,C2: nat,B6: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B6 @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A3 @ B6 ) ) ) ).
% mult_right_le_imp_le
thf(fact_741_mult__right__le__imp__le,axiom,
! [A3: real,C2: real,B6: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ C2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B6 ) ) ) ).
% mult_right_le_imp_le
thf(fact_742_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: nat,B6: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B6 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_743_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: real,B6: real,C2: real,D: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_744_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: nat,B6: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B6 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_745_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: real,B6: real,C2: real,D: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B6 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_746_mult__left__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_747_mult__right__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_748_mult__le__one,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B6 )
=> ( ( ord_less_eq_nat @ B6 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B6 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_749_mult__le__one,axiom,
! [A3: real,B6: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B6 )
=> ( ( ord_less_eq_real @ B6 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B6 ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_750_mult__left__le,axiom,
! [C2: nat,A3: nat] :
( ( ord_less_eq_nat @ C2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_751_mult__left__le,axiom,
! [C2: real,A3: real] :
( ( ord_less_eq_real @ C2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_752_linordered__field__no__lb,axiom,
! [X4: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X4 ) ).
% linordered_field_no_lb
thf(fact_753_linordered__field__no__ub,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_754_trace__adjoint__positive,axiom,
! [A: mat_complex] : ( ord_less_eq_complex @ zero_zero_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ ( schur_5982229384592763574omplex @ A ) ) ) ) ).
% trace_adjoint_positive
thf(fact_755_lowner__le__imp__trace__le,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ A @ B )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ A ) @ ( comple3184165445352484367omplex @ B ) ) ) ) ) ).
% lowner_le_imp_trace_le
thf(fact_756_lowner__le__smultc,axiom,
! [C2: complex,A: mat_complex,B: mat_complex,N: nat] :
( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ( complex_lowner_le @ A @ B )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( complex_lowner_le @ ( smult_mat_complex @ C2 @ A ) @ ( smult_mat_complex @ C2 @ B ) ) ) ) ) ) ).
% lowner_le_smultc
thf(fact_757_mult__le__cancel__left1,axiom,
! [C2: real,B6: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ C2 @ B6 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B6 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B6 @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_758_mult__le__cancel__left2,axiom,
! [C2: real,A3: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A3 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_759_mult__le__cancel__right1,axiom,
! [C2: real,B6: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ B6 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B6 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B6 @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_760_mult__le__cancel__right2,axiom,
! [A3: real,C2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A3 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_761_mult__less__cancel__left1,axiom,
! [C2: real,B6: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ C2 @ B6 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B6 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B6 @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_762_mult__less__cancel__left2,axiom,
! [C2: real,A3: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A3 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_763_mult__less__cancel__right1,axiom,
! [C2: real,B6: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ B6 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B6 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B6 @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_764_mult__less__cancel__right2,axiom,
! [A3: real,C2: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A3 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_765_mult__eq__1,axiom,
! [A3: complex,B6: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ A3 @ one_one_complex )
=> ( ( ord_less_eq_complex @ B6 @ one_one_complex )
=> ( ( ( times_times_complex @ A3 @ B6 )
= one_one_complex )
= ( ( A3 = one_one_complex )
& ( B6 = one_one_complex ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_766_mult__eq__1,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ B6 @ one_one_nat )
=> ( ( ( times_times_nat @ A3 @ B6 )
= one_one_nat )
= ( ( A3 = one_one_nat )
& ( B6 = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_767_mult__eq__1,axiom,
! [A3: real,B6: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ B6 @ one_one_real )
=> ( ( ( times_times_real @ A3 @ B6 )
= one_one_real )
= ( ( A3 = one_one_real )
& ( B6 = one_one_real ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_768_less__eq__fract__respect,axiom,
! [B6: real,B10: real,D: real,D4: real,A3: real,A6: real,C2: real,C5: real] :
( ( B6 != zero_zero_real )
=> ( ( B10 != zero_zero_real )
=> ( ( D != zero_zero_real )
=> ( ( D4 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ B10 )
= ( times_times_real @ A6 @ B6 ) )
=> ( ( ( times_times_real @ C2 @ D4 )
= ( times_times_real @ C5 @ D ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A3 @ D ) @ ( times_times_real @ B6 @ D ) ) @ ( times_times_real @ ( times_times_real @ C2 @ B6 ) @ ( times_times_real @ B6 @ D ) ) )
= ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A6 @ D4 ) @ ( times_times_real @ B10 @ D4 ) ) @ ( times_times_real @ ( times_times_real @ C5 @ B10 ) @ ( times_times_real @ B10 @ D4 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_769_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_770_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_771_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_772_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_773_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_774_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_775_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_776_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_777_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_778_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_779_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_780_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_781_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_782_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_783_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_784_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_785_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_786_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_787_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_788_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_789_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_790_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_791_positive__proj__trace,axiom,
! [P: mat_complex,R: mat_complex,N: nat] :
( ( linear5633924348262549461omplex @ P )
=> ( ( complex_positive @ R )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ P ) ) ) ) ) ) ) ).
% positive_proj_trace
thf(fact_792_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q2: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I3: nat] :
( ( Q2 @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) )
& ( ord_less_eq_real @ ( X3 @ I3 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X4: nat > real,I4: nat] : ( ord_less_eq_nat @ ( L2 @ X4 @ I4 ) @ one_one_nat )
& ! [X4: nat > real,I4: nat] :
( ( ( P @ X4 )
& ( Q2 @ I4 )
& ( ( X4 @ I4 )
= zero_zero_real ) )
=> ( ( L2 @ X4 @ I4 )
= zero_zero_nat ) )
& ! [X4: nat > real,I4: nat] :
( ( ( P @ X4 )
& ( Q2 @ I4 )
& ( ( X4 @ I4 )
= one_one_real ) )
=> ( ( L2 @ X4 @ I4 )
= one_one_nat ) )
& ! [X4: nat > real,I4: nat] :
( ( ( P @ X4 )
& ( Q2 @ I4 )
& ( ( L2 @ X4 @ I4 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X4 @ I4 ) @ ( F @ X4 @ I4 ) ) )
& ! [X4: nat > real,I4: nat] :
( ( ( P @ X4 )
& ( Q2 @ I4 )
& ( ( L2 @ X4 @ I4 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X4 @ I4 ) @ ( X4 @ I4 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_793_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B6: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B6 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_794_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_795_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_796_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_797_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_798_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_799_positive__is__hermitian,axiom,
! [A: mat_complex] :
( ( complex_positive @ A )
=> ( comple8306762464034002205omplex @ A ) ) ).
% positive_is_hermitian
thf(fact_800_Complex__Matrix_Opositive__zero,axiom,
! [N: nat] : ( complex_positive @ ( zero_mat_complex @ N @ N ) ) ).
% Complex_Matrix.positive_zero
thf(fact_801_positive__dim__eq,axiom,
! [A: mat_complex] :
( ( complex_positive @ A )
=> ( ( dim_row_complex @ A )
= ( dim_col_complex @ A ) ) ) ).
% positive_dim_eq
thf(fact_802_zero__lowner__le__positiveI,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( complex_lowner_le @ ( zero_mat_complex @ N @ N ) @ A ) ) ) ).
% zero_lowner_le_positiveI
thf(fact_803_zero__lowner__le__positiveD,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_lowner_le @ ( zero_mat_complex @ N @ N ) @ A )
=> ( complex_positive @ A ) ) ) ).
% zero_lowner_le_positiveD
thf(fact_804_lowner__le__trans__positiveI,axiom,
! [A: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( ( complex_lowner_le @ A @ B )
=> ( complex_positive @ B ) ) ) ) ).
% lowner_le_trans_positiveI
thf(fact_805_density__operator__def,axiom,
( comple5220265106149225959erator
= ( ^ [A2: mat_complex] :
( ( complex_positive @ A2 )
& ( ( comple3184165445352484367omplex @ A2 )
= one_one_complex ) ) ) ) ).
% density_operator_def
thf(fact_806_positive__if__decomp,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ? [M6: mat_complex] :
( ( times_8009071140041733218omplex @ M6 @ ( schur_5982229384592763574omplex @ M6 ) )
= A )
=> ( complex_positive @ A ) ) ) ).
% positive_if_decomp
thf(fact_807_positive__iff__decomp,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
= ( ? [X2: mat_complex] :
( ( member_mat_complex @ X2 @ ( carrier_mat_complex @ N @ N ) )
& ( ( times_8009071140041733218omplex @ X2 @ ( schur_5982229384592763574omplex @ X2 ) )
= A ) ) ) ) ) ).
% positive_iff_decomp
thf(fact_808_positive__only__if__decomp,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ? [X3: mat_complex] :
( ( member_mat_complex @ X3 @ ( carrier_mat_complex @ N @ N ) )
& ( ( times_8009071140041733218omplex @ X3 @ ( schur_5982229384592763574omplex @ X3 ) )
= A ) ) ) ) ).
% positive_only_if_decomp
thf(fact_809_positive__close__under__left__right__mult__adjoint,axiom,
! [M4: mat_complex,N: nat,A: mat_complex] :
( ( member_mat_complex @ M4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( complex_positive @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ M4 @ A ) @ ( schur_5982229384592763574omplex @ M4 ) ) ) ) ) ) ).
% positive_close_under_left_right_mult_adjoint
thf(fact_810_partial__density__operator__def,axiom,
( comple1169154605998056944erator
= ( ^ [A2: mat_complex] :
( ( complex_positive @ A2 )
& ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ A2 ) @ one_one_complex ) ) ) ) ).
% partial_density_operator_def
thf(fact_811_positive__trace,axiom,
! [A: mat_complex,N: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( comple3184165445352484367omplex @ A ) ) ) ) ).
% positive_trace
thf(fact_812_positive__smult,axiom,
! [A: mat_complex,N: nat,C2: complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( complex_positive @ ( smult_mat_complex @ C2 @ A ) ) ) ) ) ).
% positive_smult
thf(fact_813_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_814_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K3: nat] :
? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( F @ K3 @ I4 ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ! [K4: nat] :
? [K5: nat] :
( ( ord_less_eq_nat @ K4 @ K5 )
& ( F @ K5 @ I3 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_815_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_816_minf_I8_J,axiom,
! [T: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ~ ( ord_less_eq_real @ T @ X4 ) ) ).
% minf(8)
thf(fact_817_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_818_minf_I6_J,axiom,
! [T: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ord_less_eq_real @ X4 @ T ) ) ).
% minf(6)
thf(fact_819_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_820_pinf_I8_J,axiom,
! [T: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_eq_real @ T @ X4 ) ) ).
% pinf(8)
thf(fact_821_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_822_minf_I7_J,axiom,
! [T: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ~ ( ord_less_real @ T @ X4 ) ) ).
% minf(7)
thf(fact_823_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_824_minf_I5_J,axiom,
! [T: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ord_less_real @ X4 @ T ) ) ).
% minf(5)
thf(fact_825_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_826_minf_I4_J,axiom,
! [T: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_827_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_828_minf_I3_J,axiom,
! [T: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_829_minf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q2 @ X4 ) )
= ( ( P3 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_830_minf_I2_J,axiom,
! [P: real > $o,P3: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q2 @ X4 ) )
= ( ( P3 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_831_minf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q2 @ X4 ) )
= ( ( P3 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_832_minf_I1_J,axiom,
! [P: real > $o,P3: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q2 @ X4 ) )
= ( ( P3 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_833_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_834_pinf_I7_J,axiom,
! [T: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_real @ T @ X4 ) ) ).
% pinf(7)
thf(fact_835_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_836_pinf_I5_J,axiom,
! [T: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ~ ( ord_less_real @ X4 @ T ) ) ).
% pinf(5)
thf(fact_837_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_838_pinf_I4_J,axiom,
! [T: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_839_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_840_pinf_I3_J,axiom,
! [T: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_841_pinf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q2 @ X4 ) )
= ( ( P3 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_842_pinf_I2_J,axiom,
! [P: real > $o,P3: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q2 @ X4 ) )
= ( ( P3 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_843_pinf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q2 @ X4 ) )
= ( ( P3 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_844_pinf_I1_J,axiom,
! [P: real > $o,P3: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q2 @ X4 ) )
= ( ( P3 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_845_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_846_pinf_I6_J,axiom,
! [T: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ T ) ) ).
% pinf(6)
thf(fact_847_order__le__imp__less__or__eq,axiom,
! [X: complex,Y: complex] :
( ( ord_less_eq_complex @ X @ Y )
=> ( ( ord_less_complex @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_848_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_849_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_850_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_851_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_852_order__less__le__subst2,axiom,
! [A3: nat,B6: nat,F: nat > complex,C2: complex] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_eq_complex @ ( F @ B6 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_complex @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_853_order__less__le__subst2,axiom,
! [A3: real,B6: real,F: real > complex,C2: complex] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_eq_complex @ ( F @ B6 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_complex @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_854_order__less__le__subst2,axiom,
! [A3: nat,B6: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ ( F @ B6 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_855_order__less__le__subst2,axiom,
! [A3: real,B6: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ ( F @ B6 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_856_order__less__le__subst2,axiom,
! [A3: nat,B6: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_eq_real @ ( F @ B6 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_857_order__less__le__subst2,axiom,
! [A3: real,B6: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_eq_real @ ( F @ B6 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_858_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_859_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_860_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_861_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_862_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_863_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_864_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_865_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_866_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_867_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_868_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_869_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_870_order__less__subst2,axiom,
! [A3: nat,B6: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_nat @ ( F @ B6 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_871_order__less__subst2,axiom,
! [A3: nat,B6: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_real @ ( F @ B6 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_872_order__less__subst2,axiom,
! [A3: real,B6: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_nat @ ( F @ B6 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_873_order__less__subst2,axiom,
! [A3: real,B6: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_real @ ( F @ B6 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_874_order__less__subst1,axiom,
! [A3: nat,F: nat > nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_875_order__less__subst1,axiom,
! [A3: nat,F: real > nat,B6: real,C2: real] :
( ( ord_less_nat @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_real @ B6 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_876_order__less__subst1,axiom,
! [A3: real,F: nat > real,B6: nat,C2: nat] :
( ( ord_less_real @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_877_order__less__subst1,axiom,
! [A3: real,F: real > real,B6: real,C2: real] :
( ( ord_less_real @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_real @ B6 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_878_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_879_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_880_ord__less__eq__subst,axiom,
! [A3: nat,B6: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ( F @ B6 )
= C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_881_ord__less__eq__subst,axiom,
! [A3: nat,B6: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ( F @ B6 )
= C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_882_ord__less__eq__subst,axiom,
! [A3: real,B6: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ( F @ B6 )
= C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_883_ord__less__eq__subst,axiom,
! [A3: real,B6: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ( F @ B6 )
= C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_884_ord__eq__less__subst,axiom,
! [A3: nat,F: nat > nat,B6: nat,C2: nat] :
( ( A3
= ( F @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_885_ord__eq__less__subst,axiom,
! [A3: real,F: nat > real,B6: nat,C2: nat] :
( ( A3
= ( F @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_886_ord__eq__less__subst,axiom,
! [A3: nat,F: real > nat,B6: real,C2: real] :
( ( A3
= ( F @ B6 ) )
=> ( ( ord_less_real @ B6 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_887_ord__eq__less__subst,axiom,
! [A3: real,F: real > real,B6: real,C2: real] :
( ( A3
= ( F @ B6 ) )
=> ( ( ord_less_real @ B6 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_888_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_889_order__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_890_order__less__asym_H,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ~ ( ord_less_nat @ B6 @ A3 ) ) ).
% order_less_asym'
thf(fact_891_order__less__asym_H,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ A3 @ B6 )
=> ~ ( ord_less_real @ B6 @ A3 ) ) ).
% order_less_asym'
thf(fact_892_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_893_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_894_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_895_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_896_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_897_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_898_dual__order_Ostrict__implies__not__eq,axiom,
! [B6: nat,A3: nat] :
( ( ord_less_nat @ B6 @ A3 )
=> ( A3 != B6 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_899_dual__order_Ostrict__implies__not__eq,axiom,
! [B6: real,A3: real] :
( ( ord_less_real @ B6 @ A3 )
=> ( A3 != B6 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_900_order_Ostrict__implies__not__eq,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( A3 != B6 ) ) ).
% order.strict_implies_not_eq
thf(fact_901_order_Ostrict__implies__not__eq,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( A3 != B6 ) ) ).
% order.strict_implies_not_eq
thf(fact_902_dual__order_Ostrict__trans,axiom,
! [B6: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B6 @ A3 )
=> ( ( ord_less_nat @ C2 @ B6 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_903_dual__order_Ostrict__trans,axiom,
! [B6: real,A3: real,C2: real] :
( ( ord_less_real @ B6 @ A3 )
=> ( ( ord_less_real @ C2 @ B6 )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_904_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_905_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_906_order_Ostrict__trans,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_907_order_Ostrict__trans,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_real @ B6 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_908_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B6: nat] :
( ! [A7: nat,B11: nat] :
( ( ord_less_nat @ A7 @ B11 )
=> ( P @ A7 @ B11 ) )
=> ( ! [A7: nat] : ( P @ A7 @ A7 )
=> ( ! [A7: nat,B11: nat] :
( ( P @ B11 @ A7 )
=> ( P @ A7 @ B11 ) )
=> ( P @ A3 @ B6 ) ) ) ) ).
% linorder_less_wlog
thf(fact_909_linorder__less__wlog,axiom,
! [P: real > real > $o,A3: real,B6: real] :
( ! [A7: real,B11: real] :
( ( ord_less_real @ A7 @ B11 )
=> ( P @ A7 @ B11 ) )
=> ( ! [A7: real] : ( P @ A7 @ A7 )
=> ( ! [A7: real,B11: real] :
( ( P @ B11 @ A7 )
=> ( P @ A7 @ B11 ) )
=> ( P @ A3 @ B6 ) ) ) ) ).
% linorder_less_wlog
thf(fact_910_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X5: nat] : ( P5 @ X5 ) )
= ( ^ [P6: nat > $o] :
? [N4: nat] :
( ( P6 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P6 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_911_dual__order_Oirrefl,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_912_dual__order_Oirrefl,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_913_dual__order_Oasym,axiom,
! [B6: nat,A3: nat] :
( ( ord_less_nat @ B6 @ A3 )
=> ~ ( ord_less_nat @ A3 @ B6 ) ) ).
% dual_order.asym
thf(fact_914_dual__order_Oasym,axiom,
! [B6: real,A3: real] :
( ( ord_less_real @ B6 @ A3 )
=> ~ ( ord_less_real @ A3 @ B6 ) ) ).
% dual_order.asym
thf(fact_915_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_916_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_917_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_918_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_919_less__induct,axiom,
! [P: nat > $o,A3: nat] :
( ! [X3: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X3 )
=> ( P @ Y3 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ).
% less_induct
thf(fact_920_ord__less__eq__trans,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_921_ord__less__eq__trans,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_922_ord__eq__less__trans,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( A3 = B6 )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_923_ord__eq__less__trans,axiom,
! [A3: real,B6: real,C2: real] :
( ( A3 = B6 )
=> ( ( ord_less_real @ B6 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_924_order_Oasym,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ~ ( ord_less_nat @ B6 @ A3 ) ) ).
% order.asym
thf(fact_925_order_Oasym,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ A3 @ B6 )
=> ~ ( ord_less_real @ B6 @ A3 ) ) ).
% order.asym
thf(fact_926_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_927_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_928_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z2: real] :
( ( ord_less_real @ X @ Z2 )
& ( ord_less_real @ Z2 @ Y ) ) ) ).
% dense
thf(fact_929_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_930_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_931_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_932_order__trans__rules_I22_J,axiom,
! [X: complex,Y: complex,Z: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ( ord_less_eq_complex @ Y @ Z )
=> ( ord_less_complex @ X @ Z ) ) ) ).
% order_trans_rules(22)
thf(fact_933_order__trans__rules_I22_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_trans_rules(22)
thf(fact_934_order__trans__rules_I22_J,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_trans_rules(22)
thf(fact_935_order__trans__rules_I21_J,axiom,
! [X: complex,Y: complex,Z: complex] :
( ( ord_less_eq_complex @ X @ Y )
=> ( ( ord_less_complex @ Y @ Z )
=> ( ord_less_complex @ X @ Z ) ) ) ).
% order_trans_rules(21)
thf(fact_936_order__trans__rules_I21_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_trans_rules(21)
thf(fact_937_order__trans__rules_I21_J,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_trans_rules(21)
thf(fact_938_order__trans__rules_I18_J,axiom,
! [A3: complex,B6: complex] :
( ( ord_less_eq_complex @ A3 @ B6 )
=> ( ( A3 != B6 )
=> ( ord_less_complex @ A3 @ B6 ) ) ) ).
% order_trans_rules(18)
thf(fact_939_order__trans__rules_I18_J,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( A3 != B6 )
=> ( ord_less_nat @ A3 @ B6 ) ) ) ).
% order_trans_rules(18)
thf(fact_940_order__trans__rules_I18_J,axiom,
! [A3: real,B6: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( A3 != B6 )
=> ( ord_less_real @ A3 @ B6 ) ) ) ).
% order_trans_rules(18)
thf(fact_941_order__trans__rules_I17_J,axiom,
! [A3: complex,B6: complex] :
( ( A3 != B6 )
=> ( ( ord_less_eq_complex @ A3 @ B6 )
=> ( ord_less_complex @ A3 @ B6 ) ) ) ).
% order_trans_rules(17)
thf(fact_942_order__trans__rules_I17_J,axiom,
! [A3: nat,B6: nat] :
( ( A3 != B6 )
=> ( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ord_less_nat @ A3 @ B6 ) ) ) ).
% order_trans_rules(17)
thf(fact_943_order__trans__rules_I17_J,axiom,
! [A3: real,B6: real] :
( ( A3 != B6 )
=> ( ( ord_less_eq_real @ A3 @ B6 )
=> ( ord_less_real @ A3 @ B6 ) ) ) ).
% order_trans_rules(17)
thf(fact_944_order__trans__rules_I6_J,axiom,
! [A3: complex,F: complex > complex,B6: complex,C2: complex] :
( ( ord_less_complex @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_eq_complex @ B6 @ C2 )
=> ( ! [X3: complex,Y2: complex] :
( ( ord_less_eq_complex @ X3 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_945_order__trans__rules_I6_J,axiom,
! [A3: nat,F: complex > nat,B6: complex,C2: complex] :
( ( ord_less_nat @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_eq_complex @ B6 @ C2 )
=> ( ! [X3: complex,Y2: complex] :
( ( ord_less_eq_complex @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_946_order__trans__rules_I6_J,axiom,
! [A3: real,F: complex > real,B6: complex,C2: complex] :
( ( ord_less_real @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_eq_complex @ B6 @ C2 )
=> ( ! [X3: complex,Y2: complex] :
( ( ord_less_eq_complex @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_947_order__trans__rules_I6_J,axiom,
! [A3: complex,F: nat > complex,B6: nat,C2: nat] :
( ( ord_less_complex @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_948_order__trans__rules_I6_J,axiom,
! [A3: nat,F: nat > nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_949_order__trans__rules_I6_J,axiom,
! [A3: real,F: nat > real,B6: nat,C2: nat] :
( ( ord_less_real @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_950_order__trans__rules_I6_J,axiom,
! [A3: complex,F: real > complex,B6: real,C2: real] :
( ( ord_less_complex @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_951_order__trans__rules_I6_J,axiom,
! [A3: nat,F: real > nat,B6: real,C2: real] :
( ( ord_less_nat @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_952_order__trans__rules_I6_J,axiom,
! [A3: real,F: real > real,B6: real,C2: real] :
( ( ord_less_real @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_953_order__trans__rules_I4_J,axiom,
! [A3: complex,F: nat > complex,B6: nat,C2: nat] :
( ( ord_less_eq_complex @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_complex @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_954_order__trans__rules_I4_J,axiom,
! [A3: complex,F: real > complex,B6: real,C2: real] :
( ( ord_less_eq_complex @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_real @ B6 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_complex @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_955_order__trans__rules_I4_J,axiom,
! [A3: nat,F: nat > nat,B6: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_956_order__trans__rules_I4_J,axiom,
! [A3: nat,F: real > nat,B6: real,C2: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_real @ B6 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_957_order__trans__rules_I4_J,axiom,
! [A3: real,F: nat > real,B6: nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_958_order__trans__rules_I4_J,axiom,
! [A3: real,F: real > real,B6: real,C2: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B6 ) )
=> ( ( ord_less_real @ B6 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_959_order__trans__rules_I3_J,axiom,
! [A3: complex,B6: complex,F: complex > complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B6 )
=> ( ( ord_less_complex @ ( F @ B6 ) @ C2 )
=> ( ! [X3: complex,Y2: complex] :
( ( ord_less_eq_complex @ X3 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_960_order__trans__rules_I3_J,axiom,
! [A3: complex,B6: complex,F: complex > nat,C2: nat] :
( ( ord_less_eq_complex @ A3 @ B6 )
=> ( ( ord_less_nat @ ( F @ B6 ) @ C2 )
=> ( ! [X3: complex,Y2: complex] :
( ( ord_less_eq_complex @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_961_order__trans__rules_I3_J,axiom,
! [A3: complex,B6: complex,F: complex > real,C2: real] :
( ( ord_less_eq_complex @ A3 @ B6 )
=> ( ( ord_less_real @ ( F @ B6 ) @ C2 )
=> ( ! [X3: complex,Y2: complex] :
( ( ord_less_eq_complex @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_962_order__trans__rules_I3_J,axiom,
! [A3: nat,B6: nat,F: nat > complex,C2: complex] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_complex @ ( F @ B6 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_963_order__trans__rules_I3_J,axiom,
! [A3: nat,B6: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_nat @ ( F @ B6 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_964_order__trans__rules_I3_J,axiom,
! [A3: nat,B6: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_real @ ( F @ B6 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_965_order__trans__rules_I3_J,axiom,
! [A3: real,B6: real,F: real > complex,C2: complex] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_complex @ ( F @ B6 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_complex @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_966_order__trans__rules_I3_J,axiom,
! [A3: real,B6: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_nat @ ( F @ B6 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_967_order__trans__rules_I3_J,axiom,
! [A3: real,B6: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_real @ ( F @ B6 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_968_leD,axiom,
! [Y: complex,X: complex] :
( ( ord_less_eq_complex @ Y @ X )
=> ~ ( ord_less_complex @ X @ Y ) ) ).
% leD
thf(fact_969_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_970_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_971_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_972_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_973_le__less,axiom,
( ord_less_eq_complex
= ( ^ [X2: complex,Y4: complex] :
( ( ord_less_complex @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% le_less
thf(fact_974_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% le_less
thf(fact_975_le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% le_less
thf(fact_976_less__le,axiom,
( ord_less_complex
= ( ^ [X2: complex,Y4: complex] :
( ( ord_less_eq_complex @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% less_le
thf(fact_977_less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% less_le
thf(fact_978_less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% less_le
thf(fact_979_nless__le,axiom,
! [A3: complex,B6: complex] :
( ( ~ ( ord_less_complex @ A3 @ B6 ) )
= ( ~ ( ord_less_eq_complex @ A3 @ B6 )
| ( A3 = B6 ) ) ) ).
% nless_le
thf(fact_980_nless__le,axiom,
! [A3: nat,B6: nat] :
( ( ~ ( ord_less_nat @ A3 @ B6 ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B6 )
| ( A3 = B6 ) ) ) ).
% nless_le
thf(fact_981_nless__le,axiom,
! [A3: real,B6: real] :
( ( ~ ( ord_less_real @ A3 @ B6 ) )
= ( ~ ( ord_less_eq_real @ A3 @ B6 )
| ( A3 = B6 ) ) ) ).
% nless_le
thf(fact_982_not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% not_le
thf(fact_983_not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% not_le
thf(fact_984_not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% not_less
thf(fact_985_not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% not_less
thf(fact_986_antisym__conv1,axiom,
! [X: complex,Y: complex] :
( ~ ( ord_less_complex @ X @ Y )
=> ( ( ord_less_eq_complex @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_987_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_988_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_989_antisym__conv2,axiom,
! [X: complex,Y: complex] :
( ( ord_less_eq_complex @ X @ Y )
=> ( ( ~ ( ord_less_complex @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_990_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_991_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_992_less__imp__le,axiom,
! [X: complex,Y: complex] :
( ( ord_less_complex @ X @ Y )
=> ( ord_less_eq_complex @ X @ Y ) ) ).
% less_imp_le
thf(fact_993_less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% less_imp_le
thf(fact_994_less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% less_imp_le
thf(fact_995_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_996_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_997_less__le__not__le,axiom,
( ord_less_complex
= ( ^ [X2: complex,Y4: complex] :
( ( ord_less_eq_complex @ X2 @ Y4 )
& ~ ( ord_less_eq_complex @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_998_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_999_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1000_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1001_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1002_order_Oorder__iff__strict,axiom,
( ord_less_eq_complex
= ( ^ [A5: complex,B7: complex] :
( ( ord_less_complex @ A5 @ B7 )
| ( A5 = B7 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1003_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B7: nat] :
( ( ord_less_nat @ A5 @ B7 )
| ( A5 = B7 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1004_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B7: real] :
( ( ord_less_real @ A5 @ B7 )
| ( A5 = B7 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1005_order_Ostrict__iff__order,axiom,
( ord_less_complex
= ( ^ [A5: complex,B7: complex] :
( ( ord_less_eq_complex @ A5 @ B7 )
& ( A5 != B7 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1006_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B7: nat] :
( ( ord_less_eq_nat @ A5 @ B7 )
& ( A5 != B7 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1007_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B7: real] :
( ( ord_less_eq_real @ A5 @ B7 )
& ( A5 != B7 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1008_order_Ostrict__trans1,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B6 )
=> ( ( ord_less_complex @ B6 @ C2 )
=> ( ord_less_complex @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1009_order_Ostrict__trans1,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_nat @ B6 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1010_order_Ostrict__trans1,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_real @ B6 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1011_order_Ostrict__trans2,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( ord_less_complex @ A3 @ B6 )
=> ( ( ord_less_eq_complex @ B6 @ C2 )
=> ( ord_less_complex @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1012_order_Ostrict__trans2,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1013_order_Ostrict__trans2,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1014_order_Ostrict__iff__not,axiom,
( ord_less_complex
= ( ^ [A5: complex,B7: complex] :
( ( ord_less_eq_complex @ A5 @ B7 )
& ~ ( ord_less_eq_complex @ B7 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1015_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B7: nat] :
( ( ord_less_eq_nat @ A5 @ B7 )
& ~ ( ord_less_eq_nat @ B7 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1016_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B7: real] :
( ( ord_less_eq_real @ A5 @ B7 )
& ~ ( ord_less_eq_real @ B7 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1017_dense__ge__bounded,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_1018_dense__le__bounded,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_1019_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_complex
= ( ^ [B7: complex,A5: complex] :
( ( ord_less_complex @ B7 @ A5 )
| ( A5 = B7 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1020_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B7: nat,A5: nat] :
( ( ord_less_nat @ B7 @ A5 )
| ( A5 = B7 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1021_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B7: real,A5: real] :
( ( ord_less_real @ B7 @ A5 )
| ( A5 = B7 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1022_dual__order_Ostrict__iff__order,axiom,
( ord_less_complex
= ( ^ [B7: complex,A5: complex] :
( ( ord_less_eq_complex @ B7 @ A5 )
& ( A5 != B7 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1023_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B7: nat,A5: nat] :
( ( ord_less_eq_nat @ B7 @ A5 )
& ( A5 != B7 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1024_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B7: real,A5: real] :
( ( ord_less_eq_real @ B7 @ A5 )
& ( A5 != B7 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1025_dual__order_Ostrict__trans1,axiom,
! [B6: complex,A3: complex,C2: complex] :
( ( ord_less_eq_complex @ B6 @ A3 )
=> ( ( ord_less_complex @ C2 @ B6 )
=> ( ord_less_complex @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1026_dual__order_Ostrict__trans1,axiom,
! [B6: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B6 @ A3 )
=> ( ( ord_less_nat @ C2 @ B6 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1027_dual__order_Ostrict__trans1,axiom,
! [B6: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B6 @ A3 )
=> ( ( ord_less_real @ C2 @ B6 )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1028_dual__order_Ostrict__trans2,axiom,
! [B6: complex,A3: complex,C2: complex] :
( ( ord_less_complex @ B6 @ A3 )
=> ( ( ord_less_eq_complex @ C2 @ B6 )
=> ( ord_less_complex @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1029_dual__order_Ostrict__trans2,axiom,
! [B6: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B6 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B6 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1030_dual__order_Ostrict__trans2,axiom,
! [B6: real,A3: real,C2: real] :
( ( ord_less_real @ B6 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ B6 )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1031_dual__order_Ostrict__iff__not,axiom,
( ord_less_complex
= ( ^ [B7: complex,A5: complex] :
( ( ord_less_eq_complex @ B7 @ A5 )
& ~ ( ord_less_eq_complex @ A5 @ B7 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1032_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B7: nat,A5: nat] :
( ( ord_less_eq_nat @ B7 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B7 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1033_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B7: real,A5: real] :
( ( ord_less_eq_real @ B7 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B7 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1034_order_Ostrict__implies__order,axiom,
! [A3: complex,B6: complex] :
( ( ord_less_complex @ A3 @ B6 )
=> ( ord_less_eq_complex @ A3 @ B6 ) ) ).
% order.strict_implies_order
thf(fact_1035_order_Ostrict__implies__order,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ord_less_eq_nat @ A3 @ B6 ) ) ).
% order.strict_implies_order
thf(fact_1036_order_Ostrict__implies__order,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ord_less_eq_real @ A3 @ B6 ) ) ).
% order.strict_implies_order
thf(fact_1037_dual__order_Ostrict__implies__order,axiom,
! [B6: complex,A3: complex] :
( ( ord_less_complex @ B6 @ A3 )
=> ( ord_less_eq_complex @ B6 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_1038_dual__order_Ostrict__implies__order,axiom,
! [B6: nat,A3: nat] :
( ( ord_less_nat @ B6 @ A3 )
=> ( ord_less_eq_nat @ B6 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_1039_dual__order_Ostrict__implies__order,axiom,
! [B6: real,A3: real] :
( ( ord_less_real @ B6 @ A3 )
=> ( ord_less_eq_real @ B6 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_1040_complete__interval,axiom,
! [A3: nat,B6: nat,P: nat > $o] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B6 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A3 @ C3 )
& ( ord_less_eq_nat @ C3 @ B6 )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A3 @ X4 )
& ( ord_less_nat @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D5: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A3 @ X3 )
& ( ord_less_nat @ X3 @ D5 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D5 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1041_complete__interval,axiom,
! [A3: real,B6: real,P: real > $o] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B6 )
=> ? [C3: real] :
( ( ord_less_eq_real @ A3 @ C3 )
& ( ord_less_eq_real @ C3 @ B6 )
& ! [X4: real] :
( ( ( ord_less_eq_real @ A3 @ X4 )
& ( ord_less_real @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D5: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A3 @ X3 )
& ( ord_less_real @ X3 @ D5 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D5 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1042_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X2 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_1043_ex__gt__or__lt,axiom,
! [A3: real] :
? [B11: real] :
( ( ord_less_real @ A3 @ B11 )
| ( ord_less_real @ B11 @ A3 ) ) ).
% ex_gt_or_lt
thf(fact_1044_lowner__le__smult,axiom,
! [C2: real,A: mat_complex,B: mat_complex,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( complex_lowner_le @ A @ B )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( complex_lowner_le @ ( smult_mat_complex @ ( real_V4546457046886955230omplex @ C2 ) @ A ) @ ( smult_mat_complex @ ( real_V4546457046886955230omplex @ C2 ) @ B ) ) ) ) ) ) ).
% lowner_le_smult
thf(fact_1045_of__real__hom_Ohom__0__iff,axiom,
! [X: real] :
( ( ( real_V1803761363581548252l_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% of_real_hom.hom_0_iff
thf(fact_1046_of__real__hom_Ohom__0__iff,axiom,
! [X: real] :
( ( ( real_V4546457046886955230omplex @ X )
= zero_zero_complex )
= ( X = zero_zero_real ) ) ).
% of_real_hom.hom_0_iff
thf(fact_1047_of__real__hom_Ohom__zero,axiom,
( ( real_V1803761363581548252l_real @ zero_zero_real )
= zero_zero_real ) ).
% of_real_hom.hom_zero
thf(fact_1048_of__real__hom_Ohom__zero,axiom,
( ( real_V4546457046886955230omplex @ zero_zero_real )
= zero_zero_complex ) ).
% of_real_hom.hom_zero
thf(fact_1049_of__real__hom_Ohom__0,axiom,
! [X: real] :
( ( ( real_V1803761363581548252l_real @ X )
= zero_zero_real )
=> ( X = zero_zero_real ) ) ).
% of_real_hom.hom_0
thf(fact_1050_of__real__hom_Ohom__0,axiom,
! [X: real] :
( ( ( real_V4546457046886955230omplex @ X )
= zero_zero_complex )
=> ( X = zero_zero_real ) ) ).
% of_real_hom.hom_0
thf(fact_1051_of__real__hom_Ohom__1,axiom,
! [X: real] :
( ( ( real_V1803761363581548252l_real @ X )
= one_one_real )
=> ( X = one_one_real ) ) ).
% of_real_hom.hom_1
thf(fact_1052_of__real__hom_Ohom__1,axiom,
! [X: real] :
( ( ( real_V4546457046886955230omplex @ X )
= one_one_complex )
=> ( X = one_one_real ) ) ).
% of_real_hom.hom_1
thf(fact_1053_of__real__hom_Ohom__one,axiom,
( ( real_V1803761363581548252l_real @ one_one_real )
= one_one_real ) ).
% of_real_hom.hom_one
thf(fact_1054_of__real__hom_Ohom__one,axiom,
( ( real_V4546457046886955230omplex @ one_one_real )
= one_one_complex ) ).
% of_real_hom.hom_one
thf(fact_1055_of__real__hom_Ohom__1__iff,axiom,
! [X: real] :
( ( ( real_V1803761363581548252l_real @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% of_real_hom.hom_1_iff
thf(fact_1056_of__real__hom_Ohom__1__iff,axiom,
! [X: real] :
( ( ( real_V4546457046886955230omplex @ X )
= one_one_complex )
= ( X = one_one_real ) ) ).
% of_real_hom.hom_1_iff
thf(fact_1057_of__real__hom_Ohom__mult,axiom,
! [X: real,Y: real] :
( ( real_V1803761363581548252l_real @ ( times_times_real @ X @ Y ) )
= ( times_times_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% of_real_hom.hom_mult
thf(fact_1058_of__real__hom_Ohom__mult,axiom,
! [X: real,Y: real] :
( ( real_V4546457046886955230omplex @ ( times_times_real @ X @ Y ) )
= ( times_times_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% of_real_hom.hom_mult
thf(fact_1059_of__real__hom_Ohom__mult__eq__zero,axiom,
! [X: real,Y: real] :
( ( ( times_times_real @ X @ Y )
= zero_zero_real )
=> ( ( times_times_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) )
= zero_zero_real ) ) ).
% of_real_hom.hom_mult_eq_zero
thf(fact_1060_of__real__hom_Ohom__mult__eq__zero,axiom,
! [X: real,Y: real] :
( ( ( times_times_real @ X @ Y )
= zero_zero_real )
=> ( ( times_times_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) )
= zero_zero_complex ) ) ).
% of_real_hom.hom_mult_eq_zero
thf(fact_1061_of__real__hom_Ohom__dvd__1,axiom,
! [X: real] :
( ( dvd_dvd_real @ X @ one_one_real )
=> ( dvd_dvd_real @ ( real_V1803761363581548252l_real @ X ) @ one_one_real ) ) ).
% of_real_hom.hom_dvd_1
thf(fact_1062_of__real__hom_Ohom__dvd__1,axiom,
! [X: real] :
( ( dvd_dvd_real @ X @ one_one_real )
=> ( dvd_dvd_complex @ ( real_V4546457046886955230omplex @ X ) @ one_one_complex ) ) ).
% of_real_hom.hom_dvd_1
thf(fact_1063_positive__scale,axiom,
! [A: mat_complex,N: nat,C2: real] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N @ N ) )
=> ( ( complex_positive @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( complex_positive @ ( smult_mat_complex @ ( real_V4546457046886955230omplex @ C2 ) @ A ) ) ) ) ) ).
% positive_scale
thf(fact_1064_density__collapse__def,axiom,
( projec3470689467825365843llapse
= ( ^ [R3: mat_complex,P6: mat_complex] :
( if_mat_complex
@ ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R3 @ P6 ) )
= zero_zero_complex )
@ ( projec8360710381328234318ensity @ ( dim_row_complex @ R3 ) )
@ ( smult_mat_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ one_one_real ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R3 @ P6 ) ) ) @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P6 @ R3 ) @ P6 ) ) ) ) ) ).
% density_collapse_def
thf(fact_1065_div__by__1,axiom,
! [A3: complex] :
( ( divide1717551699836669952omplex @ A3 @ one_one_complex )
= A3 ) ).
% div_by_1
thf(fact_1066_div__by__1,axiom,
! [A3: nat] :
( ( divide_divide_nat @ A3 @ one_one_nat )
= A3 ) ).
% div_by_1
thf(fact_1067_div__by__1,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ one_one_real )
= A3 ) ).
% div_by_1
thf(fact_1068_div__div__div__same,axiom,
! [D: nat,B6: nat,A3: nat] :
( ( dvd_dvd_nat @ D @ B6 )
=> ( ( dvd_dvd_nat @ B6 @ A3 )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A3 @ D ) @ ( divide_divide_nat @ B6 @ D ) )
= ( divide_divide_nat @ A3 @ B6 ) ) ) ) ).
% div_div_div_same
thf(fact_1069_dvd__div__eq__cancel,axiom,
! [A3: complex,C2: complex,B6: complex] :
( ( ( divide1717551699836669952omplex @ A3 @ C2 )
= ( divide1717551699836669952omplex @ B6 @ C2 ) )
=> ( ( dvd_dvd_complex @ C2 @ A3 )
=> ( ( dvd_dvd_complex @ C2 @ B6 )
=> ( A3 = B6 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1070_dvd__div__eq__cancel,axiom,
! [A3: nat,C2: nat,B6: nat] :
( ( ( divide_divide_nat @ A3 @ C2 )
= ( divide_divide_nat @ B6 @ C2 ) )
=> ( ( dvd_dvd_nat @ C2 @ A3 )
=> ( ( dvd_dvd_nat @ C2 @ B6 )
=> ( A3 = B6 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1071_dvd__div__eq__cancel,axiom,
! [A3: real,C2: real,B6: real] :
( ( ( divide_divide_real @ A3 @ C2 )
= ( divide_divide_real @ B6 @ C2 ) )
=> ( ( dvd_dvd_real @ C2 @ A3 )
=> ( ( dvd_dvd_real @ C2 @ B6 )
=> ( A3 = B6 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1072_dvd__div__eq__iff,axiom,
! [C2: complex,A3: complex,B6: complex] :
( ( dvd_dvd_complex @ C2 @ A3 )
=> ( ( dvd_dvd_complex @ C2 @ B6 )
=> ( ( ( divide1717551699836669952omplex @ A3 @ C2 )
= ( divide1717551699836669952omplex @ B6 @ C2 ) )
= ( A3 = B6 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1073_dvd__div__eq__iff,axiom,
! [C2: nat,A3: nat,B6: nat] :
( ( dvd_dvd_nat @ C2 @ A3 )
=> ( ( dvd_dvd_nat @ C2 @ B6 )
=> ( ( ( divide_divide_nat @ A3 @ C2 )
= ( divide_divide_nat @ B6 @ C2 ) )
= ( A3 = B6 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1074_dvd__div__eq__iff,axiom,
! [C2: real,A3: real,B6: real] :
( ( dvd_dvd_real @ C2 @ A3 )
=> ( ( dvd_dvd_real @ C2 @ B6 )
=> ( ( ( divide_divide_real @ A3 @ C2 )
= ( divide_divide_real @ B6 @ C2 ) )
= ( A3 = B6 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1075_div__dvd__div,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( dvd_dvd_nat @ A3 @ B6 )
=> ( ( dvd_dvd_nat @ A3 @ C2 )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B6 @ A3 ) @ ( divide_divide_nat @ C2 @ A3 ) )
= ( dvd_dvd_nat @ B6 @ C2 ) ) ) ) ).
% div_dvd_div
thf(fact_1076_divide__divide__eq__left_H,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A3 @ B6 ) @ C2 )
= ( divide1717551699836669952omplex @ A3 @ ( times_times_complex @ C2 @ B6 ) ) ) ).
% divide_divide_eq_left'
thf(fact_1077_divide__divide__eq__left_H,axiom,
! [A3: real,B6: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A3 @ B6 ) @ C2 )
= ( divide_divide_real @ A3 @ ( times_times_real @ C2 @ B6 ) ) ) ).
% divide_divide_eq_left'
thf(fact_1078_times__divide__eq__right,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( times_times_complex @ A3 @ ( divide1717551699836669952omplex @ B6 @ C2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ B6 ) @ C2 ) ) ).
% times_divide_eq_right
thf(fact_1079_times__divide__eq__right,axiom,
! [A3: real,B6: real,C2: real] :
( ( times_times_real @ A3 @ ( divide_divide_real @ B6 @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A3 @ B6 ) @ C2 ) ) ).
% times_divide_eq_right
thf(fact_1080_divide__divide__times__eq,axiom,
! [X: complex,Y: complex,Z: complex,W2: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W2 ) @ ( times_times_complex @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_1081_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W2: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X @ W2 ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_1082_divide__divide__eq__right,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( divide1717551699836669952omplex @ A3 @ ( divide1717551699836669952omplex @ B6 @ C2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ C2 ) @ B6 ) ) ).
% divide_divide_eq_right
thf(fact_1083_divide__divide__eq__right,axiom,
! [A3: real,B6: real,C2: real] :
( ( divide_divide_real @ A3 @ ( divide_divide_real @ B6 @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A3 @ C2 ) @ B6 ) ) ).
% divide_divide_eq_right
thf(fact_1084_times__divide__times__eq,axiom,
! [X: complex,Y: complex,Z: complex,W2: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_1085_times__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W2: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_1086_divide__divide__eq__left,axiom,
! [A3: complex,B6: complex,C2: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A3 @ B6 ) @ C2 )
= ( divide1717551699836669952omplex @ A3 @ ( times_times_complex @ B6 @ C2 ) ) ) ).
% divide_divide_eq_left
thf(fact_1087_divide__divide__eq__left,axiom,
! [A3: real,B6: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A3 @ B6 ) @ C2 )
= ( divide_divide_real @ A3 @ ( times_times_real @ B6 @ C2 ) ) ) ).
% divide_divide_eq_left
thf(fact_1088_times__divide__eq__left,axiom,
! [B6: complex,C2: complex,A3: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ B6 @ C2 ) @ A3 )
= ( divide1717551699836669952omplex @ ( times_times_complex @ B6 @ A3 ) @ C2 ) ) ).
% times_divide_eq_left
thf(fact_1089_times__divide__eq__left,axiom,
! [B6: real,C2: real,A3: real] :
( ( times_times_real @ ( divide_divide_real @ B6 @ C2 ) @ A3 )
= ( divide_divide_real @ ( times_times_real @ B6 @ A3 ) @ C2 ) ) ).
% times_divide_eq_left
thf(fact_1090_div__by__0,axiom,
! [A3: complex] :
( ( divide1717551699836669952omplex @ A3 @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_1091_div__by__0,axiom,
! [A3: nat] :
( ( divide_divide_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1092_div__by__0,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_1093_div__0,axiom,
! [A3: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A3 )
= zero_zero_complex ) ).
% div_0
thf(fact_1094_div__0,axiom,
! [A3: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% div_0
thf(fact_1095_div__0,axiom,
! [A3: real] :
( ( divide_divide_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% div_0
thf(fact_1096_divide__eq__0__iff,axiom,
! [A3: complex,B6: complex] :
( ( ( divide1717551699836669952omplex @ A3 @ B6 )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( B6 = zero_zero_complex ) ) ) ).
% divide_eq_0_iff
thf(fact_1097_divide__eq__0__iff,axiom,
! [A3: real,B6: real] :
( ( ( divide_divide_real @ A3 @ B6 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B6 = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_1098_divide__cancel__left,axiom,
! [C2: complex,A3: complex,B6: complex] :
( ( ( divide1717551699836669952omplex @ C2 @ A3 )
= ( divide1717551699836669952omplex @ C2 @ B6 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B6 ) ) ) ).
% divide_cancel_left
thf(fact_1099_divide__cancel__left,axiom,
! [C2: real,A3: real,B6: real] :
( ( ( divide_divide_real @ C2 @ A3 )
= ( divide_divide_real @ C2 @ B6 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B6 ) ) ) ).
% divide_cancel_left
thf(fact_1100_divide__cancel__right,axiom,
! [A3: complex,C2: complex,B6: complex] :
( ( ( divide1717551699836669952omplex @ A3 @ C2 )
= ( divide1717551699836669952omplex @ B6 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B6 ) ) ) ).
% divide_cancel_right
thf(fact_1101_divide__cancel__right,axiom,
! [A3: real,C2: real,B6: real] :
( ( ( divide_divide_real @ A3 @ C2 )
= ( divide_divide_real @ B6 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B6 ) ) ) ).
% divide_cancel_right
thf(fact_1102_division__ring__divide__zero,axiom,
! [A3: complex] :
( ( divide1717551699836669952omplex @ A3 @ zero_zero_complex )
= zero_zero_complex ) ).
% division_ring_divide_zero
thf(fact_1103_division__ring__divide__zero,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_1104_unit__div__1__div__1,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A3 ) )
= A3 ) ) ).
% unit_div_1_div_1
thf(fact_1105_dvd__div__unit__iff,axiom,
! [B6: nat,A3: nat,C2: nat] :
( ( dvd_dvd_nat @ B6 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A3 @ ( divide_divide_nat @ C2 @ B6 ) )
= ( dvd_dvd_nat @ A3 @ C2 ) ) ) ).
% dvd_div_unit_iff
thf(fact_1106_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1107_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_1108_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1109_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1110_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1111_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1112_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1113_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1114_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1115_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1116_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1117_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1118_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1119_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1120_less__eq__div__iff__mult__less__eq,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1121_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1122_trace__proj__pos__real,axiom,
! [P: mat_complex,R: mat_complex,N: nat] :
( ( linear5633924348262549461omplex @ P )
=> ( ( complex_positive @ R )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N @ N ) )
=> ( ( real_V4546457046886955230omplex @ ( re @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ P ) ) ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ P ) ) ) ) ) ) ) ).
% trace_proj_pos_real
thf(fact_1123_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1124_real__divide__square__eq,axiom,
! [R4: real,A3: real] :
( ( divide_divide_real @ ( times_times_real @ R4 @ A3 ) @ ( times_times_real @ R4 @ R4 ) )
= ( divide_divide_real @ A3 @ R4 ) ) ).
% real_divide_square_eq
thf(fact_1125_reals__power__lt__ex,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ Y )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_real @ ( power_power_real @ ( divide_divide_real @ one_one_real @ Y ) @ K3 ) @ X ) ) ) ) ).
% reals_power_lt_ex
thf(fact_1126_cpx__sq__mat_Otrace__hermitian__pos__real,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex,A: mat_complex,R: mat_complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( ( comple8306762464034002205omplex @ A )
=> ( ( complex_positive @ R )
=> ( ( member_mat_complex @ A @ Fc_mats )
=> ( ( member_mat_complex @ R @ Fc_mats )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ A ) )
= ( real_V4546457046886955230omplex @ ( re @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ A ) ) ) ) ) ) ) ) ) ) ).
% cpx_sq_mat.trace_hermitian_pos_real
thf(fact_1127_cpx__sq__mat_Ocpx__sq__mat__mult,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex,A: mat_complex,B: mat_complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( ( member_mat_complex @ A @ Fc_mats )
=> ( ( member_mat_complex @ B @ Fc_mats )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A @ B ) @ Fc_mats ) ) ) ) ).
% cpx_sq_mat.cpx_sq_mat_mult
thf(fact_1128_cpx__sq__mat_Onpos,axiom,
! [DimR: nat,DimC: nat,Fc_mats: set_mat_complex] :
( ( linear7199532782703566157sq_mat @ DimR @ DimC @ Fc_mats )
=> ( ord_less_nat @ zero_zero_nat @ DimR ) ) ).
% cpx_sq_mat.npos
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Matrix__Omat_It__Complex__Ocomplex_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Matrix__Omat_It__Complex__Ocomplex_J_T,axiom,
! [X: mat_complex,Y: mat_complex] :
( ( if_mat_complex @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Matrix__Omat_It__Complex__Ocomplex_J_T,axiom,
! [X: mat_complex,Y: mat_complex] :
( ( if_mat_complex @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( dim_row_a @ b1 )
= ( dim_col_a @ b1 ) ) ).
%------------------------------------------------------------------------------