TPTP Problem File: SLH0775^1.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_00741_029735__19400546_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1255 ( 460 unt; 149 typ; 0 def)
% Number of atoms : 3257 (1096 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 10492 ( 236 ~; 85 |; 179 &;8496 @)
% ( 0 <=>;1496 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Number of types : 17 ( 16 usr)
% Number of type conns : 488 ( 488 >; 0 *; 0 +; 0 <<)
% Number of symbols : 136 ( 133 usr; 12 con; 0-6 aty)
% Number of variables : 3157 ( 161 ^;2902 !; 94 ?;3157 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:36:55.379
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
thf(ty_n_t__List__Olist_It__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
list_l5436439031154120755omplex: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J_J,type,
list_l3981933317855906654omplex: $tType ).
thf(ty_n_t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
list_mat_complex: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J,type,
list_list_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__List__Olist_It__List__Olist_It__Real__Oreal_J_J,type,
list_list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Real__Oreal_J_J,type,
set_mat_real: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $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__Set__Oset_It__Real__Oreal_J,type,
set_real: $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 ).
% Explicit typings (133)
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_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_Ounitary_001t__Complex__Ocomplex,type,
comple6660659447773130958omplex: mat_complex > $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_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
plus_p8323303612493835998omplex: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Real__Oreal_J,type,
plus_plus_mat_real: mat_real > mat_real > mat_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__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_It__Nat__Onat_J,type,
times_times_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Real__Oreal_J,type,
times_times_mat_real: mat_real > mat_real > mat_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__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_If_001t__Complex__Ocomplex,type,
if_complex: $o > complex > complex > 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_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_List_Ogen__length_001t__Complex__Ocomplex,type,
gen_length_complex: nat > list_complex > nat ).
thf(sy_c_List_Ogen__length_001t__List__Olist_It__Complex__Ocomplex_J,type,
gen_le1671510949261875563omplex: nat > list_list_complex > nat ).
thf(sy_c_List_Ogen__length_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
gen_le107826107610854458omplex: nat > list_mat_complex > nat ).
thf(sy_c_List_Olist_ONil_001t__Complex__Ocomplex,type,
nil_complex: list_complex ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Complex__Ocomplex_J,type,
nil_list_complex: list_list_complex ).
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_Olist__all_001t__Complex__Ocomplex,type,
list_all_complex: ( complex > $o ) > list_complex > $o ).
thf(sy_c_List_Olist_Olist__all_001t__List__Olist_It__Complex__Ocomplex_J,type,
list_a4212339457297940234omplex: ( list_complex > $o ) > list_list_complex > $o ).
thf(sy_c_List_Olist_Olist__all_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
list_all_mat_complex: ( mat_complex > $o ) > list_mat_complex > $o ).
thf(sy_c_List_Olist__ex_001t__Complex__Ocomplex,type,
list_ex_complex: ( complex > $o ) > list_complex > $o ).
thf(sy_c_List_Olist__ex_001t__List__Olist_It__Complex__Ocomplex_J,type,
list_ex_list_complex: ( list_complex > $o ) > list_list_complex > $o ).
thf(sy_c_List_Olist__ex_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
list_ex_mat_complex: ( mat_complex > $o ) > list_mat_complex > $o ).
thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
nth_complex: list_complex > nat > complex ).
thf(sy_c_List_Onth_001t__List__Olist_It__Complex__Ocomplex_J,type,
nth_list_complex: list_list_complex > nat > list_complex ).
thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J,type,
nth_li53272486250751239omplex: list_l3981933317855906654omplex > nat > list_list_complex ).
thf(sy_c_List_Onth_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
nth_list_mat_complex: list_l5436439031154120755omplex > nat > list_mat_complex ).
thf(sy_c_List_Onth_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
nth_mat_complex: list_mat_complex > nat > mat_complex ).
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_Odiag__block__mat_001t__Complex__Ocomplex,type,
diag_b9145358668110806138omplex: list_mat_complex > mat_complex ).
thf(sy_c_Matrix_Odim__col_001t__Complex__Ocomplex,type,
dim_col_complex: mat_complex > nat ).
thf(sy_c_Matrix_Odim__row_001t__Complex__Ocomplex,type,
dim_row_complex: mat_complex > nat ).
thf(sy_c_Matrix_Oinverts__mat_001t__Complex__Ocomplex,type,
inverts_mat_complex: mat_complex > mat_complex > $o ).
thf(sy_c_Matrix_Omat__diag_001t__Complex__Ocomplex,type,
mat_diag_complex: nat > ( nat > complex ) > mat_complex ).
thf(sy_c_Matrix_Omat__diag_001t__Nat__Onat,type,
mat_diag_nat: nat > ( nat > nat ) > mat_nat ).
thf(sy_c_Matrix_Omat__diag_001t__Real__Oreal,type,
mat_diag_real: nat > ( nat > real ) > mat_real ).
thf(sy_c_Matrix_Omk__diagonal_001t__Complex__Ocomplex,type,
mk_diagonal_complex: list_complex > mat_complex ).
thf(sy_c_Matrix_Oone__mat_001t__Complex__Ocomplex,type,
one_mat_complex: nat > mat_complex ).
thf(sy_c_Matrix_Oone__mat_001t__Nat__Onat,type,
one_mat_nat: nat > mat_nat ).
thf(sy_c_Matrix_Oone__mat_001t__Real__Oreal,type,
one_mat_real: nat > mat_real ).
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_Osquare__mat_001t__Complex__Ocomplex,type,
square_mat_complex: mat_complex > $o ).
thf(sy_c_Matrix_Oundef__vec_001t__Complex__Ocomplex,type,
undef_vec_complex: nat > complex ).
thf(sy_c_Matrix_Oundef__vec_001t__List__Olist_It__Complex__Ocomplex_J,type,
undef_8020164207840674868omplex: nat > list_complex ).
thf(sy_c_Matrix_Oundef__vec_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
undef_2495355514574404529omplex: nat > mat_complex ).
thf(sy_c_Matrix__Legacy_Ocol_001t__Complex__Ocomplex,type,
matrix_col_complex: list_list_complex > nat > list_complex ).
thf(sy_c_Matrix__Legacy_Ocol_001t__List__Olist_It__Complex__Ocomplex_J,type,
matrix9199650144385845897omplex: list_l3981933317855906654omplex > nat > list_list_complex ).
thf(sy_c_Matrix__Legacy_Ocol_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
matrix725289766534482588omplex: list_l5436439031154120755omplex > nat > list_mat_complex ).
thf(sy_c_Matrix__Legacy_Omat_001t__Complex__Ocomplex,type,
matrix_mat_complex: nat > nat > list_list_complex > $o ).
thf(sy_c_Matrix__Legacy_Omat_001t__List__Olist_It__Complex__Ocomplex_J,type,
matrix6976670468949791273omplex: nat > nat > list_l3981933317855906654omplex > $o ).
thf(sy_c_Matrix__Legacy_Omat_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
matrix6216835547647503100omplex: nat > nat > list_l5436439031154120755omplex > $o ).
thf(sy_c_Matrix__Legacy_Omat_001t__Nat__Onat,type,
matrix_mat_nat: nat > nat > list_list_nat > $o ).
thf(sy_c_Matrix__Legacy_Omat_001t__Real__Oreal,type,
matrix_mat_real: nat > nat > list_list_real > $o ).
thf(sy_c_Matrix__Legacy_Omat__map_001t__Complex__Ocomplex,type,
matrix553418799951983617omplex: ( complex > complex ) > list_list_complex > list_list_complex ).
thf(sy_c_Matrix__Legacy_Omat__map_001t__List__Olist_It__Complex__Ocomplex_J,type,
matrix9216512725699604625omplex: ( list_complex > list_complex ) > list_l3981933317855906654omplex > list_l3981933317855906654omplex ).
thf(sy_c_Matrix__Legacy_Omat__map_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
matrix89116945220183700omplex: ( mat_complex > mat_complex ) > list_l5436439031154120755omplex > list_l5436439031154120755omplex ).
thf(sy_c_Matrix__Legacy_Omat__multI_001t__Complex__Ocomplex,type,
matrix2876711754638949054omplex: complex > ( complex > complex > complex ) > ( complex > complex > complex ) > nat > list_list_complex > list_list_complex > list_list_complex ).
thf(sy_c_Matrix__Legacy_Omat__multI_001t__Nat__Onat,type,
matrix_mat_multI_nat: nat > ( nat > nat > nat ) > ( nat > nat > nat ) > nat > list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Matrix__Legacy_Omat__multI_001t__Real__Oreal,type,
matrix6810070775276521660I_real: real > ( real > real > real ) > ( real > real > real ) > nat > list_list_real > list_list_real > list_list_real ).
thf(sy_c_Matrix__Legacy_Omat__plusI_001t__Complex__Ocomplex,type,
matrix6097015163314587732omplex: ( complex > complex > complex ) > list_list_complex > list_list_complex > list_list_complex ).
thf(sy_c_Matrix__Legacy_Omat__plusI_001t__List__Olist_It__Complex__Ocomplex_J,type,
matrix2250674347687116644omplex: ( list_complex > list_complex > list_complex ) > list_l3981933317855906654omplex > list_l3981933317855906654omplex > list_l3981933317855906654omplex ).
thf(sy_c_Matrix__Legacy_Omat__plusI_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
matrix1578179600178973825omplex: ( mat_complex > mat_complex > mat_complex ) > list_l5436439031154120755omplex > list_l5436439031154120755omplex > list_l5436439031154120755omplex ).
thf(sy_c_Matrix__Legacy_Omat__plusI_001t__Nat__Onat,type,
matrix_mat_plusI_nat: ( nat > nat > nat ) > list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Matrix__Legacy_Omat__plusI_001t__Real__Oreal,type,
matrix4160047064883189586I_real: ( real > real > real ) > list_list_real > list_list_real > list_list_real ).
thf(sy_c_Matrix__Legacy_Orow_001t__Complex__Ocomplex,type,
matrix_row_complex: list_list_complex > nat > list_complex ).
thf(sy_c_Matrix__Legacy_Orow_001t__List__Olist_It__Complex__Ocomplex_J,type,
matrix9134808206163378723omplex: list_l3981933317855906654omplex > nat > list_list_complex ).
thf(sy_c_Matrix__Legacy_Orow_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
matrix8356865337816758658omplex: list_l5436439031154120755omplex > nat > list_mat_complex ).
thf(sy_c_Matrix__Legacy_Osub__mat_001t__Complex__Ocomplex,type,
matrix742113920429806117omplex: nat > nat > list_list_complex > list_list_complex ).
thf(sy_c_Matrix__Legacy_Osub__mat_001t__List__Olist_It__Complex__Ocomplex_J,type,
matrix1439577319965977781omplex: nat > nat > list_l3981933317855906654omplex > list_l3981933317855906654omplex ).
thf(sy_c_Matrix__Legacy_Osub__mat_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
matrix4057538736392336240omplex: nat > nat > list_l5436439031154120755omplex > list_l5436439031154120755omplex ).
thf(sy_c_Matrix__Legacy_Otranspose_001t__Complex__Ocomplex,type,
matrix1433782295178676338omplex: nat > list_list_complex > list_list_complex ).
thf(sy_c_Matrix__Legacy_Otranspose_001t__List__Olist_It__Complex__Ocomplex_J,type,
matrix7733108353310726274omplex: nat > list_l3981933317855906654omplex > list_l3981933317855906654omplex ).
thf(sy_c_Matrix__Legacy_Otranspose_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
matrix6044053461928786659omplex: nat > list_l5436439031154120755omplex > list_l5436439031154120755omplex ).
thf(sy_c_Matrix__Legacy_Ovec__plusI_001t__Complex__Ocomplex,type,
matrix6198229848702844640omplex: ( complex > complex > complex ) > list_complex > list_complex > list_complex ).
thf(sy_c_Matrix__Tensor_Omult_OTensor_001t__Complex__Ocomplex,type,
matrix1305980297522496462omplex: ( complex > complex > complex ) > list_list_complex > list_list_complex > list_list_complex ).
thf(sy_c_Matrix__Tensor_Omult_Orow__length_001t__Complex__Ocomplex,type,
matrix1515831402840476169omplex: list_list_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J,type,
size_s7907857696548412130omplex: list_list_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
size_s5969786470865220249omplex: list_mat_complex > nat ).
thf(sy_c_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_001_062_It__Complex__Ocomplex_M_Eo_J,type,
ord_le4573692005234683329plex_o: ( complex > $o ) > ( complex > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Complex__Ocomplex_J_M_Eo_J,type,
ord_le6360058522932223793plex_o: ( list_complex > $o ) > ( list_complex > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J_M_Eo_J,type,
ord_le8334822129778880289plex_o: ( list_list_complex > $o ) > ( list_list_complex > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J_M_Eo_J,type,
ord_le1186661314528880752plex_o: ( list_mat_complex > $o ) > ( list_mat_complex > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Matrix__Omat_It__Complex__Ocomplex_J_M_Eo_J,type,
ord_le2790225379703085046plex_o: ( mat_complex > $o ) > ( mat_complex > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
ord_le3632134057777142183omplex: set_mat_complex > set_mat_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Projective__Measurements_Odensity__collapse,type,
projec3470689467825365843llapse: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Projective__Measurements_Ohermitian__decomp_001t__Complex__Ocomplex,type,
projec5943904436471448624omplex: mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_Projective__Measurements_Omax__mix__density,type,
projec8360710381328234318ensity: nat > mat_complex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
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_Set_OCollect_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
collect_mat_complex: ( mat_complex > $o ) > set_mat_complex ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
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_Oreal__diag__decomp_001t__Complex__Ocomplex,type,
spectr5409772854192057952omplex: mat_complex > mat_complex > mat_complex > $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_Ounitary__diag_001t__Complex__Ocomplex,type,
spectr532731689276696518omplex: mat_complex > mat_complex > mat_complex > $o ).
thf(sy_c_Tensor_Omat__of__cols__list,type,
mat_of_cols_list: nat > list_list_complex > mat_complex ).
thf(sy_c_Tensor_Omat__to__cols__list,type,
mat_to_cols_list: mat_complex > list_list_complex ).
thf(sy_c_Tensor_Otensor__mat,type,
tensor_mat: mat_complex > mat_complex > mat_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__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_Al,type,
al: list_mat_complex ).
% Relevant facts (1098)
thf(fact_0_assms_I1_J,axiom,
al != nil_mat_complex ).
% assms(1)
thf(fact_1_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_2_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_3_nth__equalityI,axiom,
! [Xs: list_mat_complex,Ys: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ( nth_mat_complex @ Xs @ I )
= ( nth_mat_complex @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_4_nth__equalityI,axiom,
! [Xs: list_list_complex,Ys: list_list_complex] :
( ( ( size_s7907857696548412130omplex @ Xs )
= ( size_s7907857696548412130omplex @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s7907857696548412130omplex @ Xs ) )
=> ( ( nth_list_complex @ Xs @ I )
= ( nth_list_complex @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_5_nth__equalityI,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( ( nth_complex @ Xs @ I )
= ( nth_complex @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_6_Skolem__list__nth,axiom,
! [K: nat,P: nat > mat_complex > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X: mat_complex] : ( P @ I2 @ X ) ) )
= ( ? [Xs2: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_mat_complex @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_7_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_complex > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X: list_complex] : ( P @ I2 @ X ) ) )
= ( ? [Xs2: list_list_complex] :
( ( ( size_s7907857696548412130omplex @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_list_complex @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_8_Skolem__list__nth,axiom,
! [K: nat,P: nat > complex > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X: complex] : ( P @ I2 @ X ) ) )
= ( ? [Xs2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_complex @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_9_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_mat_complex,Z: list_mat_complex] : ( Y = Z ) )
= ( ^ [Xs2: list_mat_complex,Ys2: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs2 )
= ( size_s5969786470865220249omplex @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5969786470865220249omplex @ Xs2 ) )
=> ( ( nth_mat_complex @ Xs2 @ I2 )
= ( nth_mat_complex @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_10_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_list_complex,Z: list_list_complex] : ( Y = Z ) )
= ( ^ [Xs2: list_list_complex,Ys2: list_list_complex] :
( ( ( size_s7907857696548412130omplex @ Xs2 )
= ( size_s7907857696548412130omplex @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s7907857696548412130omplex @ Xs2 ) )
=> ( ( nth_list_complex @ Xs2 @ I2 )
= ( nth_list_complex @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_11_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_complex,Z: list_complex] : ( Y = Z ) )
= ( ^ [Xs2: list_complex,Ys2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( ( nth_complex @ Xs2 @ I2 )
= ( nth_complex @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_12_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_13_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_14_length__induct,axiom,
! [P: list_mat_complex > $o,Xs: list_mat_complex] :
( ! [Xs3: list_mat_complex] :
( ! [Ys3: list_mat_complex] :
( ( ord_less_nat @ ( size_s5969786470865220249omplex @ Ys3 ) @ ( size_s5969786470865220249omplex @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_15_length__induct,axiom,
! [P: list_list_complex > $o,Xs: list_list_complex] :
( ! [Xs3: list_list_complex] :
( ! [Ys3: list_list_complex] :
( ( ord_less_nat @ ( size_s7907857696548412130omplex @ Ys3 ) @ ( size_s7907857696548412130omplex @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_16_length__induct,axiom,
! [P: list_complex > $o,Xs: list_complex] :
( ! [Xs3: list_complex] :
( ! [Ys3: list_complex] :
( ( ord_less_nat @ ( size_s3451745648224563538omplex @ Ys3 ) @ ( size_s3451745648224563538omplex @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_17_list__all__length,axiom,
( list_all_mat_complex
= ( ^ [P2: mat_complex > $o,Xs2: list_mat_complex] :
! [N: nat] :
( ( ord_less_nat @ N @ ( size_s5969786470865220249omplex @ Xs2 ) )
=> ( P2 @ ( nth_mat_complex @ Xs2 @ N ) ) ) ) ) ).
% list_all_length
thf(fact_18_list__all__length,axiom,
( list_a4212339457297940234omplex
= ( ^ [P2: list_complex > $o,Xs2: list_list_complex] :
! [N: nat] :
( ( ord_less_nat @ N @ ( size_s7907857696548412130omplex @ Xs2 ) )
=> ( P2 @ ( nth_list_complex @ Xs2 @ N ) ) ) ) ) ).
% list_all_length
thf(fact_19_list__all__length,axiom,
( list_all_complex
= ( ^ [P2: complex > $o,Xs2: list_complex] :
! [N: nat] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( P2 @ ( nth_complex @ Xs2 @ N ) ) ) ) ) ).
% list_all_length
thf(fact_20_hermitian__real__diag__decomp,axiom,
! [A: mat_complex,N2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( comple8306762464034002205omplex @ A )
=> ~ ! [B: mat_complex,U: mat_complex] :
~ ( spectr5409772854192057952omplex @ A @ B @ U ) ) ) ) ).
% hermitian_real_diag_decomp
thf(fact_21_assms_I2_J,axiom,
( list_all_mat_complex
@ ^ [A2: mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ A2 ) )
& ( comple8306762464034002205omplex @ A2 ) )
@ al ) ).
% assms(2)
thf(fact_22_hermitian__square,axiom,
! [M2: mat_complex] :
( ( comple8306762464034002205omplex @ M2 )
=> ( member_mat_complex @ M2 @ ( carrier_mat_complex @ ( dim_row_complex @ M2 ) @ ( dim_row_complex @ M2 ) ) ) ) ).
% hermitian_square
thf(fact_23_list__all__simps_I2_J,axiom,
! [P: mat_complex > $o] : ( list_all_mat_complex @ P @ nil_mat_complex ) ).
% list_all_simps(2)
thf(fact_24_list__all__simps_I2_J,axiom,
! [P: complex > $o] : ( list_all_complex @ P @ nil_complex ) ).
% list_all_simps(2)
thf(fact_25_list__all__simps_I2_J,axiom,
! [P: list_complex > $o] : ( list_a4212339457297940234omplex @ P @ nil_list_complex ) ).
% list_all_simps(2)
thf(fact_26_list_Opred__inject_I1_J,axiom,
! [P: mat_complex > $o] : ( list_all_mat_complex @ P @ nil_mat_complex ) ).
% list.pred_inject(1)
thf(fact_27_list_Opred__inject_I1_J,axiom,
! [P: complex > $o] : ( list_all_complex @ P @ nil_complex ) ).
% list.pred_inject(1)
thf(fact_28_list_Opred__inject_I1_J,axiom,
! [P: list_complex > $o] : ( list_a4212339457297940234omplex @ P @ nil_list_complex ) ).
% list.pred_inject(1)
thf(fact_29_list_Opred__True,axiom,
( ( list_all_mat_complex
@ ^ [Uu: mat_complex] : $true )
= ( ^ [Uu: list_mat_complex] : $true ) ) ).
% list.pred_True
thf(fact_30_list_Opred__True,axiom,
( ( list_all_complex
@ ^ [Uu: complex] : $true )
= ( ^ [Uu: list_complex] : $true ) ) ).
% list.pred_True
thf(fact_31_list_Opred__True,axiom,
( ( list_a4212339457297940234omplex
@ ^ [Uu: list_complex] : $true )
= ( ^ [Uu: list_list_complex] : $true ) ) ).
% list.pred_True
thf(fact_32_carrier__mat__def,axiom,
( carrier_mat_complex
= ( ^ [Nr2: nat,Nc2: nat] :
( collect_mat_complex
@ ^ [M3: mat_complex] :
( ( ( dim_row_complex @ M3 )
= Nr2 )
& ( ( dim_col_complex @ M3 )
= Nc2 ) ) ) ) ) ).
% carrier_mat_def
thf(fact_33_length__0__conv,axiom,
! [Xs: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs )
= zero_zero_nat )
= ( Xs = nil_mat_complex ) ) ).
% length_0_conv
thf(fact_34_length__0__conv,axiom,
! [Xs: list_list_complex] :
( ( ( size_s7907857696548412130omplex @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_complex ) ) ).
% length_0_conv
thf(fact_35_length__0__conv,axiom,
! [Xs: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= zero_zero_nat )
= ( Xs = nil_complex ) ) ).
% length_0_conv
thf(fact_36_list_Osize_I3_J,axiom,
( ( size_s5969786470865220249omplex @ nil_mat_complex )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_37_list_Osize_I3_J,axiom,
( ( size_s7907857696548412130omplex @ nil_list_complex )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_38_list_Osize_I3_J,axiom,
( ( size_s3451745648224563538omplex @ nil_complex )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_39_length__greater__0__conv,axiom,
! [Xs: list_mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5969786470865220249omplex @ Xs ) )
= ( Xs != nil_mat_complex ) ) ).
% length_greater_0_conv
thf(fact_40_length__greater__0__conv,axiom,
! [Xs: list_list_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s7907857696548412130omplex @ Xs ) )
= ( Xs != nil_list_complex ) ) ).
% length_greater_0_conv
thf(fact_41_length__greater__0__conv,axiom,
! [Xs: list_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) )
= ( Xs != nil_complex ) ) ).
% length_greater_0_conv
thf(fact_42_real__diag__decomp__hermitian,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr5409772854192057952omplex @ A @ B2 @ U2 )
=> ( comple8306762464034002205omplex @ A ) ) ).
% real_diag_decomp_hermitian
thf(fact_43_neq__if__length__neq,axiom,
! [Xs: list_mat_complex,Ys: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs )
!= ( size_s5969786470865220249omplex @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_44_neq__if__length__neq,axiom,
! [Xs: list_list_complex,Ys: list_list_complex] :
( ( ( size_s7907857696548412130omplex @ Xs )
!= ( size_s7907857696548412130omplex @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_45_neq__if__length__neq,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
!= ( size_s3451745648224563538omplex @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_46_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs3: list_mat_complex] :
( ( size_s5969786470865220249omplex @ Xs3 )
= N2 ) ).
% Ex_list_of_length
thf(fact_47_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs3: list_list_complex] :
( ( size_s7907857696548412130omplex @ Xs3 )
= N2 ) ).
% Ex_list_of_length
thf(fact_48_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs3: list_complex] :
( ( size_s3451745648224563538omplex @ Xs3 )
= N2 ) ).
% Ex_list_of_length
thf(fact_49_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_50_undef__vec__def,axiom,
( undef_vec_complex
= ( nth_complex @ nil_complex ) ) ).
% undef_vec_def
thf(fact_51_undef__vec__def,axiom,
( undef_2495355514574404529omplex
= ( nth_mat_complex @ nil_mat_complex ) ) ).
% undef_vec_def
thf(fact_52_undef__vec__def,axiom,
( undef_8020164207840674868omplex
= ( nth_list_complex @ nil_list_complex ) ) ).
% undef_vec_def
thf(fact_53_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_54_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_55_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_56_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_57_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_58_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_59_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_60_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_61_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_62_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_63_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_64_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_65_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_66_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_67_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_68_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_69_mem__Collect__eq,axiom,
! [A3: real,P: real > $o] :
( ( member_real @ A3 @ ( collect_real @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_70_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_71_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_72_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_73_linorder__neqE__nat,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_74_size__neq__size__imp__neq,axiom,
! [X3: list_mat_complex,Y2: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ X3 )
!= ( size_s5969786470865220249omplex @ Y2 ) )
=> ( X3 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_75_size__neq__size__imp__neq,axiom,
! [X3: list_list_complex,Y2: list_list_complex] :
( ( ( size_s7907857696548412130omplex @ X3 )
!= ( size_s7907857696548412130omplex @ Y2 ) )
=> ( X3 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_76_size__neq__size__imp__neq,axiom,
! [X3: list_complex,Y2: list_complex] :
( ( ( size_s3451745648224563538omplex @ X3 )
!= ( size_s3451745648224563538omplex @ Y2 ) )
=> ( X3 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_77_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_78_rel__simps_I70_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% rel_simps(70)
thf(fact_79_rel__simps_I70_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% rel_simps(70)
thf(fact_80_zero__order_I5_J,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% zero_order(5)
thf(fact_81_zero__order_I4_J,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_order(4)
thf(fact_82_zero__order_I3_J,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% zero_order(3)
thf(fact_83_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_84_gr__implies__not__zero,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_85_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_86_zero__reorient,axiom,
! [X3: complex] :
( ( zero_zero_complex = X3 )
= ( X3 = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_87_zero__reorient,axiom,
! [X3: real] :
( ( zero_zero_real = X3 )
= ( X3 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_88_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_89_density__collapse__carrier,axiom,
! [R: mat_complex,P: mat_complex,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ R ) )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( member_mat_complex @ ( projec3470689467825365843llapse @ R @ P ) @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ) ).
% density_collapse_carrier
thf(fact_90_mk__diagonal__dim_I2_J,axiom,
! [As: list_complex] :
( ( dim_col_complex @ ( mk_diagonal_complex @ As ) )
= ( size_s3451745648224563538omplex @ As ) ) ).
% mk_diagonal_dim(2)
thf(fact_91_mk__diagonal__dim_I1_J,axiom,
! [As: list_complex] :
( ( dim_row_complex @ ( mk_diagonal_complex @ As ) )
= ( size_s3451745648224563538omplex @ As ) ) ).
% mk_diagonal_dim(1)
thf(fact_92_length__code,axiom,
( size_s5969786470865220249omplex
= ( gen_le107826107610854458omplex @ zero_zero_nat ) ) ).
% length_code
thf(fact_93_length__code,axiom,
( size_s7907857696548412130omplex
= ( gen_le1671510949261875563omplex @ zero_zero_nat ) ) ).
% length_code
thf(fact_94_length__code,axiom,
( size_s3451745648224563538omplex
= ( gen_length_complex @ zero_zero_nat ) ) ).
% length_code
thf(fact_95_unitary__is__corthogonal,axiom,
! [U2: mat_complex,N2: nat] :
( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple6660659447773130958omplex @ U2 )
=> ( schur_549222400177443379omplex @ U2 ) ) ) ).
% unitary_is_corthogonal
thf(fact_96_cpx__sq__mat__axioms__def,axiom,
( linear2040860143340867312axioms
= ( ^ [DimR: nat,DimC: nat] :
( ( DimR = DimC )
& ( ord_less_nat @ zero_zero_nat @ DimR ) ) ) ) ).
% cpx_sq_mat_axioms_def
thf(fact_97_cpx__sq__mat__axioms_Ointro,axiom,
! [DimR2: nat,DimC2: nat] :
( ( DimR2 = DimC2 )
=> ( ( ord_less_nat @ zero_zero_nat @ DimR2 )
=> ( linear2040860143340867312axioms @ DimR2 @ DimC2 ) ) ) ).
% cpx_sq_mat_axioms.intro
thf(fact_98_tensor__mat__unitary,axiom,
! [U2: mat_complex,V: mat_complex] :
( ( comple6660659447773130958omplex @ U2 )
=> ( ( comple6660659447773130958omplex @ V )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ U2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ V ) )
=> ( comple6660659447773130958omplex @ ( tensor_mat @ U2 @ V ) ) ) ) ) ) ).
% tensor_mat_unitary
thf(fact_99_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_le107826107610854458omplex @ N2 @ nil_mat_complex )
= N2 ) ).
% gen_length_code(1)
thf(fact_100_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_le1671510949261875563omplex @ N2 @ nil_list_complex )
= N2 ) ).
% gen_length_code(1)
thf(fact_101_tensor__mat__hermitian,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,N4: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N4 @ N4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( comple8306762464034002205omplex @ A )
=> ( ( comple8306762464034002205omplex @ B2 )
=> ( comple8306762464034002205omplex @ ( tensor_mat @ A @ B2 ) ) ) ) ) ) ) ) ).
% tensor_mat_hermitian
thf(fact_102_hermitian__decomp__dim__carrier,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B2 @ U2 )
=> ( member_mat_complex @ B2 @ ( carrier_mat_complex @ ( dim_row_complex @ A ) @ ( dim_col_complex @ A ) ) ) ) ).
% hermitian_decomp_dim_carrier
thf(fact_103_tensor__mat__carrier,axiom,
! [U2: mat_complex,V: mat_complex] : ( member_mat_complex @ ( tensor_mat @ U2 @ V ) @ ( carrier_mat_complex @ ( times_times_nat @ ( dim_row_complex @ U2 ) @ ( dim_row_complex @ V ) ) @ ( times_times_nat @ ( dim_col_complex @ U2 ) @ ( dim_col_complex @ V ) ) ) ) ).
% tensor_mat_carrier
thf(fact_104_density__collapse__operator,axiom,
! [P: mat_complex,R: mat_complex,N2: nat] :
( ( linear5633924348262549461omplex @ P )
=> ( ( comple5220265106149225959erator @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_row_complex @ R ) )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( comple5220265106149225959erator @ ( projec3470689467825365843llapse @ R @ P ) ) ) ) ) ) ) ).
% density_collapse_operator
thf(fact_105_list__ex__length,axiom,
( list_ex_mat_complex
= ( ^ [P2: mat_complex > $o,Xs2: list_mat_complex] :
? [N: nat] :
( ( ord_less_nat @ N @ ( size_s5969786470865220249omplex @ Xs2 ) )
& ( P2 @ ( nth_mat_complex @ Xs2 @ N ) ) ) ) ) ).
% list_ex_length
thf(fact_106_list__ex__length,axiom,
( list_ex_list_complex
= ( ^ [P2: list_complex > $o,Xs2: list_list_complex] :
? [N: nat] :
( ( ord_less_nat @ N @ ( size_s7907857696548412130omplex @ Xs2 ) )
& ( P2 @ ( nth_list_complex @ Xs2 @ N ) ) ) ) ) ).
% list_ex_length
thf(fact_107_list__ex__length,axiom,
( list_ex_complex
= ( ^ [P2: complex > $o,Xs2: list_complex] :
? [N: nat] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
& ( P2 @ ( nth_complex @ Xs2 @ N ) ) ) ) ) ).
% list_ex_length
thf(fact_108_mult__distr__tensor,axiom,
! [A: mat_complex,B2: mat_complex,C: mat_complex,D: mat_complex] :
( ( ( dim_col_complex @ A )
= ( dim_row_complex @ B2 ) )
=> ( ( ( dim_col_complex @ C )
= ( dim_row_complex @ D ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ D ) )
=> ( ( tensor_mat @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( times_8009071140041733218omplex @ C @ D ) )
= ( times_8009071140041733218omplex @ ( tensor_mat @ A @ C ) @ ( tensor_mat @ B2 @ D ) ) ) ) ) ) ) ) ) ).
% mult_distr_tensor
thf(fact_109_tensor__mat__is__assoc,axiom,
! [A: mat_complex,B2: mat_complex,C: mat_complex] :
( ( tensor_mat @ A @ ( tensor_mat @ B2 @ C ) )
= ( tensor_mat @ ( tensor_mat @ A @ B2 ) @ C ) ) ).
% tensor_mat_is_assoc
thf(fact_110_projector__square__eq,axiom,
! [M2: mat_complex] :
( ( linear5633924348262549461omplex @ M2 )
=> ( ( times_8009071140041733218omplex @ M2 @ M2 )
= M2 ) ) ).
% projector_square_eq
thf(fact_111_mult_Oleft__commute,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( times_times_nat @ B3 @ ( times_times_nat @ A3 @ C2 ) )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_112_mult_Oleft__commute,axiom,
! [B3: complex,A3: complex,C2: complex] :
( ( times_times_complex @ B3 @ ( times_times_complex @ A3 @ C2 ) )
= ( times_times_complex @ A3 @ ( times_times_complex @ B3 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_113_mult_Oleft__commute,axiom,
! [B3: real,A3: real,C2: real] :
( ( times_times_real @ B3 @ ( times_times_real @ A3 @ C2 ) )
= ( times_times_real @ A3 @ ( times_times_real @ B3 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_114_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_115_mult_Ocommute,axiom,
( times_times_complex
= ( ^ [A4: complex,B4: complex] : ( times_times_complex @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_116_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_117_semigroup__mult__class_Omult_Oassoc,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B3 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% semigroup_mult_class.mult.assoc
thf(fact_118_semigroup__mult__class_Omult_Oassoc,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( times_times_complex @ ( times_times_complex @ A3 @ B3 ) @ C2 )
= ( times_times_complex @ A3 @ ( times_times_complex @ B3 @ C2 ) ) ) ).
% semigroup_mult_class.mult.assoc
thf(fact_119_semigroup__mult__class_Omult_Oassoc,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B3 ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B3 @ C2 ) ) ) ).
% semigroup_mult_class.mult.assoc
thf(fact_120_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B3 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_121_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( times_times_complex @ ( times_times_complex @ A3 @ B3 ) @ C2 )
= ( times_times_complex @ A3 @ ( times_times_complex @ B3 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_122_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B3 ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B3 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_123_projector__def,axiom,
( linear5633924348262549461omplex
= ( ^ [M5: mat_complex] :
( ( comple8306762464034002205omplex @ M5 )
& ( ( times_8009071140041733218omplex @ M5 @ M5 )
= M5 ) ) ) ) ).
% projector_def
thf(fact_124_arith__simps_I63_J,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% arith_simps(63)
thf(fact_125_arith__simps_I63_J,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ zero_zero_complex )
= zero_zero_complex ) ).
% arith_simps(63)
thf(fact_126_arith__simps_I63_J,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% arith_simps(63)
thf(fact_127_arith__simps_I62_J,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% arith_simps(62)
thf(fact_128_arith__simps_I62_J,axiom,
! [A3: complex] :
( ( times_times_complex @ zero_zero_complex @ A3 )
= zero_zero_complex ) ).
% arith_simps(62)
thf(fact_129_arith__simps_I62_J,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% arith_simps(62)
thf(fact_130_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_131_times__nat_Osimps_I1_J,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% times_nat.simps(1)
thf(fact_132_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_133_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_134_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_135_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_136_mult__carrier__mat,axiom,
! [A: mat_complex,Nr: nat,N2: nat,B2: mat_complex,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ Nc ) )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_137_assoc__mult__mat,axiom,
! [A: mat_complex,N_1: nat,N_2: nat,B2: 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 @ B2 @ ( carrier_mat_complex @ N_2 @ N_3 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N_3 @ N_4 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ C )
= ( times_8009071140041733218omplex @ A @ ( times_8009071140041733218omplex @ B2 @ C ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_138_index__mult__mat_I2_J,axiom,
! [A: mat_complex,B2: mat_complex] :
( ( dim_row_complex @ ( times_8009071140041733218omplex @ A @ B2 ) )
= ( dim_row_complex @ A ) ) ).
% index_mult_mat(2)
thf(fact_139_index__mult__mat_I3_J,axiom,
! [A: mat_complex,B2: mat_complex] :
( ( dim_col_complex @ ( times_8009071140041733218omplex @ A @ B2 ) )
= ( dim_col_complex @ B2 ) ) ).
% index_mult_mat(3)
thf(fact_140_mat__assoc__test_I1_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( times_8009071140041733218omplex @ C @ D ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ C ) @ D ) ) ) ) ) ) ).
% mat_assoc_test(1)
thf(fact_141_hermitian__square__hermitian,axiom,
! [A: mat_complex] :
( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( times_8009071140041733218omplex @ A @ A ) ) ) ).
% hermitian_square_hermitian
thf(fact_142_dim__col__tensor__mat,axiom,
! [A: mat_complex,B2: mat_complex] :
( ( dim_col_complex @ ( tensor_mat @ A @ B2 ) )
= ( times_times_nat @ ( dim_col_complex @ A ) @ ( dim_col_complex @ B2 ) ) ) ).
% dim_col_tensor_mat
thf(fact_143_dim__row__tensor__mat,axiom,
! [A: mat_complex,B2: mat_complex] :
( ( dim_row_complex @ ( tensor_mat @ A @ B2 ) )
= ( times_times_nat @ ( dim_row_complex @ A ) @ ( dim_row_complex @ B2 ) ) ) ).
% dim_row_tensor_mat
thf(fact_144_list__ex__simps_I2_J,axiom,
! [P: mat_complex > $o] :
~ ( list_ex_mat_complex @ P @ nil_mat_complex ) ).
% list_ex_simps(2)
thf(fact_145_list__ex__simps_I2_J,axiom,
! [P: list_complex > $o] :
~ ( list_ex_list_complex @ P @ nil_list_complex ) ).
% list_ex_simps(2)
thf(fact_146_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_147_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_148_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( M = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_149_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_150_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_151_mult__less__mono2,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_152_mult__less__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_153_unitary__times__unitary,axiom,
! [P: mat_complex,N2: nat,Q: mat_complex] :
( ( member_mat_complex @ P @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ Q @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple6660659447773130958omplex @ P )
=> ( ( comple6660659447773130958omplex @ Q )
=> ( comple6660659447773130958omplex @ ( times_8009071140041733218omplex @ P @ Q ) ) ) ) ) ) ).
% unitary_times_unitary
thf(fact_154_hermitian__decomp__unitary,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B2 @ U2 )
=> ( comple6660659447773130958omplex @ U2 ) ) ).
% hermitian_decomp_unitary
thf(fact_155_projector__hermitian,axiom,
! [M2: mat_complex] :
( ( linear5633924348262549461omplex @ M2 )
=> ( comple8306762464034002205omplex @ M2 ) ) ).
% projector_hermitian
thf(fact_156_hermitian__decomp__decomp_H,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B2 @ U2 )
=> ( spectr5409772854192057952omplex @ A @ B2 @ U2 ) ) ).
% hermitian_decomp_decomp'
thf(fact_157_mult__less__iff1,axiom,
! [Z2: real,X3: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y2 @ Z2 ) )
= ( ord_less_real @ X3 @ Y2 ) ) ) ).
% mult_less_iff1
thf(fact_158_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_159_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_160_mult__less__cancel__right__disj,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A3 @ B3 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_161_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_162_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_163_mult__strict__right__mono__neg,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_164_mult__less__cancel__left__disj,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A3 @ B3 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_165_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_166_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_167_mult__strict__left__mono__neg,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_168_linorder__neqE__linordered__idom,axiom,
! [X3: real,Y2: real] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ Y2 @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_169_mult__right__cancel,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_170_mult__right__cancel,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_171_mult__right__cancel,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_172_mult__cancel__right,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B3 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_173_mult__cancel__right,axiom,
! [A3: complex,C2: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B3 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_174_mult__cancel__right,axiom,
! [A3: real,C2: real,B3: real] :
( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B3 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_175_mult__left__cancel,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_176_mult__left__cancel,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ C2 @ A3 )
= ( times_times_complex @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_177_mult__left__cancel,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_178_mult__cancel__left,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B3 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_179_mult__cancel__left,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( ( times_times_complex @ C2 @ A3 )
= ( times_times_complex @ C2 @ B3 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_180_mult__cancel__left,axiom,
! [C2: real,A3: real,B3: real] :
( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B3 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_181_no__zero__divisors,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B3 != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B3 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_182_no__zero__divisors,axiom,
! [A3: complex,B3: complex] :
( ( A3 != zero_zero_complex )
=> ( ( B3 != zero_zero_complex )
=> ( ( times_times_complex @ A3 @ B3 )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_183_no__zero__divisors,axiom,
! [A3: real,B3: real] :
( ( A3 != zero_zero_real )
=> ( ( B3 != zero_zero_real )
=> ( ( times_times_real @ A3 @ B3 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_184_mult__eq__0__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_185_mult__eq__0__iff,axiom,
! [A3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ B3 )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( B3 = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_186_mult__eq__0__iff,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_187_divisors__zero,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_188_divisors__zero,axiom,
! [A3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ B3 )
= zero_zero_complex )
=> ( ( A3 = zero_zero_complex )
| ( B3 = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_189_divisors__zero,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_190_mult__not__zero,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B3 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_191_mult__not__zero,axiom,
! [A3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ B3 )
!= zero_zero_complex )
=> ( ( A3 != zero_zero_complex )
& ( B3 != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_192_mult__not__zero,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B3 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_193_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_194_lambda__zero,axiom,
( ( ^ [H: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_195_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_196_mult__sign__intros_I8_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(8)
thf(fact_197_mult__sign__intros_I7_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(7)
thf(fact_198_mult__sign__intros_I7_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(7)
thf(fact_199_mult__sign__intros_I6_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(6)
thf(fact_200_mult__sign__intros_I6_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(6)
thf(fact_201_mult__sign__intros_I5_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_202_mult__sign__intros_I5_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_203_not__square__less__zero,axiom,
! [A3: real] :
~ ( ord_less_real @ ( times_times_real @ A3 @ A3 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_204_mult__less__0__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).
% mult_less_0_iff
thf(fact_205_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B3 @ A3 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_206_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B3 @ A3 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_207_zero__less__mult__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_208_zero__less__mult__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_209_zero__less__mult__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_210_zero__less__mult__pos2,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B3 @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_211_zero__less__mult__pos2,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B3 @ A3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_212_mult__less__cancel__left__neg,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( ord_less_real @ B3 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_213_mult__less__cancel__left__pos,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_214_max__mix__is__density,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( comple5220265106149225959erator @ ( projec8360710381328234318ensity @ N2 ) ) ) ).
% max_mix_is_density
thf(fact_215_mult__delta__right,axiom,
! [B3: $o,X3: nat,Y2: nat] :
( ( B3
=> ( ( times_times_nat @ X3 @ ( if_nat @ B3 @ Y2 @ zero_zero_nat ) )
= ( times_times_nat @ X3 @ Y2 ) ) )
& ( ~ B3
=> ( ( times_times_nat @ X3 @ ( if_nat @ B3 @ Y2 @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_216_mult__delta__right,axiom,
! [B3: $o,X3: complex,Y2: complex] :
( ( B3
=> ( ( times_times_complex @ X3 @ ( if_complex @ B3 @ Y2 @ zero_zero_complex ) )
= ( times_times_complex @ X3 @ Y2 ) ) )
& ( ~ B3
=> ( ( times_times_complex @ X3 @ ( if_complex @ B3 @ Y2 @ zero_zero_complex ) )
= zero_zero_complex ) ) ) ).
% mult_delta_right
thf(fact_217_mult__delta__right,axiom,
! [B3: $o,X3: real,Y2: real] :
( ( B3
=> ( ( times_times_real @ X3 @ ( if_real @ B3 @ Y2 @ zero_zero_real ) )
= ( times_times_real @ X3 @ Y2 ) ) )
& ( ~ B3
=> ( ( times_times_real @ X3 @ ( if_real @ B3 @ Y2 @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_218_mult__delta__left,axiom,
! [B3: $o,X3: nat,Y2: nat] :
( ( B3
=> ( ( times_times_nat @ ( if_nat @ B3 @ X3 @ zero_zero_nat ) @ Y2 )
= ( times_times_nat @ X3 @ Y2 ) ) )
& ( ~ B3
=> ( ( times_times_nat @ ( if_nat @ B3 @ X3 @ zero_zero_nat ) @ Y2 )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_219_mult__delta__left,axiom,
! [B3: $o,X3: complex,Y2: complex] :
( ( B3
=> ( ( times_times_complex @ ( if_complex @ B3 @ X3 @ zero_zero_complex ) @ Y2 )
= ( times_times_complex @ X3 @ Y2 ) ) )
& ( ~ B3
=> ( ( times_times_complex @ ( if_complex @ B3 @ X3 @ zero_zero_complex ) @ Y2 )
= zero_zero_complex ) ) ) ).
% mult_delta_left
thf(fact_220_mult__delta__left,axiom,
! [B3: $o,X3: real,Y2: real] :
( ( B3
=> ( ( times_times_real @ ( if_real @ B3 @ X3 @ zero_zero_real ) @ Y2 )
= ( times_times_real @ X3 @ Y2 ) ) )
& ( ~ B3
=> ( ( times_times_real @ ( if_real @ B3 @ X3 @ zero_zero_real ) @ Y2 )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_221_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X3: complex,A3: complex,B3: complex] :
( ( X3 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ X3 )
= ( times_times_complex @ B3 @ X3 ) )
=> ( A3 = B3 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_222_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X3: real,A3: real,B3: real] :
( ( X3 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X3 )
= ( times_times_real @ B3 @ X3 ) )
=> ( A3 = B3 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_223_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: complex,X3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ X3 )
= ( times_times_complex @ B3 @ X3 ) )
= ( ( A3 = B3 )
| ( X3 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_224_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: real,X3: real,B3: real] :
( ( ( times_times_real @ A3 @ X3 )
= ( times_times_real @ B3 @ X3 ) )
= ( ( A3 = B3 )
| ( X3 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_225_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: complex,X3: complex,Y2: complex] :
( ( A3 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ X3 )
= ( times_times_complex @ A3 @ Y2 ) )
=> ( X3 = Y2 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_226_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: real,X3: real,Y2: real] :
( ( A3 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X3 )
= ( times_times_real @ A3 @ Y2 ) )
=> ( X3 = Y2 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_227_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: complex,X3: complex,Y2: complex] :
( ( ( times_times_complex @ A3 @ X3 )
= ( times_times_complex @ A3 @ Y2 ) )
= ( ( X3 = Y2 )
| ( A3 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_228_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: real,X3: real,Y2: real] :
( ( ( times_times_real @ A3 @ X3 )
= ( times_times_real @ A3 @ Y2 ) )
= ( ( X3 = Y2 )
| ( A3 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_229_max__mix__density__carrier,axiom,
! [N2: nat] : ( member_mat_complex @ ( projec8360710381328234318ensity @ N2 ) @ ( carrier_mat_complex @ N2 @ N2 ) ) ).
% max_mix_density_carrier
thf(fact_230_vector__space__over__itself_Ovector__space__assms_I3_J,axiom,
! [A3: complex,B3: complex,X3: complex] :
( ( times_times_complex @ A3 @ ( times_times_complex @ B3 @ X3 ) )
= ( times_times_complex @ ( times_times_complex @ A3 @ B3 ) @ X3 ) ) ).
% vector_space_over_itself.vector_space_assms(3)
thf(fact_231_vector__space__over__itself_Ovector__space__assms_I3_J,axiom,
! [A3: real,B3: real,X3: real] :
( ( times_times_real @ A3 @ ( times_times_real @ B3 @ X3 ) )
= ( times_times_real @ ( times_times_real @ A3 @ B3 ) @ X3 ) ) ).
% vector_space_over_itself.vector_space_assms(3)
thf(fact_232_vector__space__over__itself_Oscale__left__commute,axiom,
! [A3: complex,B3: complex,X3: complex] :
( ( times_times_complex @ A3 @ ( times_times_complex @ B3 @ X3 ) )
= ( times_times_complex @ B3 @ ( times_times_complex @ A3 @ X3 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_233_vector__space__over__itself_Oscale__left__commute,axiom,
! [A3: real,B3: real,X3: real] :
( ( times_times_real @ A3 @ ( times_times_real @ B3 @ X3 ) )
= ( times_times_real @ B3 @ ( times_times_real @ A3 @ X3 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_234_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: complex,X3: complex] :
( ( ( times_times_complex @ A3 @ X3 )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( X3 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_235_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: real,X3: real] :
( ( ( times_times_real @ A3 @ X3 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( X3 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_236_vector__space__over__itself_Oscale__zero__left,axiom,
! [X3: complex] :
( ( times_times_complex @ zero_zero_complex @ X3 )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_237_vector__space__over__itself_Oscale__zero__left,axiom,
! [X3: real] :
( ( times_times_real @ zero_zero_real @ X3 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_238_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_239_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_240_mult__hom_Ohom__zero,axiom,
! [C2: nat] :
( ( times_times_nat @ C2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_241_mult__hom_Ohom__zero,axiom,
! [C2: complex] :
( ( times_times_complex @ C2 @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_hom.hom_zero
thf(fact_242_mult__hom_Ohom__zero,axiom,
! [C2: real] :
( ( times_times_real @ C2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_243_mat__diag__diag,axiom,
! [N2: nat,F: nat > nat,G: nat > nat] :
( ( times_times_mat_nat @ ( mat_diag_nat @ N2 @ F ) @ ( mat_diag_nat @ N2 @ G ) )
= ( mat_diag_nat @ N2
@ ^ [I2: nat] : ( times_times_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) ) ).
% mat_diag_diag
thf(fact_244_mat__diag__diag,axiom,
! [N2: nat,F: nat > real,G: nat > real] :
( ( times_times_mat_real @ ( mat_diag_real @ N2 @ F ) @ ( mat_diag_real @ N2 @ G ) )
= ( mat_diag_real @ N2
@ ^ [I2: nat] : ( times_times_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) ) ).
% mat_diag_diag
thf(fact_245_mat__diag__diag,axiom,
! [N2: nat,F: nat > complex,G: nat > complex] :
( ( times_8009071140041733218omplex @ ( mat_diag_complex @ N2 @ F ) @ ( mat_diag_complex @ N2 @ G ) )
= ( mat_diag_complex @ N2
@ ^ [I2: nat] : ( times_times_complex @ ( F @ I2 ) @ ( G @ I2 ) ) ) ) ).
% mat_diag_diag
thf(fact_246_mat__conj__unit__commute,axiom,
! [U2: mat_complex,A: mat_complex,N2: nat] :
( ( comple6660659447773130958omplex @ U2 )
=> ( ( ( times_8009071140041733218omplex @ U2 @ A )
= ( times_8009071140041733218omplex @ A @ U2 ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( spectr5699176650994449695omplex @ U2 @ A )
= A ) ) ) ) ) ).
% mat_conj_unit_commute
thf(fact_247_mult__commute__abs,axiom,
! [C2: nat] :
( ( ^ [X2: nat] : ( times_times_nat @ X2 @ C2 ) )
= ( times_times_nat @ C2 ) ) ).
% mult_commute_abs
thf(fact_248_mult__commute__abs,axiom,
! [C2: complex] :
( ( ^ [X2: complex] : ( times_times_complex @ X2 @ C2 ) )
= ( times_times_complex @ C2 ) ) ).
% mult_commute_abs
thf(fact_249_mult__commute__abs,axiom,
! [C2: real] :
( ( ^ [X2: real] : ( times_times_real @ X2 @ C2 ) )
= ( times_times_real @ C2 ) ) ).
% mult_commute_abs
thf(fact_250_unitary__density,axiom,
! [R: mat_complex,U2: mat_complex,N2: nat] :
( ( comple5220265106149225959erator @ R )
=> ( ( comple6660659447773130958omplex @ U2 )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( comple5220265106149225959erator @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U2 @ R ) @ ( schur_5982229384592763574omplex @ U2 ) ) ) ) ) ) ) ).
% unitary_density
thf(fact_251_mat__to__cols__list__is__not__Nil,axiom,
! [A: mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ A ) )
=> ( ( mat_to_cols_list @ A )
!= nil_list_complex ) ) ).
% mat_to_cols_list_is_not_Nil
thf(fact_252_Complex__Matrix_Oadjoint__adjoint,axiom,
! [A: mat_complex] :
( ( schur_5982229384592763574omplex @ ( schur_5982229384592763574omplex @ A ) )
= A ) ).
% Complex_Matrix.adjoint_adjoint
thf(fact_253_mat__conj__adjoint,axiom,
! [U2: mat_complex,V: mat_complex] :
( ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U2 ) @ V )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U2 ) @ V ) @ U2 ) ) ).
% mat_conj_adjoint
thf(fact_254_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_255_adjoint__dim,axiom,
! [A: mat_complex,N2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( member_mat_complex @ ( schur_5982229384592763574omplex @ A ) @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ).
% adjoint_dim
thf(fact_256_adjoint__dim_H,axiom,
! [A: mat_complex,N2: nat,M: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ M ) )
=> ( member_mat_complex @ ( schur_5982229384592763574omplex @ A ) @ ( carrier_mat_complex @ M @ N2 ) ) ) ).
% adjoint_dim'
thf(fact_257_hermitian__mat__conj_H,axiom,
! [A: mat_complex,N2: nat,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U2 ) @ A ) ) ) ) ) ).
% hermitian_mat_conj'
thf(fact_258_hermitian__def,axiom,
( comple8306762464034002205omplex
= ( ^ [A2: mat_complex] :
( ( schur_5982229384592763574omplex @ A2 )
= A2 ) ) ) ).
% hermitian_def
thf(fact_259_length__mat__to__cols__list,axiom,
! [A: mat_complex] :
( ( size_s7907857696548412130omplex @ ( mat_to_cols_list @ A ) )
= ( dim_col_complex @ A ) ) ).
% length_mat_to_cols_list
thf(fact_260_mat__conj__commute,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple6660659447773130958omplex @ U2 )
=> ( ( ( times_8009071140041733218omplex @ A @ B2 )
= ( times_8009071140041733218omplex @ B2 @ A ) )
=> ( ( times_8009071140041733218omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U2 ) @ A ) @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U2 ) @ B2 ) )
= ( times_8009071140041733218omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U2 ) @ B2 ) @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ U2 ) @ A ) ) ) ) ) ) ) ) ).
% mat_conj_commute
thf(fact_261_unitary__mult__conjugate,axiom,
! [A: mat_complex,N2: nat,V: mat_complex,U2: mat_complex,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ V @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple6660659447773130958omplex @ V )
=> ( ( ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ V ) @ A )
= ( spectr5699176650994449695omplex @ U2 @ B2 ) )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ V @ U2 ) @ B2 ) @ ( schur_5982229384592763574omplex @ ( times_8009071140041733218omplex @ V @ U2 ) ) ) ) ) ) ) ) ) ) ).
% unitary_mult_conjugate
thf(fact_262_adjoint__mult,axiom,
! [A: mat_complex,N2: nat,M: nat,B2: mat_complex,L: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ M ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ M @ L ) )
=> ( ( schur_5982229384592763574omplex @ ( times_8009071140041733218omplex @ A @ B2 ) )
= ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ B2 ) @ ( schur_5982229384592763574omplex @ A ) ) ) ) ) ).
% adjoint_mult
thf(fact_263_adjoint__dim__col,axiom,
! [A: mat_complex] :
( ( dim_col_complex @ ( schur_5982229384592763574omplex @ A ) )
= ( dim_row_complex @ A ) ) ).
% adjoint_dim_col
thf(fact_264_adjoint__dim__row,axiom,
! [A: mat_complex] :
( ( dim_row_complex @ ( schur_5982229384592763574omplex @ A ) )
= ( dim_col_complex @ A ) ) ).
% adjoint_dim_row
thf(fact_265_mat__assoc__test_I2_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ ( times_8009071140041733218omplex @ A @ ( schur_5982229384592763574omplex @ B2 ) ) ) @ C )
= ( times_8009071140041733218omplex @ B2 @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ A ) @ C ) ) ) ) ) ) ) ).
% mat_assoc_test(2)
thf(fact_266_unitary__adjoint,axiom,
! [A: mat_complex,N2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple6660659447773130958omplex @ A )
=> ( comple6660659447773130958omplex @ ( schur_5982229384592763574omplex @ A ) ) ) ) ).
% unitary_adjoint
thf(fact_267_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_268_mat__diag__dim,axiom,
! [N2: nat,F: nat > complex] : ( member_mat_complex @ ( mat_diag_complex @ N2 @ F ) @ ( carrier_mat_complex @ N2 @ N2 ) ) ).
% mat_diag_dim
thf(fact_269_unitary__conjugate__real__diag__decomp,axiom,
! [A: mat_complex,N2: nat,Us: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ Us @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple6660659447773130958omplex @ Us )
=> ( ( spectr5409772854192057952omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ Us ) @ A ) @ B2 @ U2 )
=> ( spectr5409772854192057952omplex @ A @ B2 @ ( times_8009071140041733218omplex @ Us @ U2 ) ) ) ) ) ) ).
% unitary_conjugate_real_diag_decomp
thf(fact_270_mult__adjoint__hermitian,axiom,
! [A: mat_complex,N2: nat,M: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ M ) )
=> ( comple8306762464034002205omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ A ) @ A ) ) ) ).
% mult_adjoint_hermitian
thf(fact_271_unitary__elim,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,P: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple6660659447773130958omplex @ P )
=> ( ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ A ) @ ( schur_5982229384592763574omplex @ P ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ B2 ) @ ( schur_5982229384592763574omplex @ P ) ) )
=> ( A = B2 ) ) ) ) ) ) ).
% unitary_elim
thf(fact_272_hermitian__mat__conj,axiom,
! [A: mat_complex,N2: nat,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple8306762464034002205omplex @ A )
=> ( comple8306762464034002205omplex @ ( spectr5699176650994449695omplex @ U2 @ A ) ) ) ) ) ).
% hermitian_mat_conj
thf(fact_273_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_274_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_275_unitarily__equiv__conjugate,axiom,
! [A: mat_complex,N2: nat,V: mat_complex,U2: mat_complex,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ V @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( spectr6340060708231679580omplex @ ( spectr5699176650994449695omplex @ ( schur_5982229384592763574omplex @ V ) @ A ) @ B2 @ U2 )
=> ( ( comple6660659447773130958omplex @ V )
=> ( spectr6340060708231679580omplex @ A @ B2 @ ( times_8009071140041733218omplex @ V @ U2 ) ) ) ) ) ) ) ) ).
% unitarily_equiv_conjugate
thf(fact_276_unitaryD2,axiom,
! [A: mat_complex,N2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple6660659447773130958omplex @ A )
=> ( inverts_mat_complex @ ( schur_5982229384592763574omplex @ A ) @ A ) ) ) ).
% unitaryD2
thf(fact_277_row__length__mat__to__cols__list,axiom,
! [A: mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ A ) )
=> ( ( matrix1515831402840476169omplex @ ( mat_to_cols_list @ A ) )
= ( dim_row_complex @ A ) ) ) ).
% row_length_mat_to_cols_list
thf(fact_278_conjugate__eq__unitarily__equiv,axiom,
! [A: mat_complex,N2: nat,V: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ V @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( spectr6340060708231679580omplex @ A @ B2 @ U2 )
=> ( ( comple6660659447773130958omplex @ V )
=> ( ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ V @ B2 ) @ ( schur_5982229384592763574omplex @ V ) )
= B2 )
=> ( spectr6340060708231679580omplex @ A @ B2 @ ( times_8009071140041733218omplex @ U2 @ V ) ) ) ) ) ) ) ).
% conjugate_eq_unitarily_equiv
thf(fact_279_length__col__mat__to__cols__list,axiom,
! [J: nat,A: mat_complex] :
( ( ord_less_nat @ J @ ( dim_col_complex @ A ) )
=> ( ( size_s3451745648224563538omplex @ ( matrix_col_complex @ ( mat_to_cols_list @ A ) @ J ) )
= ( dim_row_complex @ A ) ) ) ).
% length_col_mat_to_cols_list
thf(fact_280_length__row__mat__to__cols__list,axiom,
! [I3: nat,A: mat_complex] :
( ( ord_less_nat @ I3 @ ( dim_row_complex @ A ) )
=> ( ( size_s3451745648224563538omplex @ ( matrix_row_complex @ ( mat_to_cols_list @ A ) @ I3 ) )
= ( dim_col_complex @ A ) ) ) ).
% length_row_mat_to_cols_list
thf(fact_281_unitary__operator__keep__trace,axiom,
! [U2: mat_complex,N2: nat,A: mat_complex] :
( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple6660659447773130958omplex @ U2 )
=> ( ( comple3184165445352484367omplex @ A )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U2 ) @ A ) @ U2 ) ) ) ) ) ) ).
% unitary_operator_keep_trace
thf(fact_282_unitarily__equiv__trace,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( spectr6340060708231679580omplex @ A @ B2 @ U2 )
=> ( ( comple3184165445352484367omplex @ A )
= ( comple3184165445352484367omplex @ B2 ) ) ) ) ).
% unitarily_equiv_trace
thf(fact_283_unitarily__equiv__carrier_I1_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( spectr6340060708231679580omplex @ A @ B2 @ U2 )
=> ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ).
% unitarily_equiv_carrier(1)
thf(fact_284_unitarily__equiv__carrier_I2_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( spectr6340060708231679580omplex @ A @ B2 @ U2 )
=> ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ).
% unitarily_equiv_carrier(2)
thf(fact_285_unitarily__equiv__adjoint,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B2 @ U2 )
=> ( spectr6340060708231679580omplex @ B2 @ A @ ( schur_5982229384592763574omplex @ U2 ) ) ) ).
% unitarily_equiv_adjoint
thf(fact_286_unitarily__equivD_I1_J,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B2 @ U2 )
=> ( comple6660659447773130958omplex @ U2 ) ) ).
% unitarily_equivD(1)
thf(fact_287_inverts__mat__unique,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( inverts_mat_complex @ A @ B2 )
=> ( ( inverts_mat_complex @ A @ C )
=> ( B2 = C ) ) ) ) ) ) ).
% inverts_mat_unique
thf(fact_288_inverts__mat__symm,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( inverts_mat_complex @ A @ B2 )
=> ( inverts_mat_complex @ B2 @ A ) ) ) ) ).
% inverts_mat_symm
thf(fact_289_mat__assoc__test_I10_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ C ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ B2 @ C ) @ A ) ) ) ) ) ) ) ).
% mat_assoc_test(10)
thf(fact_290_mat__assoc__test_I11_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ C ) @ D ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ C @ D ) @ A ) @ B2 ) ) ) ) ) ) ) ).
% mat_assoc_test(11)
thf(fact_291_unitarily__equiv__carrier_H_I1_J,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B2 @ U2 )
=> ( member_mat_complex @ A @ ( carrier_mat_complex @ ( dim_row_complex @ A ) @ ( dim_row_complex @ A ) ) ) ) ).
% unitarily_equiv_carrier'(1)
thf(fact_292_unitarily__equiv__carrier_H_I2_J,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B2 @ U2 )
=> ( member_mat_complex @ B2 @ ( carrier_mat_complex @ ( dim_row_complex @ A ) @ ( dim_row_complex @ A ) ) ) ) ).
% unitarily_equiv_carrier'(2)
thf(fact_293_unitarily__equiv__carrier_H_I3_J,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B2 @ U2 )
=> ( member_mat_complex @ U2 @ ( carrier_mat_complex @ ( dim_row_complex @ A ) @ ( dim_row_complex @ A ) ) ) ) ).
% unitarily_equiv_carrier'(3)
thf(fact_294_unitarily__equiv__square,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( spectr6340060708231679580omplex @ A @ B2 @ U2 )
=> ( spectr6340060708231679580omplex @ ( times_8009071140041733218omplex @ A @ A ) @ ( times_8009071140041733218omplex @ B2 @ B2 ) @ U2 ) ) ) ).
% unitarily_equiv_square
thf(fact_295_unitarily__equiv__eq,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B2 @ U2 )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ U2 @ B2 ) @ ( schur_5982229384592763574omplex @ U2 ) ) ) ) ).
% unitarily_equiv_eq
thf(fact_296_unitarily__equiv__commute,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex,C: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B2 @ U2 )
=> ( ( ( times_8009071140041733218omplex @ A @ C )
= ( times_8009071140041733218omplex @ C @ A ) )
=> ( ( times_8009071140041733218omplex @ B2 @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U2 ) @ C ) @ U2 ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U2 ) @ C ) @ U2 ) @ B2 ) ) ) ) ).
% unitarily_equiv_commute
thf(fact_297_trace__comm,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ B2 ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B2 @ A ) ) ) ) ) ).
% trace_comm
thf(fact_298_projector__collapse__trace,axiom,
! [P: mat_complex,N2: nat,R: mat_complex] :
( ( linear5633924348262549461omplex @ P )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ R ) @ P ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ P ) ) ) ) ) ) ).
% projector_collapse_trace
thf(fact_299_unitarily__equivI_H,axiom,
! [A: mat_complex,U2: mat_complex,B2: mat_complex,N2: nat] :
( ( A
= ( spectr5699176650994449695omplex @ U2 @ B2 ) )
=> ( ( comple6660659447773130958omplex @ U2 )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( spectr6340060708231679580omplex @ A @ B2 @ U2 ) ) ) ) ) ).
% unitarily_equivI'
thf(fact_300_trace__pdo__eq__imp__eq,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ! [Rho: mat_complex] :
( ( member_mat_complex @ Rho @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple1169154605998056944erator @ Rho )
=> ( ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ Rho ) )
= ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B2 @ Rho ) ) ) ) )
=> ( A = B2 ) ) ) ) ).
% trace_pdo_eq_imp_eq
thf(fact_301_mat__to__cols__list__is__mat,axiom,
! [A: mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ A ) )
=> ( matrix_mat_complex @ ( matrix1515831402840476169omplex @ ( mat_to_cols_list @ A ) ) @ ( size_s7907857696548412130omplex @ ( mat_to_cols_list @ A ) ) @ ( mat_to_cols_list @ A ) ) ) ).
% mat_to_cols_list_is_mat
thf(fact_302_mat__to__cols__list__to__mat,axiom,
! [A: mat_complex] :
( ( mat_of_cols_list @ ( dim_row_complex @ A ) @ ( mat_to_cols_list @ A ) )
= A ) ).
% mat_to_cols_list_to_mat
thf(fact_303_lowner__le__keep__under__measurement,axiom,
! [M2: mat_complex,N2: nat,A: mat_complex,B2: mat_complex] :
( ( member_mat_complex @ M2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( complex_lowner_le @ A @ B2 )
=> ( complex_lowner_le @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ M2 ) @ A ) @ M2 ) @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ M2 ) @ B2 ) @ M2 ) ) ) ) ) ) ).
% lowner_le_keep_under_measurement
thf(fact_304_unitary__mult__square__eq,axiom,
! [A: mat_complex,N2: nat,U2: mat_complex,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( A
= ( spectr5699176650994449695omplex @ U2 @ B2 ) )
=> ( ( ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ U2 ) @ U2 )
= ( one_mat_complex @ N2 ) )
=> ( ( times_8009071140041733218omplex @ A @ A )
= ( spectr5699176650994449695omplex @ U2 @ ( times_8009071140041733218omplex @ B2 @ B2 ) ) ) ) ) ) ) ) ).
% unitary_mult_square_eq
thf(fact_305_unitarily__equivI,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B2 @ U2 @ ( schur_5982229384592763574omplex @ U2 ) )
=> ( ( comple6660659447773130958omplex @ U2 )
=> ( spectr6340060708231679580omplex @ A @ B2 @ U2 ) ) ) ).
% unitarily_equivI
thf(fact_306_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_307_list__to__mat__to__cols__list,axiom,
! [Nr: nat,Nc: nat,L: list_list_complex] :
( ( matrix_mat_complex @ Nr @ Nc @ L )
=> ( ( mat_to_cols_list @ ( mat_of_cols_list @ Nr @ L ) )
= L ) ) ).
% list_to_mat_to_cols_list
thf(fact_308_similar__mat__wit__refl,axiom,
! [A: mat_complex,N2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( simila5774310414453981135omplex @ A @ A @ ( one_mat_complex @ N2 ) @ ( one_mat_complex @ N2 ) ) ) ).
% similar_mat_wit_refl
thf(fact_309_similar__mat__wit__sym,axiom,
! [A: mat_complex,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( simila5774310414453981135omplex @ B2 @ A @ Q @ P ) ) ).
% similar_mat_wit_sym
thf(fact_310_similar__mat__witD2_I2_J,axiom,
! [A: mat_complex,N2: nat,M: nat,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( ( times_8009071140041733218omplex @ Q @ P )
= ( one_mat_complex @ N2 ) ) ) ) ).
% similar_mat_witD2(2)
thf(fact_311_similar__mat__witD2_I1_J,axiom,
! [A: mat_complex,N2: nat,M: nat,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( ( times_8009071140041733218omplex @ P @ Q )
= ( one_mat_complex @ N2 ) ) ) ) ).
% similar_mat_witD2(1)
thf(fact_312_similar__mat__witI,axiom,
! [P: mat_complex,Q: mat_complex,N2: nat,A: mat_complex,B2: mat_complex] :
( ( ( times_8009071140041733218omplex @ P @ Q )
= ( one_mat_complex @ N2 ) )
=> ( ( ( times_8009071140041733218omplex @ Q @ P )
= ( one_mat_complex @ N2 ) )
=> ( ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ B2 ) @ Q ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ Q @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( simila5774310414453981135omplex @ A @ B2 @ P @ Q ) ) ) ) ) ) ) ) ).
% similar_mat_witI
thf(fact_313_similar__mat__witD_I2_J,axiom,
! [N2: nat,A: mat_complex,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N2
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( ( times_8009071140041733218omplex @ Q @ P )
= ( one_mat_complex @ N2 ) ) ) ) ).
% similar_mat_witD(2)
thf(fact_314_similar__mat__witD_I1_J,axiom,
! [N2: nat,A: mat_complex,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N2
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( ( times_8009071140041733218omplex @ P @ Q )
= ( one_mat_complex @ N2 ) ) ) ) ).
% similar_mat_witD(1)
thf(fact_315_one__carrier__mat,axiom,
! [N2: nat] : ( member_mat_complex @ ( one_mat_complex @ N2 ) @ ( carrier_mat_complex @ N2 @ N2 ) ) ).
% one_carrier_mat
thf(fact_316_index__one__mat_I2_J,axiom,
! [N2: nat] :
( ( dim_row_complex @ ( one_mat_complex @ N2 ) )
= N2 ) ).
% index_one_mat(2)
thf(fact_317_index__one__mat_I3_J,axiom,
! [N2: nat] :
( ( dim_col_complex @ ( one_mat_complex @ N2 ) )
= N2 ) ).
% index_one_mat(3)
thf(fact_318_hermitian__one,axiom,
! [N2: nat] : ( comple8306762464034002205omplex @ ( one_mat_complex @ N2 ) ) ).
% hermitian_one
thf(fact_319_unitary__one,axiom,
! [N2: nat] : ( comple6660659447773130958omplex @ ( one_mat_complex @ N2 ) ) ).
% unitary_one
thf(fact_320_adjoint__one,axiom,
! [N2: nat] :
( ( schur_5982229384592763574omplex @ ( one_mat_complex @ N2 ) )
= ( one_mat_complex @ N2 ) ) ).
% adjoint_one
thf(fact_321_similar__mat__witD2_I7_J,axiom,
! [A: mat_complex,N2: nat,M: nat,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( member_mat_complex @ Q @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ).
% similar_mat_witD2(7)
thf(fact_322_similar__mat__witD2_I6_J,axiom,
! [A: mat_complex,N2: nat,M: nat,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( member_mat_complex @ P @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ).
% similar_mat_witD2(6)
thf(fact_323_similar__mat__witD2_I5_J,axiom,
! [A: mat_complex,N2: nat,M: nat,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ).
% similar_mat_witD2(5)
thf(fact_324_similar__mat__witD2_I4_J,axiom,
! [A: mat_complex,N2: nat,M: nat,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ).
% similar_mat_witD2(4)
thf(fact_325_similar__mat__wit__dim__row,axiom,
! [A: mat_complex,B2: mat_complex,Q: mat_complex,R: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B2 @ Q @ R )
=> ( ( dim_row_complex @ B2 )
= ( dim_row_complex @ A ) ) ) ).
% similar_mat_wit_dim_row
thf(fact_326_similar__mat__wit__trans,axiom,
! [A: mat_complex,B2: mat_complex,P: mat_complex,Q: mat_complex,C: mat_complex,P3: mat_complex,Q2: mat_complex] :
( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( ( simila5774310414453981135omplex @ B2 @ C @ P3 @ Q2 )
=> ( simila5774310414453981135omplex @ A @ C @ ( times_8009071140041733218omplex @ P @ P3 ) @ ( times_8009071140041733218omplex @ Q2 @ Q ) ) ) ) ).
% similar_mat_wit_trans
thf(fact_327_lowner__le__refl,axiom,
! [A: mat_complex,N2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( complex_lowner_le @ A @ A ) ) ).
% lowner_le_refl
thf(fact_328_lowner__le__trans,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,C: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( complex_lowner_le @ A @ B2 )
=> ( ( complex_lowner_le @ B2 @ C )
=> ( complex_lowner_le @ A @ C ) ) ) ) ) ) ).
% lowner_le_trans
thf(fact_329_lowner__le__antisym,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( complex_lowner_le @ A @ B2 )
=> ( ( complex_lowner_le @ B2 @ A )
=> ( A = B2 ) ) ) ) ) ).
% lowner_le_antisym
thf(fact_330_left__mult__one__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( one_mat_complex @ Nr ) @ A )
= A ) ) ).
% left_mult_one_mat
thf(fact_331_right__mult__one__mat,axiom,
! [A: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A @ ( one_mat_complex @ Nc ) )
= A ) ) ).
% right_mult_one_mat
thf(fact_332_left__mult__one__mat_H,axiom,
! [A: mat_complex,N2: nat] :
( ( ( dim_row_complex @ A )
= N2 )
=> ( ( times_8009071140041733218omplex @ ( one_mat_complex @ N2 ) @ A )
= A ) ) ).
% left_mult_one_mat'
thf(fact_333_right__mult__one__mat_H,axiom,
! [A: mat_complex,N2: nat] :
( ( ( dim_col_complex @ A )
= N2 )
=> ( ( times_8009071140041733218omplex @ A @ ( one_mat_complex @ N2 ) )
= A ) ) ).
% right_mult_one_mat'
thf(fact_334_mat__assoc__test_I3_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A @ ( one_mat_complex @ N2 ) ) @ ( one_mat_complex @ N2 ) ) @ B2 ) @ ( one_mat_complex @ N2 ) )
= ( times_8009071140041733218omplex @ A @ B2 ) ) ) ) ) ) ).
% mat_assoc_test(3)
thf(fact_335_similar__mat__witD_I7_J,axiom,
! [N2: nat,A: mat_complex,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N2
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( member_mat_complex @ Q @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ).
% similar_mat_witD(7)
thf(fact_336_similar__mat__witD_I6_J,axiom,
! [N2: nat,A: mat_complex,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N2
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( member_mat_complex @ P @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ).
% similar_mat_witD(6)
thf(fact_337_similar__mat__witD_I5_J,axiom,
! [N2: nat,A: mat_complex,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N2
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ).
% similar_mat_witD(5)
thf(fact_338_similar__mat__witD_I4_J,axiom,
! [N2: nat,A: mat_complex,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N2
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ).
% similar_mat_witD(4)
thf(fact_339_similar__mat__witD2_I3_J,axiom,
! [A: mat_complex,N2: nat,M: nat,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ M ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ B2 ) @ Q ) ) ) ) ).
% similar_mat_witD2(3)
thf(fact_340_similar__mat__witD_I3_J,axiom,
! [N2: nat,A: mat_complex,B2: mat_complex,P: mat_complex,Q: mat_complex] :
( ( N2
= ( dim_row_complex @ A ) )
=> ( ( simila5774310414453981135omplex @ A @ B2 @ P @ Q )
=> ( A
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ P @ B2 ) @ Q ) ) ) ) ).
% similar_mat_witD(3)
thf(fact_341_unitarily__equivD_I2_J,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr6340060708231679580omplex @ A @ B2 @ U2 )
=> ( simila5774310414453981135omplex @ A @ B2 @ U2 @ ( schur_5982229384592763574omplex @ U2 ) ) ) ).
% unitarily_equivD(2)
thf(fact_342_hermitian__decomp__sim,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( projec5943904436471448624omplex @ A @ B2 @ U2 )
=> ( simila5774310414453981135omplex @ A @ B2 @ U2 @ ( schur_5982229384592763574omplex @ U2 ) ) ) ).
% hermitian_decomp_sim
thf(fact_343_inverts__mat__def,axiom,
( inverts_mat_complex
= ( ^ [A2: mat_complex,B5: mat_complex] :
( ( times_8009071140041733218omplex @ A2 @ B5 )
= ( one_mat_complex @ ( dim_row_complex @ A2 ) ) ) ) ) ).
% inverts_mat_def
thf(fact_344_tensor__mat__id,axiom,
! [D1: nat,D2: nat] :
( ( ord_less_nat @ zero_zero_nat @ D1 )
=> ( ( ord_less_nat @ zero_zero_nat @ D2 )
=> ( ( tensor_mat @ ( one_mat_complex @ D1 ) @ ( one_mat_complex @ D2 ) )
= ( one_mat_complex @ ( times_times_nat @ D1 @ D2 ) ) ) ) ) ).
% tensor_mat_id
thf(fact_345_unitary__simps_I2_J,axiom,
! [A: mat_complex,N2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple6660659447773130958omplex @ A )
=> ( ( times_8009071140041733218omplex @ A @ ( schur_5982229384592763574omplex @ A ) )
= ( one_mat_complex @ N2 ) ) ) ) ).
% unitary_simps(2)
thf(fact_346_unitary__simps_I1_J,axiom,
! [A: mat_complex,N2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple6660659447773130958omplex @ A )
=> ( ( times_8009071140041733218omplex @ ( schur_5982229384592763574omplex @ A ) @ A )
= ( one_mat_complex @ N2 ) ) ) ) ).
% unitary_simps(1)
thf(fact_347_row__col,axiom,
! [Nr: nat,Nc: nat,M: list_l5436439031154120755omplex,I3: nat,J: nat] :
( ( matrix6216835547647503100omplex @ Nr @ Nc @ M )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( nth_mat_complex @ ( matrix8356865337816758658omplex @ M @ I3 ) @ J )
= ( nth_mat_complex @ ( matrix725289766534482588omplex @ M @ J ) @ I3 ) ) ) ) ) ).
% row_col
thf(fact_348_row__col,axiom,
! [Nr: nat,Nc: nat,M: list_l3981933317855906654omplex,I3: nat,J: nat] :
( ( matrix6976670468949791273omplex @ Nr @ Nc @ M )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( nth_list_complex @ ( matrix9134808206163378723omplex @ M @ I3 ) @ J )
= ( nth_list_complex @ ( matrix9199650144385845897omplex @ M @ J ) @ I3 ) ) ) ) ) ).
% row_col
thf(fact_349_row__col,axiom,
! [Nr: nat,Nc: nat,M: list_list_complex,I3: nat,J: nat] :
( ( matrix_mat_complex @ Nr @ Nc @ M )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( nth_complex @ ( matrix_row_complex @ M @ I3 ) @ J )
= ( nth_complex @ ( matrix_col_complex @ M @ J ) @ I3 ) ) ) ) ) ).
% row_col
thf(fact_350_mat__col__eq,axiom,
! [Nr: nat,Nc: nat,M1: list_list_complex,M22: list_list_complex] :
( ( matrix_mat_complex @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_complex @ Nr @ Nc @ M22 )
=> ( ( M1 = M22 )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ Nc )
=> ( ( matrix_col_complex @ M1 @ I2 )
= ( matrix_col_complex @ M22 @ I2 ) ) ) ) ) ) ) ).
% mat_col_eq
thf(fact_351_Matrix__Legacy_Omat__col__eqI,axiom,
! [Nr: nat,Nc: nat,M1: list_list_complex,M22: list_list_complex] :
( ( matrix_mat_complex @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_complex @ Nr @ Nc @ M22 )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ Nc )
=> ( ( matrix_col_complex @ M1 @ I )
= ( matrix_col_complex @ M22 @ I ) ) )
=> ( M1 = M22 ) ) ) ) ).
% Matrix_Legacy.mat_col_eqI
thf(fact_352_mat__row__eq,axiom,
! [Nr: nat,Nc: nat,M1: list_list_complex,M22: list_list_complex] :
( ( matrix_mat_complex @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_complex @ Nr @ Nc @ M22 )
=> ( ( M1 = M22 )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ Nr )
=> ( ( matrix_row_complex @ M1 @ I2 )
= ( matrix_row_complex @ M22 @ I2 ) ) ) ) ) ) ) ).
% mat_row_eq
thf(fact_353_mat__row__eqI,axiom,
! [Nr: nat,Nc: nat,M1: list_list_complex,M22: list_list_complex] :
( ( matrix_mat_complex @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_complex @ Nr @ Nc @ M22 )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ Nr )
=> ( ( matrix_row_complex @ M1 @ I )
= ( matrix_row_complex @ M22 @ I ) ) )
=> ( M1 = M22 ) ) ) ) ).
% mat_row_eqI
thf(fact_354_mat__mult__left__right__inverse,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( ( times_8009071140041733218omplex @ A @ B2 )
= ( one_mat_complex @ N2 ) )
=> ( ( times_8009071140041733218omplex @ B2 @ A )
= ( one_mat_complex @ N2 ) ) ) ) ) ).
% mat_mult_left_right_inverse
thf(fact_355_mat__eq,axiom,
! [Nr: nat,Nc: nat,M1: list_l5436439031154120755omplex,M22: list_l5436439031154120755omplex] :
( ( matrix6216835547647503100omplex @ Nr @ Nc @ M1 )
=> ( ( matrix6216835547647503100omplex @ Nr @ Nc @ M22 )
=> ( ( M1 = M22 )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ Nc )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ Nr )
=> ( ( nth_mat_complex @ ( nth_list_mat_complex @ M1 @ I2 ) @ J2 )
= ( nth_mat_complex @ ( nth_list_mat_complex @ M22 @ I2 ) @ J2 ) ) ) ) ) ) ) ) ).
% mat_eq
thf(fact_356_mat__eq,axiom,
! [Nr: nat,Nc: nat,M1: list_l3981933317855906654omplex,M22: list_l3981933317855906654omplex] :
( ( matrix6976670468949791273omplex @ Nr @ Nc @ M1 )
=> ( ( matrix6976670468949791273omplex @ Nr @ Nc @ M22 )
=> ( ( M1 = M22 )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ Nc )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ Nr )
=> ( ( nth_list_complex @ ( nth_li53272486250751239omplex @ M1 @ I2 ) @ J2 )
= ( nth_list_complex @ ( nth_li53272486250751239omplex @ M22 @ I2 ) @ J2 ) ) ) ) ) ) ) ) ).
% mat_eq
thf(fact_357_mat__eq,axiom,
! [Nr: nat,Nc: nat,M1: list_list_complex,M22: list_list_complex] :
( ( matrix_mat_complex @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_complex @ Nr @ Nc @ M22 )
=> ( ( M1 = M22 )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ Nc )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ Nr )
=> ( ( nth_complex @ ( nth_list_complex @ M1 @ I2 ) @ J2 )
= ( nth_complex @ ( nth_list_complex @ M22 @ I2 ) @ J2 ) ) ) ) ) ) ) ) ).
% mat_eq
thf(fact_358_mat__eqI,axiom,
! [Nr: nat,Nc: nat,M1: list_l5436439031154120755omplex,M22: list_l5436439031154120755omplex] :
( ( matrix6216835547647503100omplex @ Nr @ Nc @ M1 )
=> ( ( matrix6216835547647503100omplex @ Nr @ Nc @ M22 )
=> ( ! [I: nat,J3: nat] :
( ( ord_less_nat @ I @ Nc )
=> ( ( ord_less_nat @ J3 @ Nr )
=> ( ( nth_mat_complex @ ( nth_list_mat_complex @ M1 @ I ) @ J3 )
= ( nth_mat_complex @ ( nth_list_mat_complex @ M22 @ I ) @ J3 ) ) ) )
=> ( M1 = M22 ) ) ) ) ).
% mat_eqI
thf(fact_359_mat__eqI,axiom,
! [Nr: nat,Nc: nat,M1: list_l3981933317855906654omplex,M22: list_l3981933317855906654omplex] :
( ( matrix6976670468949791273omplex @ Nr @ Nc @ M1 )
=> ( ( matrix6976670468949791273omplex @ Nr @ Nc @ M22 )
=> ( ! [I: nat,J3: nat] :
( ( ord_less_nat @ I @ Nc )
=> ( ( ord_less_nat @ J3 @ Nr )
=> ( ( nth_list_complex @ ( nth_li53272486250751239omplex @ M1 @ I ) @ J3 )
= ( nth_list_complex @ ( nth_li53272486250751239omplex @ M22 @ I ) @ J3 ) ) ) )
=> ( M1 = M22 ) ) ) ) ).
% mat_eqI
thf(fact_360_mat__eqI,axiom,
! [Nr: nat,Nc: nat,M1: list_list_complex,M22: list_list_complex] :
( ( matrix_mat_complex @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_complex @ Nr @ Nc @ M22 )
=> ( ! [I: nat,J3: nat] :
( ( ord_less_nat @ I @ Nc )
=> ( ( ord_less_nat @ J3 @ Nr )
=> ( ( nth_complex @ ( nth_list_complex @ M1 @ I ) @ J3 )
= ( nth_complex @ ( nth_list_complex @ M22 @ I ) @ J3 ) ) ) )
=> ( M1 = M22 ) ) ) ) ).
% mat_eqI
thf(fact_361_mat__map__index,axiom,
! [Nr: nat,Nc: nat,M: list_l5436439031154120755omplex,I3: nat,J: nat,F: mat_complex > mat_complex] :
( ( matrix6216835547647503100omplex @ Nr @ Nc @ M )
=> ( ( ord_less_nat @ I3 @ Nc )
=> ( ( ord_less_nat @ J @ Nr )
=> ( ( nth_mat_complex @ ( nth_list_mat_complex @ ( matrix89116945220183700omplex @ F @ M ) @ I3 ) @ J )
= ( F @ ( nth_mat_complex @ ( nth_list_mat_complex @ M @ I3 ) @ J ) ) ) ) ) ) ).
% mat_map_index
thf(fact_362_mat__map__index,axiom,
! [Nr: nat,Nc: nat,M: list_l3981933317855906654omplex,I3: nat,J: nat,F: list_complex > list_complex] :
( ( matrix6976670468949791273omplex @ Nr @ Nc @ M )
=> ( ( ord_less_nat @ I3 @ Nc )
=> ( ( ord_less_nat @ J @ Nr )
=> ( ( nth_list_complex @ ( nth_li53272486250751239omplex @ ( matrix9216512725699604625omplex @ F @ M ) @ I3 ) @ J )
= ( F @ ( nth_list_complex @ ( nth_li53272486250751239omplex @ M @ I3 ) @ J ) ) ) ) ) ) ).
% mat_map_index
thf(fact_363_mat__map__index,axiom,
! [Nr: nat,Nc: nat,M: list_list_complex,I3: nat,J: nat,F: complex > complex] :
( ( matrix_mat_complex @ Nr @ Nc @ M )
=> ( ( ord_less_nat @ I3 @ Nc )
=> ( ( ord_less_nat @ J @ Nr )
=> ( ( nth_complex @ ( nth_list_complex @ ( matrix553418799951983617omplex @ F @ M ) @ I3 ) @ J )
= ( F @ ( nth_complex @ ( nth_list_complex @ M @ I3 ) @ J ) ) ) ) ) ) ).
% mat_map_index
thf(fact_364_row__transpose__is__col,axiom,
! [Nr: nat,Nc: nat,M: list_list_complex,I3: nat] :
( ( matrix_mat_complex @ Nr @ Nc @ M )
=> ( ( ord_less_nat @ I3 @ Nc )
=> ( ( matrix_row_complex @ ( matrix1433782295178676338omplex @ Nr @ M ) @ I3 )
= ( matrix_col_complex @ M @ I3 ) ) ) ) ).
% row_transpose_is_col
thf(fact_365_col__transpose__is__row,axiom,
! [Nr: nat,Nc: nat,M: list_list_complex,I3: nat] :
( ( matrix_mat_complex @ Nr @ Nc @ M )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( matrix_col_complex @ ( matrix1433782295178676338omplex @ Nr @ M ) @ I3 )
= ( matrix_row_complex @ M @ I3 ) ) ) ) ).
% col_transpose_is_row
thf(fact_366_lowner__le__traceI,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ! [Rho: mat_complex] :
( ( member_mat_complex @ Rho @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple1169154605998056944erator @ Rho )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ Rho ) ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B2 @ Rho ) ) ) ) )
=> ( complex_lowner_le @ A @ B2 ) ) ) ) ).
% lowner_le_traceI
thf(fact_367_lowner__le__traceD,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,Rho2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ Rho2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( complex_lowner_le @ A @ B2 )
=> ( ( comple1169154605998056944erator @ Rho2 )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ Rho2 ) ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B2 @ Rho2 ) ) ) ) ) ) ) ) ).
% lowner_le_traceD
thf(fact_368_lowner__le__trace,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( complex_lowner_le @ A @ B2 )
= ( ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple1169154605998056944erator @ X2 )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ A @ X2 ) ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ B2 @ X2 ) ) ) ) ) ) ) ) ) ).
% lowner_le_trace
thf(fact_369_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_370_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_371_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_372_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_373_mult__sign__intros_I4_J,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B3 @ zero_zero_complex )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_374_mult__sign__intros_I4_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_375_mult__sign__intros_I3_J,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B3 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B3 ) @ zero_zero_complex ) ) ) ).
% mult_sign_intros(3)
thf(fact_376_mult__sign__intros_I3_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(3)
thf(fact_377_mult__sign__intros_I3_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(3)
thf(fact_378_mult__sign__intros_I2_J,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ B3 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B3 ) @ zero_zero_complex ) ) ) ).
% mult_sign_intros(2)
thf(fact_379_mult__sign__intros_I2_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(2)
thf(fact_380_mult__sign__intros_I2_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(2)
thf(fact_381_mult__sign__intros_I1_J,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B3 )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_382_mult__sign__intros_I1_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_383_mult__sign__intros_I1_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_384_mult__mono,axiom,
! [A3: complex,B3: complex,C2: complex,D3: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ C2 @ D3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B3 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_385_mult__mono,axiom,
! [A3: nat,B3: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_386_mult__mono,axiom,
! [A3: real,B3: real,C2: real,D3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C2 @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_387_mult__mono_H,axiom,
! [A3: complex,B3: complex,C2: complex,D3: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ C2 @ D3 )
=> ( ( 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 @ B3 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_388_mult__mono_H,axiom,
! [A3: nat,B3: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ( 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 @ B3 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_389_mult__mono_H,axiom,
! [A3: real,B3: real,C2: real,D3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C2 @ D3 )
=> ( ( 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 @ B3 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_390_zero__le__square,axiom,
! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_391_split__mult__pos__le,axiom,
! [A3: complex,B3: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
& ( ord_less_eq_complex @ zero_zero_complex @ B3 ) )
| ( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
& ( ord_less_eq_complex @ B3 @ zero_zero_complex ) ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B3 ) ) ) ).
% split_mult_pos_le
thf(fact_392_split__mult__pos__le,axiom,
! [A3: real,B3: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ).
% split_mult_pos_le
thf(fact_393_mult__left__mono__neg,axiom,
! [B3: complex,A3: complex,C2: complex] :
( ( ord_less_eq_complex @ B3 @ A3 )
=> ( ( ord_less_eq_complex @ C2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B3 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_394_mult__left__mono__neg,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_395_mult__left__mono,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_396_mult__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_397_mult__left__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_398_mult__right__mono__neg,axiom,
! [B3: complex,A3: complex,C2: complex] :
( ( ord_less_eq_complex @ B3 @ A3 )
=> ( ( ord_less_eq_complex @ C2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B3 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_399_mult__right__mono__neg,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_400_mult__right__mono,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B3 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_401_mult__right__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_402_mult__right__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_403_mult__le__0__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).
% mult_le_0_iff
thf(fact_404_split__mult__neg__le,axiom,
! [A3: complex,B3: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
& ( ord_less_eq_complex @ B3 @ zero_zero_complex ) )
| ( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
& ( ord_less_eq_complex @ zero_zero_complex @ B3 ) ) )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B3 ) @ zero_zero_complex ) ) ).
% split_mult_neg_le
thf(fact_405_split__mult__neg__le,axiom,
! [A3: nat,B3: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
& ( ord_less_eq_nat @ B3 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_406_split__mult__neg__le,axiom,
! [A3: real,B3: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_407_mult__nonneg__nonpos2,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ B3 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ B3 @ A3 ) @ zero_zero_complex ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_408_mult__nonneg__nonpos2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B3 @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_409_mult__nonneg__nonpos2,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B3 @ A3 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_410_zero__le__mult__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_411_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_412_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_413_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_414_mult__le__cancel__iff2,axiom,
! [Z2: real,X3: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X3 ) @ ( times_times_real @ Z2 @ Y2 ) )
= ( ord_less_eq_real @ X3 @ Y2 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_415_mult__le__cancel__iff1,axiom,
! [Z2: real,X3: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y2 @ Z2 ) )
= ( ord_less_eq_real @ X3 @ Y2 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_416_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: nat,B3: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ( 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 @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_417_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: real,B3: real,C2: real,D3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C2 @ D3 )
=> ( ( 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 @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_418_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: nat,B3: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ( 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 @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_419_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: real,B3: real,C2: real,D3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ C2 @ D3 )
=> ( ( 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 @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_420_mult__right__le__imp__le,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_421_mult__right__le__imp__le,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_422_mult__left__le__imp__le,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_423_mult__left__le__imp__le,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_424_mult__le__cancel__left__pos,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_425_mult__le__cancel__left__neg,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( ord_less_eq_real @ B3 @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_426_mult__less__cancel__right,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B3 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_427_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: nat,B3: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ( 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 @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_428_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: real,B3: real,C2: real,D3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C2 @ D3 )
=> ( ( 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 @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_429_mult__right__less__imp__less,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_430_mult__right__less__imp__less,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_431_mult__less__cancel__left,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B3 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_432_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: nat,B3: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_433_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: real,B3: real,C2: real,D3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C2 @ D3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_434_mult__left__less__imp__less,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_435_mult__left__less__imp__less,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_436_mult__le__cancel__right,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B3 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_437_mult__le__cancel__left,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B3 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_438_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_439_lowner__le__imp__trace__le,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( complex_lowner_le @ A @ B2 )
=> ( ord_less_eq_complex @ ( comple3184165445352484367omplex @ A ) @ ( comple3184165445352484367omplex @ B2 ) ) ) ) ) ).
% lowner_le_imp_trace_le
thf(fact_440_transpose__index,axiom,
! [Nr: nat,Nc: nat,M: list_l5436439031154120755omplex,I3: nat,J: nat] :
( ( matrix6216835547647503100omplex @ Nr @ Nc @ M )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( nth_mat_complex @ ( nth_list_mat_complex @ ( matrix6044053461928786659omplex @ Nr @ M ) @ I3 ) @ J )
= ( nth_mat_complex @ ( nth_list_mat_complex @ M @ J ) @ I3 ) ) ) ) ) ).
% transpose_index
thf(fact_441_transpose__index,axiom,
! [Nr: nat,Nc: nat,M: list_l3981933317855906654omplex,I3: nat,J: nat] :
( ( matrix6976670468949791273omplex @ Nr @ Nc @ M )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( nth_list_complex @ ( nth_li53272486250751239omplex @ ( matrix7733108353310726274omplex @ Nr @ M ) @ I3 ) @ J )
= ( nth_list_complex @ ( nth_li53272486250751239omplex @ M @ J ) @ I3 ) ) ) ) ) ).
% transpose_index
thf(fact_442_transpose__index,axiom,
! [Nr: nat,Nc: nat,M: list_list_complex,I3: nat,J: nat] :
( ( matrix_mat_complex @ Nr @ Nc @ M )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J @ Nc )
=> ( ( nth_complex @ ( nth_list_complex @ ( matrix1433782295178676338omplex @ Nr @ M ) @ I3 ) @ J )
= ( nth_complex @ ( nth_list_complex @ M @ J ) @ I3 ) ) ) ) ) ).
% transpose_index
thf(fact_443_less__eq__fract__respect,axiom,
! [B3: real,B6: real,D3: real,D4: real,A3: real,A5: real,C2: real,C3: real] :
( ( B3 != zero_zero_real )
=> ( ( B6 != zero_zero_real )
=> ( ( D3 != zero_zero_real )
=> ( ( D4 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ B6 )
= ( times_times_real @ A5 @ B3 ) )
=> ( ( ( times_times_real @ C2 @ D4 )
= ( times_times_real @ C3 @ D3 ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A3 @ D3 ) @ ( times_times_real @ B3 @ D3 ) ) @ ( times_times_real @ ( times_times_real @ C2 @ B3 ) @ ( times_times_real @ B3 @ D3 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A5 @ D4 ) @ ( times_times_real @ B6 @ D4 ) ) @ ( times_times_real @ ( times_times_real @ C3 @ B6 ) @ ( times_times_real @ B6 @ D4 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_444_positive__proj__trace,axiom,
! [P: mat_complex,R: mat_complex,N2: nat] :
( ( linear5633924348262549461omplex @ P )
=> ( ( complex_positive @ R )
=> ( ( member_mat_complex @ P @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ R @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ R @ P ) ) ) ) ) ) ) ).
% positive_proj_trace
thf(fact_445_list_Opred__mono,axiom,
! [P: mat_complex > $o,Pa: mat_complex > $o] :
( ( ord_le2790225379703085046plex_o @ P @ Pa )
=> ( ord_le1186661314528880752plex_o @ ( list_all_mat_complex @ P ) @ ( list_all_mat_complex @ Pa ) ) ) ).
% list.pred_mono
thf(fact_446_list_Opred__mono,axiom,
! [P: complex > $o,Pa: complex > $o] :
( ( ord_le4573692005234683329plex_o @ P @ Pa )
=> ( ord_le6360058522932223793plex_o @ ( list_all_complex @ P ) @ ( list_all_complex @ Pa ) ) ) ).
% list.pred_mono
thf(fact_447_list_Opred__mono,axiom,
! [P: list_complex > $o,Pa: list_complex > $o] :
( ( ord_le6360058522932223793plex_o @ P @ Pa )
=> ( ord_le8334822129778880289plex_o @ ( list_a4212339457297940234omplex @ P ) @ ( list_a4212339457297940234omplex @ Pa ) ) ) ).
% list.pred_mono
thf(fact_448_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_449_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_450_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_451_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_452_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_453_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_454_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J: nat] :
( ! [I: nat,J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_455_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_456_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_457_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_458_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_459_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
& ( M3 != N ) ) ) ) ).
% nat_less_le
thf(fact_460_mult__le__mono2,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_461_mult__le__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_462_mult__le__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_463_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_464_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_465_positive__is__hermitian,axiom,
! [A: mat_complex] :
( ( complex_positive @ A )
=> ( comple8306762464034002205omplex @ A ) ) ).
% positive_is_hermitian
thf(fact_466_positive__one,axiom,
! [N2: nat] : ( complex_positive @ ( one_mat_complex @ N2 ) ) ).
% positive_one
thf(fact_467_projector__positive,axiom,
! [M2: mat_complex] :
( ( linear5633924348262549461omplex @ M2 )
=> ( complex_positive @ M2 ) ) ).
% projector_positive
thf(fact_468_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_469_positive__dim__eq,axiom,
! [A: mat_complex] :
( ( complex_positive @ A )
=> ( ( dim_row_complex @ A )
= ( dim_col_complex @ A ) ) ) ).
% positive_dim_eq
thf(fact_470_lowner__le__trans__positiveI,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( complex_positive @ A )
=> ( ( complex_lowner_le @ A @ B2 )
=> ( complex_positive @ B2 ) ) ) ) ).
% lowner_le_trans_positiveI
thf(fact_471_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_472_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_473_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_474_positive__if__decomp,axiom,
! [A: mat_complex,N2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ? [M6: mat_complex] :
( ( times_8009071140041733218omplex @ M6 @ ( schur_5982229384592763574omplex @ M6 ) )
= A )
=> ( complex_positive @ A ) ) ) ).
% positive_if_decomp
thf(fact_475_positive__iff__decomp,axiom,
! [A: mat_complex,N2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( complex_positive @ A )
= ( ? [X2: mat_complex] :
( ( member_mat_complex @ X2 @ ( carrier_mat_complex @ N2 @ N2 ) )
& ( ( times_8009071140041733218omplex @ X2 @ ( schur_5982229384592763574omplex @ X2 ) )
= A ) ) ) ) ) ).
% positive_iff_decomp
thf(fact_476_positive__only__if__decomp,axiom,
! [A: mat_complex,N2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( complex_positive @ A )
=> ? [X4: mat_complex] :
( ( member_mat_complex @ X4 @ ( carrier_mat_complex @ N2 @ N2 ) )
& ( ( times_8009071140041733218omplex @ X4 @ ( schur_5982229384592763574omplex @ X4 ) )
= A ) ) ) ) ).
% positive_only_if_decomp
thf(fact_477_positive__close__under__left__right__mult__adjoint,axiom,
! [M2: mat_complex,N2: nat,A: mat_complex] :
( ( member_mat_complex @ M2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( complex_positive @ A )
=> ( complex_positive @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ M2 @ A ) @ ( schur_5982229384592763574omplex @ M2 ) ) ) ) ) ) ).
% positive_close_under_left_right_mult_adjoint
thf(fact_478_positive__trace,axiom,
! [A: mat_complex,N2: nat] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( complex_positive @ A )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( comple3184165445352484367omplex @ A ) ) ) ) ).
% positive_trace
thf(fact_479_sub__mat__index,axiom,
! [Nr: nat,Nc: nat,M: list_l5436439031154120755omplex,Sr: nat,Sc: nat,J: nat,I3: nat] :
( ( matrix6216835547647503100omplex @ Nr @ Nc @ M )
=> ( ( ord_less_eq_nat @ Sr @ Nr )
=> ( ( ord_less_eq_nat @ Sc @ Nc )
=> ( ( ord_less_nat @ J @ Sr )
=> ( ( ord_less_nat @ I3 @ Sc )
=> ( ( nth_mat_complex @ ( nth_list_mat_complex @ ( matrix4057538736392336240omplex @ Sr @ Sc @ M ) @ I3 ) @ J )
= ( nth_mat_complex @ ( nth_list_mat_complex @ M @ I3 ) @ J ) ) ) ) ) ) ) ).
% sub_mat_index
thf(fact_480_sub__mat__index,axiom,
! [Nr: nat,Nc: nat,M: list_l3981933317855906654omplex,Sr: nat,Sc: nat,J: nat,I3: nat] :
( ( matrix6976670468949791273omplex @ Nr @ Nc @ M )
=> ( ( ord_less_eq_nat @ Sr @ Nr )
=> ( ( ord_less_eq_nat @ Sc @ Nc )
=> ( ( ord_less_nat @ J @ Sr )
=> ( ( ord_less_nat @ I3 @ Sc )
=> ( ( nth_list_complex @ ( nth_li53272486250751239omplex @ ( matrix1439577319965977781omplex @ Sr @ Sc @ M ) @ I3 ) @ J )
= ( nth_list_complex @ ( nth_li53272486250751239omplex @ M @ I3 ) @ J ) ) ) ) ) ) ) ).
% sub_mat_index
thf(fact_481_sub__mat__index,axiom,
! [Nr: nat,Nc: nat,M: list_list_complex,Sr: nat,Sc: nat,J: nat,I3: nat] :
( ( matrix_mat_complex @ Nr @ Nc @ M )
=> ( ( ord_less_eq_nat @ Sr @ Nr )
=> ( ( ord_less_eq_nat @ Sc @ Nc )
=> ( ( ord_less_nat @ J @ Sr )
=> ( ( ord_less_nat @ I3 @ Sc )
=> ( ( nth_complex @ ( nth_list_complex @ ( matrix742113920429806117omplex @ Sr @ Sc @ M ) @ I3 ) @ J )
= ( nth_complex @ ( nth_list_complex @ M @ I3 ) @ J ) ) ) ) ) ) ) ).
% sub_mat_index
thf(fact_482_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_483_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% minf(8)
thf(fact_484_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_485_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ord_less_eq_real @ X5 @ T ) ) ).
% minf(6)
thf(fact_486_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_487_le__trans,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_488_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_489_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_490_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_491_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B3 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_492_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_493_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ~ ( ord_less_real @ T @ X5 ) ) ).
% minf(7)
thf(fact_494_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_495_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ord_less_real @ X5 @ T ) ) ).
% minf(5)
thf(fact_496_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_497_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_498_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_499_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_500_minf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_501_minf_I2_J,axiom,
! [P: real > $o,P3: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_502_minf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_503_minf_I1_J,axiom,
! [P: real > $o,P3: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_504_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_505_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ord_less_real @ T @ X5 ) ) ).
% pinf(7)
thf(fact_506_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_507_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ~ ( ord_less_real @ X5 @ T ) ) ).
% pinf(5)
thf(fact_508_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_509_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_510_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_511_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_512_pinf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_513_pinf_I2_J,axiom,
! [P: real > $o,P3: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_514_pinf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_515_pinf_I1_J,axiom,
! [P: real > $o,P3: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_516_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_517_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% pinf(6)
thf(fact_518_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_519_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ord_less_eq_real @ T @ X5 ) ) ).
% pinf(8)
thf(fact_520_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_521_inf__pigeonhole__principle,axiom,
! [N2: nat,F: nat > nat > $o] :
( ! [K2: nat] :
? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( F @ K2 @ I4 ) )
=> ? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F @ K4 @ I ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_522_order__le__imp__less__or__eq,axiom,
! [X3: complex,Y2: complex] :
( ( ord_less_eq_complex @ X3 @ Y2 )
=> ( ( ord_less_complex @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_523_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_524_order__le__imp__less__or__eq,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_real @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_525_linorder__le__less__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_526_linorder__le__less__linear,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
| ( ord_less_real @ Y2 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_527_order__less__imp__not__less,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_528_order__less__imp__not__less,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_529_order__less__imp__not__eq2,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_530_order__less__imp__not__eq2,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_531_order__less__imp__not__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_532_order__less__imp__not__eq,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_533_linorder__less__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ).
% linorder_less_linear
thf(fact_534_linorder__less__linear,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less_real @ Y2 @ X3 ) ) ).
% linorder_less_linear
thf(fact_535_order__less__imp__triv,axiom,
! [X3: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_536_order__less__imp__triv,axiom,
! [X3: real,Y2: real,P: $o] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ( ord_less_real @ Y2 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_537_order__less__not__sym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% order_less_not_sym
thf(fact_538_order__less__not__sym,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X3 ) ) ).
% order_less_not_sym
thf(fact_539_order__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_540_order__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_541_order__less__subst2,axiom,
! [A3: real,B3: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_542_order__less__subst2,axiom,
! [A3: real,B3: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_543_order__less__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_544_order__less__subst1,axiom,
! [A3: nat,F: real > nat,B3: real,C2: real] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_545_order__less__subst1,axiom,
! [A3: real,F: nat > real,B3: nat,C2: nat] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_546_order__less__subst1,axiom,
! [A3: real,F: real > real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_547_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_548_order__less__irrefl,axiom,
! [X3: real] :
~ ( ord_less_real @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_549_ord__less__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_550_ord__less__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_551_ord__less__eq__subst,axiom,
! [A3: real,B3: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_552_ord__less__eq__subst,axiom,
! [A3: real,B3: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_553_ord__eq__less__subst,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_554_ord__eq__less__subst,axiom,
! [A3: real,F: nat > real,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_555_ord__eq__less__subst,axiom,
! [A3: nat,F: real > nat,B3: real,C2: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_556_ord__eq__less__subst,axiom,
! [A3: real,F: real > real,B3: real,C2: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_557_order__less__trans,axiom,
! [X3: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_558_order__less__trans,axiom,
! [X3: real,Y2: real,Z2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X3 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_559_order__less__asym_H,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_560_order__less__asym_H,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( ord_less_real @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_561_linorder__neq__iff,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
= ( ( ord_less_nat @ X3 @ Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_562_linorder__neq__iff,axiom,
! [X3: real,Y2: real] :
( ( X3 != Y2 )
= ( ( ord_less_real @ X3 @ Y2 )
| ( ord_less_real @ Y2 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_563_order__less__asym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% order_less_asym
thf(fact_564_order__less__asym,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X3 ) ) ).
% order_less_asym
thf(fact_565_linorder__neqE,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_566_linorder__neqE,axiom,
! [X3: real,Y2: real] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ Y2 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_567_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_568_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_569_order_Ostrict__implies__not__eq,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_570_order_Ostrict__implies__not__eq,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_571_dual__order_Ostrict__trans,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_less_nat @ C2 @ B3 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_572_dual__order_Ostrict__trans,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C2 @ B3 )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_573_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_574_not__less__iff__gr__or__eq,axiom,
! [X3: real,Y2: real] :
( ( ~ ( ord_less_real @ X3 @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_575_order_Ostrict__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_576_order_Ostrict__trans,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_577_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A6: nat,B7: nat] :
( ( ord_less_nat @ A6 @ B7 )
=> ( P @ A6 @ B7 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B7: nat] :
( ( P @ B7 @ A6 )
=> ( P @ A6 @ B7 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_578_linorder__less__wlog,axiom,
! [P: real > real > $o,A3: real,B3: real] :
( ! [A6: real,B7: real] :
( ( ord_less_real @ A6 @ B7 )
=> ( P @ A6 @ B7 ) )
=> ( ! [A6: real] : ( P @ A6 @ A6 )
=> ( ! [A6: real,B7: real] :
( ( P @ B7 @ A6 )
=> ( P @ A6 @ B7 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_579_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P2: nat > $o] :
? [N: nat] :
( ( P2 @ N )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ~ ( P2 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_580_dual__order_Oirrefl,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_581_dual__order_Oirrefl,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_582_dual__order_Oasym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ~ ( ord_less_nat @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_583_dual__order_Oasym,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ~ ( ord_less_real @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_584_linorder__cases,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_cases
thf(fact_585_linorder__cases,axiom,
! [X3: real,Y2: real] :
( ~ ( ord_less_real @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less_real @ Y2 @ X3 ) ) ) ).
% linorder_cases
thf(fact_586_antisym__conv3,axiom,
! [Y2: nat,X3: nat] :
( ~ ( ord_less_nat @ Y2 @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_587_antisym__conv3,axiom,
! [Y2: real,X3: real] :
( ~ ( ord_less_real @ Y2 @ X3 )
=> ( ( ~ ( ord_less_real @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_588_less__induct,axiom,
! [P: nat > $o,A3: nat] :
( ! [X4: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X4 )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ).
% less_induct
thf(fact_589_ord__less__eq__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_590_ord__less__eq__trans,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_591_ord__eq__less__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( A3 = B3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_592_ord__eq__less__trans,axiom,
! [A3: real,B3: real,C2: real] :
( ( A3 = B3 )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_593_order_Oasym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% order.asym
thf(fact_594_order_Oasym,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( ord_less_real @ B3 @ A3 ) ) ).
% order.asym
thf(fact_595_less__imp__neq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_596_less__imp__neq,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_597_dense,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ? [Z3: real] :
( ( ord_less_real @ X3 @ Z3 )
& ( ord_less_real @ Z3 @ Y2 ) ) ) ).
% dense
thf(fact_598_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_599_gt__ex,axiom,
! [X3: real] :
? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% gt_ex
thf(fact_600_lt__ex,axiom,
! [X3: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X3 ) ).
% lt_ex
thf(fact_601_order__trans__rules_I22_J,axiom,
! [X3: complex,Y2: complex,Z2: complex] :
( ( ord_less_complex @ X3 @ Y2 )
=> ( ( ord_less_eq_complex @ Y2 @ Z2 )
=> ( ord_less_complex @ X3 @ Z2 ) ) ) ).
% order_trans_rules(22)
thf(fact_602_order__trans__rules_I22_J,axiom,
! [X3: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% order_trans_rules(22)
thf(fact_603_order__trans__rules_I22_J,axiom,
! [X3: real,Y2: real,Z2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_real @ X3 @ Z2 ) ) ) ).
% order_trans_rules(22)
thf(fact_604_order__trans__rules_I21_J,axiom,
! [X3: complex,Y2: complex,Z2: complex] :
( ( ord_less_eq_complex @ X3 @ Y2 )
=> ( ( ord_less_complex @ Y2 @ Z2 )
=> ( ord_less_complex @ X3 @ Z2 ) ) ) ).
% order_trans_rules(21)
thf(fact_605_order__trans__rules_I21_J,axiom,
! [X3: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% order_trans_rules(21)
thf(fact_606_order__trans__rules_I21_J,axiom,
! [X3: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X3 @ Z2 ) ) ) ).
% order_trans_rules(21)
thf(fact_607_order__trans__rules_I18_J,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_complex @ A3 @ B3 ) ) ) ).
% order_trans_rules(18)
thf(fact_608_order__trans__rules_I18_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% order_trans_rules(18)
thf(fact_609_order__trans__rules_I18_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% order_trans_rules(18)
thf(fact_610_order__trans__rules_I17_J,axiom,
! [A3: complex,B3: complex] :
( ( A3 != B3 )
=> ( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ord_less_complex @ A3 @ B3 ) ) ) ).
% order_trans_rules(17)
thf(fact_611_order__trans__rules_I17_J,axiom,
! [A3: nat,B3: nat] :
( ( A3 != B3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% order_trans_rules(17)
thf(fact_612_order__trans__rules_I17_J,axiom,
! [A3: real,B3: real] :
( ( A3 != B3 )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% order_trans_rules(17)
thf(fact_613_order__trans__rules_I6_J,axiom,
! [A3: complex,F: complex > complex,B3: complex,C2: complex] :
( ( ord_less_complex @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C2 )
=> ( ! [X4: complex,Y3: complex] :
( ( ord_less_eq_complex @ X4 @ Y3 )
=> ( ord_less_eq_complex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_614_order__trans__rules_I6_J,axiom,
! [A3: nat,F: complex > nat,B3: complex,C2: complex] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C2 )
=> ( ! [X4: complex,Y3: complex] :
( ( ord_less_eq_complex @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_615_order__trans__rules_I6_J,axiom,
! [A3: real,F: complex > real,B3: complex,C2: complex] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C2 )
=> ( ! [X4: complex,Y3: complex] :
( ( ord_less_eq_complex @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_616_order__trans__rules_I6_J,axiom,
! [A3: complex,F: nat > complex,B3: nat,C2: nat] :
( ( ord_less_complex @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_complex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_617_order__trans__rules_I6_J,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_618_order__trans__rules_I6_J,axiom,
! [A3: real,F: nat > real,B3: nat,C2: nat] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_619_order__trans__rules_I6_J,axiom,
! [A3: complex,F: real > complex,B3: real,C2: real] :
( ( ord_less_complex @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_complex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_620_order__trans__rules_I6_J,axiom,
! [A3: nat,F: real > nat,B3: real,C2: real] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_621_order__trans__rules_I6_J,axiom,
! [A3: real,F: real > real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(6)
thf(fact_622_order__trans__rules_I5_J,axiom,
! [A3: nat,B3: nat,F: nat > complex,C2: complex] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ ( F @ B3 ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_complex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(5)
thf(fact_623_order__trans__rules_I5_J,axiom,
! [A3: real,B3: real,F: real > complex,C2: complex] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ ( F @ B3 ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_complex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(5)
thf(fact_624_order__trans__rules_I5_J,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(5)
thf(fact_625_order__trans__rules_I5_J,axiom,
! [A3: real,B3: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(5)
thf(fact_626_order__trans__rules_I5_J,axiom,
! [A3: nat,B3: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(5)
thf(fact_627_order__trans__rules_I5_J,axiom,
! [A3: real,B3: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(5)
thf(fact_628_order__trans__rules_I4_J,axiom,
! [A3: complex,F: nat > complex,B3: nat,C2: nat] :
( ( ord_less_eq_complex @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_complex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_629_order__trans__rules_I4_J,axiom,
! [A3: complex,F: real > complex,B3: real,C2: real] :
( ( ord_less_eq_complex @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_complex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_630_order__trans__rules_I4_J,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_631_order__trans__rules_I4_J,axiom,
! [A3: nat,F: real > nat,B3: real,C2: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_632_order__trans__rules_I4_J,axiom,
! [A3: real,F: nat > real,B3: nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_633_order__trans__rules_I4_J,axiom,
! [A3: real,F: real > real,B3: real,C2: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(4)
thf(fact_634_order__trans__rules_I3_J,axiom,
! [A3: complex,B3: complex,F: complex > complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_complex @ ( F @ B3 ) @ C2 )
=> ( ! [X4: complex,Y3: complex] :
( ( ord_less_eq_complex @ X4 @ Y3 )
=> ( ord_less_eq_complex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_635_order__trans__rules_I3_J,axiom,
! [A3: complex,B3: complex,F: complex > nat,C2: nat] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X4: complex,Y3: complex] :
( ( ord_less_eq_complex @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_636_order__trans__rules_I3_J,axiom,
! [A3: complex,B3: complex,F: complex > real,C2: real] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C2 )
=> ( ! [X4: complex,Y3: complex] :
( ( ord_less_eq_complex @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_637_order__trans__rules_I3_J,axiom,
! [A3: nat,B3: nat,F: nat > complex,C2: complex] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_complex @ ( F @ B3 ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_complex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_638_order__trans__rules_I3_J,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_639_order__trans__rules_I3_J,axiom,
! [A3: nat,B3: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_640_order__trans__rules_I3_J,axiom,
! [A3: real,B3: real,F: real > complex,C2: complex] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_complex @ ( F @ B3 ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_complex @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_641_order__trans__rules_I3_J,axiom,
! [A3: real,B3: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_642_order__trans__rules_I3_J,axiom,
! [A3: real,B3: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_trans_rules(3)
thf(fact_643_leD,axiom,
! [Y2: complex,X3: complex] :
( ( ord_less_eq_complex @ Y2 @ X3 )
=> ~ ( ord_less_complex @ X3 @ Y2 ) ) ).
% leD
thf(fact_644_leD,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y2 ) ) ).
% leD
thf(fact_645_leD,axiom,
! [Y2: real,X3: real] :
( ( ord_less_eq_real @ Y2 @ X3 )
=> ~ ( ord_less_real @ X3 @ Y2 ) ) ).
% leD
thf(fact_646_leI,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% leI
thf(fact_647_leI,axiom,
! [X3: real,Y2: real] :
( ~ ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% leI
thf(fact_648_le__less,axiom,
( ord_less_eq_complex
= ( ^ [X2: complex,Y5: complex] :
( ( ord_less_complex @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% le_less
thf(fact_649_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% le_less
thf(fact_650_le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% le_less
thf(fact_651_less__le,axiom,
( ord_less_complex
= ( ^ [X2: complex,Y5: complex] :
( ( ord_less_eq_complex @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% less_le
thf(fact_652_less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% less_le
thf(fact_653_less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% less_le
thf(fact_654_nless__le,axiom,
! [A3: complex,B3: complex] :
( ( ~ ( ord_less_complex @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_complex @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_655_nless__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_656_nless__le,axiom,
! [A3: real,B3: real] :
( ( ~ ( ord_less_real @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_657_not__le,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X3 ) ) ).
% not_le
thf(fact_658_not__le,axiom,
! [X3: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X3 @ Y2 ) )
= ( ord_less_real @ Y2 @ X3 ) ) ).
% not_le
thf(fact_659_not__less,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% not_less
thf(fact_660_not__less,axiom,
! [X3: real,Y2: real] :
( ( ~ ( ord_less_real @ X3 @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% not_less
thf(fact_661_antisym__conv1,axiom,
! [X3: complex,Y2: complex] :
( ~ ( ord_less_complex @ X3 @ Y2 )
=> ( ( ord_less_eq_complex @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_662_antisym__conv1,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_663_antisym__conv1,axiom,
! [X3: real,Y2: real] :
( ~ ( ord_less_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_664_antisym__conv2,axiom,
! [X3: complex,Y2: complex] :
( ( ord_less_eq_complex @ X3 @ Y2 )
=> ( ( ~ ( ord_less_complex @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_665_antisym__conv2,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_666_antisym__conv2,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ~ ( ord_less_real @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_667_less__imp__le,axiom,
! [X3: complex,Y2: complex] :
( ( ord_less_complex @ X3 @ Y2 )
=> ( ord_less_eq_complex @ X3 @ Y2 ) ) ).
% less_imp_le
thf(fact_668_less__imp__le,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% less_imp_le
thf(fact_669_less__imp__le,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ X3 @ Y2 ) ) ).
% less_imp_le
thf(fact_670_dense__ge,axiom,
! [Z2: real,Y2: real] :
( ! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_eq_real @ Y2 @ X4 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_ge
thf(fact_671_dense__le,axiom,
! [Y2: real,Z2: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ X4 @ Z2 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_le
thf(fact_672_less__le__not__le,axiom,
( ord_less_complex
= ( ^ [X2: complex,Y5: complex] :
( ( ord_less_eq_complex @ X2 @ Y5 )
& ~ ( ord_less_eq_complex @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_673_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_674_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_675_not__le__imp__less,axiom,
! [Y2: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ord_less_nat @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_676_not__le__imp__less,axiom,
! [Y2: real,X3: real] :
( ~ ( ord_less_eq_real @ Y2 @ X3 )
=> ( ord_less_real @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_677_order_Oorder__iff__strict,axiom,
( ord_less_eq_complex
= ( ^ [A4: complex,B4: complex] :
( ( ord_less_complex @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_678_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_679_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_680_order_Ostrict__iff__order,axiom,
( ord_less_complex
= ( ^ [A4: complex,B4: complex] :
( ( ord_less_eq_complex @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_681_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_682_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_683_order_Ostrict__trans1,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_complex @ B3 @ C2 )
=> ( ord_less_complex @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_684_order_Ostrict__trans1,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_685_order_Ostrict__trans1,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ B3 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_686_order_Ostrict__trans2,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ord_less_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ B3 @ C2 )
=> ( ord_less_complex @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_687_order_Ostrict__trans2,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_688_order_Ostrict__trans2,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_689_order_Ostrict__iff__not,axiom,
( ord_less_complex
= ( ^ [A4: complex,B4: complex] :
( ( ord_less_eq_complex @ A4 @ B4 )
& ~ ( ord_less_eq_complex @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_690_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_691_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_692_dense__ge__bounded,axiom,
! [Z2: real,X3: real,Y2: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X3 )
=> ( ord_less_eq_real @ Y2 @ W ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_693_dense__le__bounded,axiom,
! [X3: real,Y2: real,Z2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ! [W: real] :
( ( ord_less_real @ X3 @ W )
=> ( ( ord_less_real @ W @ Y2 )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_694_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_complex
= ( ^ [B4: complex,A4: complex] :
( ( ord_less_complex @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_695_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_696_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_697_dual__order_Ostrict__iff__order,axiom,
( ord_less_complex
= ( ^ [B4: complex,A4: complex] :
( ( ord_less_eq_complex @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_698_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_699_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_700_dual__order_Ostrict__trans1,axiom,
! [B3: complex,A3: complex,C2: complex] :
( ( ord_less_eq_complex @ B3 @ A3 )
=> ( ( ord_less_complex @ C2 @ B3 )
=> ( ord_less_complex @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_701_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_nat @ C2 @ B3 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_702_dual__order_Ostrict__trans1,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_real @ C2 @ B3 )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_703_dual__order_Ostrict__trans2,axiom,
! [B3: complex,A3: complex,C2: complex] :
( ( ord_less_complex @ B3 @ A3 )
=> ( ( ord_less_eq_complex @ C2 @ B3 )
=> ( ord_less_complex @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_704_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B3 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_705_dual__order_Ostrict__trans2,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ B3 )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_706_dual__order_Ostrict__iff__not,axiom,
( ord_less_complex
= ( ^ [B4: complex,A4: complex] :
( ( ord_less_eq_complex @ B4 @ A4 )
& ~ ( ord_less_eq_complex @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_707_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_708_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_709_order_Ostrict__implies__order,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_complex @ A3 @ B3 )
=> ( ord_less_eq_complex @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_710_order_Ostrict__implies__order,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_711_order_Ostrict__implies__order,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_712_dual__order_Ostrict__implies__order,axiom,
! [B3: complex,A3: complex] :
( ( ord_less_complex @ B3 @ A3 )
=> ( ord_less_eq_complex @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_713_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_714_dual__order_Ostrict__implies__order,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_715_less__eq__set__def,axiom,
( ord_le3632134057777142183omplex
= ( ^ [A2: set_mat_complex,B5: set_mat_complex] :
( ord_le2790225379703085046plex_o
@ ^ [X2: mat_complex] : ( member_mat_complex @ X2 @ A2 )
@ ^ [X2: mat_complex] : ( member_mat_complex @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_716_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A2: set_real,B5: set_real] :
( ord_less_eq_real_o
@ ^ [X2: real] : ( member_real @ X2 @ A2 )
@ ^ [X2: real] : ( member_real @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_717_complete__interval,axiom,
! [A3: nat,B3: nat,P: nat > $o] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A3 @ C4 )
& ( ord_less_eq_nat @ C4 @ B3 )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A3 @ X5 )
& ( ord_less_nat @ X5 @ C4 ) )
=> ( P @ X5 ) )
& ! [D5: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A3 @ X4 )
& ( ord_less_nat @ X4 @ D5 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D5 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_718_complete__interval,axiom,
! [A3: real,B3: real,P: real > $o] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C4: real] :
( ( ord_less_eq_real @ A3 @ C4 )
& ( ord_less_eq_real @ C4 @ B3 )
& ! [X5: real] :
( ( ( ord_less_eq_real @ A3 @ X5 )
& ( ord_less_real @ X5 @ C4 ) )
=> ( P @ X5 ) )
& ! [D5: real] :
( ! [X4: real] :
( ( ( ord_less_eq_real @ A3 @ X4 )
& ( ord_less_real @ X4 @ D5 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_real @ D5 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_719_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_720_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A5 ) )
= ( ord_less_nat @ A5 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_721_verit__comp__simplify1_I3_J,axiom,
! [B6: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B6 @ A5 ) )
= ( ord_less_real @ A5 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_722_Collect__subset,axiom,
! [A: set_mat_complex,P: mat_complex > $o] :
( ord_le3632134057777142183omplex
@ ( collect_mat_complex
@ ^ [X2: mat_complex] :
( ( member_mat_complex @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_723_Collect__subset,axiom,
! [A: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_724_ex__gt__or__lt,axiom,
! [A3: real] :
? [B7: real] :
( ( ord_less_real @ A3 @ B7 )
| ( ord_less_real @ B7 @ A3 ) ) ).
% ex_gt_or_lt
thf(fact_725_verit__comp__simplify1_I1_J,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_726_verit__comp__simplify1_I1_J,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_727_pred__subset__eq,axiom,
! [R: set_mat_complex,S2: set_mat_complex] :
( ( ord_le2790225379703085046plex_o
@ ^ [X2: mat_complex] : ( member_mat_complex @ X2 @ R )
@ ^ [X2: mat_complex] : ( member_mat_complex @ X2 @ S2 ) )
= ( ord_le3632134057777142183omplex @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_728_pred__subset__eq,axiom,
! [R: set_real,S2: set_real] :
( ( ord_less_eq_real_o
@ ^ [X2: real] : ( member_real @ X2 @ R )
@ ^ [X2: real] : ( member_real @ X2 @ S2 ) )
= ( ord_less_eq_set_real @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_729_hermitian__square__similar__mat__wit,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,U2: mat_complex] :
( ( comple8306762464034002205omplex @ A )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( spectr532731689276696518omplex @ A @ B2 @ U2 )
=> ( simila5774310414453981135omplex @ ( times_8009071140041733218omplex @ A @ A ) @ ( times_8009071140041733218omplex @ B2 @ B2 ) @ U2 @ ( schur_5982229384592763574omplex @ U2 ) ) ) ) ) ).
% hermitian_square_similar_mat_wit
thf(fact_730_unitary__diag__carrier_I2_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( spectr532731689276696518omplex @ A @ B2 @ U2 )
=> ( member_mat_complex @ U2 @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ).
% unitary_diag_carrier(2)
thf(fact_731_unitary__diag__carrier_I1_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,U2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( spectr532731689276696518omplex @ A @ B2 @ U2 )
=> ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ).
% unitary_diag_carrier(1)
thf(fact_732_unitary__diagD_I3_J,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr532731689276696518omplex @ A @ B2 @ U2 )
=> ( comple6660659447773130958omplex @ U2 ) ) ).
% unitary_diagD(3)
thf(fact_733_unitary__diag__imp__unitarily__equiv,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr532731689276696518omplex @ A @ B2 @ U2 )
=> ( spectr6340060708231679580omplex @ A @ B2 @ U2 ) ) ).
% unitary_diag_imp_unitarily_equiv
thf(fact_734_real__diag__decompD_I1_J,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr5409772854192057952omplex @ A @ B2 @ U2 )
=> ( spectr532731689276696518omplex @ A @ B2 @ U2 ) ) ).
% real_diag_decompD(1)
thf(fact_735_unitary__diagD_I1_J,axiom,
! [A: mat_complex,B2: mat_complex,U2: mat_complex] :
( ( spectr532731689276696518omplex @ A @ B2 @ U2 )
=> ( simila5774310414453981135omplex @ A @ B2 @ U2 @ ( schur_5982229384592763574omplex @ U2 ) ) ) ).
% unitary_diagD(1)
thf(fact_736_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_737_mat__plus__index,axiom,
! [Nr: nat,Nc: nat,M1: list_l5436439031154120755omplex,M22: list_l5436439031154120755omplex,I3: nat,J: nat,Pl: mat_complex > mat_complex > mat_complex] :
( ( matrix6216835547647503100omplex @ Nr @ Nc @ M1 )
=> ( ( matrix6216835547647503100omplex @ Nr @ Nc @ M22 )
=> ( ( ord_less_nat @ I3 @ Nc )
=> ( ( ord_less_nat @ J @ Nr )
=> ( ( nth_mat_complex @ ( nth_list_mat_complex @ ( matrix1578179600178973825omplex @ Pl @ M1 @ M22 ) @ I3 ) @ J )
= ( Pl @ ( nth_mat_complex @ ( nth_list_mat_complex @ M1 @ I3 ) @ J ) @ ( nth_mat_complex @ ( nth_list_mat_complex @ M22 @ I3 ) @ J ) ) ) ) ) ) ) ).
% mat_plus_index
thf(fact_738_mat__plus__index,axiom,
! [Nr: nat,Nc: nat,M1: list_l3981933317855906654omplex,M22: list_l3981933317855906654omplex,I3: nat,J: nat,Pl: list_complex > list_complex > list_complex] :
( ( matrix6976670468949791273omplex @ Nr @ Nc @ M1 )
=> ( ( matrix6976670468949791273omplex @ Nr @ Nc @ M22 )
=> ( ( ord_less_nat @ I3 @ Nc )
=> ( ( ord_less_nat @ J @ Nr )
=> ( ( nth_list_complex @ ( nth_li53272486250751239omplex @ ( matrix2250674347687116644omplex @ Pl @ M1 @ M22 ) @ I3 ) @ J )
= ( Pl @ ( nth_list_complex @ ( nth_li53272486250751239omplex @ M1 @ I3 ) @ J ) @ ( nth_list_complex @ ( nth_li53272486250751239omplex @ M22 @ I3 ) @ J ) ) ) ) ) ) ) ).
% mat_plus_index
thf(fact_739_mat__plus__index,axiom,
! [Nr: nat,Nc: nat,M1: list_list_complex,M22: list_list_complex,I3: nat,J: nat,Pl: complex > complex > complex] :
( ( matrix_mat_complex @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_complex @ Nr @ Nc @ M22 )
=> ( ( ord_less_nat @ I3 @ Nc )
=> ( ( ord_less_nat @ J @ Nr )
=> ( ( nth_complex @ ( nth_list_complex @ ( matrix6097015163314587732omplex @ Pl @ M1 @ M22 ) @ I3 ) @ J )
= ( Pl @ ( nth_complex @ ( nth_list_complex @ M1 @ I3 ) @ J ) @ ( nth_complex @ ( nth_list_complex @ M22 @ I3 ) @ J ) ) ) ) ) ) ) ).
% mat_plus_index
thf(fact_740_tensor__mat__def,axiom,
( tensor_mat
= ( ^ [A2: mat_complex,B5: mat_complex] : ( mat_of_cols_list @ ( times_times_nat @ ( dim_row_complex @ A2 ) @ ( dim_row_complex @ B5 ) ) @ ( matrix1305980297522496462omplex @ times_times_complex @ ( mat_to_cols_list @ A2 ) @ ( mat_to_cols_list @ B5 ) ) ) ) ) ).
% tensor_mat_def
thf(fact_741_one__reorient,axiom,
! [X3: complex] :
( ( one_one_complex = X3 )
= ( X3 = one_one_complex ) ) ).
% one_reorient
thf(fact_742_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_743_one__reorient,axiom,
! [X3: real] :
( ( one_one_real = X3 )
= ( X3 = one_one_real ) ) ).
% one_reorient
thf(fact_744_rel__simps_I71_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% rel_simps(71)
thf(fact_745_rel__simps_I71_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% rel_simps(71)
thf(fact_746_arithmetic__simps_I79_J,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% arithmetic_simps(79)
thf(fact_747_arithmetic__simps_I79_J,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% arithmetic_simps(79)
thf(fact_748_arithmetic__simps_I79_J,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% arithmetic_simps(79)
thf(fact_749_arithmetic__simps_I78_J,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% arithmetic_simps(78)
thf(fact_750_arithmetic__simps_I78_J,axiom,
! [A3: complex] :
( ( times_times_complex @ one_one_complex @ A3 )
= A3 ) ).
% arithmetic_simps(78)
thf(fact_751_arithmetic__simps_I78_J,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% arithmetic_simps(78)
thf(fact_752_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_753_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_754_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_755_mult_Ocomm__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.comm_neutral
thf(fact_756_mult_Ocomm__neutral,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% mult.comm_neutral
thf(fact_757_mult_Ocomm__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.comm_neutral
thf(fact_758_vector__space__over__itself_Ovector__space__assms_I4_J,axiom,
! [X3: complex] :
( ( times_times_complex @ one_one_complex @ X3 )
= X3 ) ).
% vector_space_over_itself.vector_space_assms(4)
thf(fact_759_vector__space__over__itself_Ovector__space__assms_I4_J,axiom,
! [X3: real] :
( ( times_times_real @ one_one_real @ X3 )
= X3 ) ).
% vector_space_over_itself.vector_space_assms(4)
thf(fact_760_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_761_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_762_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_763_lambda__one,axiom,
( ( ^ [X2: nat] : X2 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_764_lambda__one,axiom,
( ( ^ [X2: complex] : X2 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_765_lambda__one,axiom,
( ( ^ [X2: real] : X2 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_766_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_767_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_768_verit__comp__simplify_I29_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% verit_comp_simplify(29)
thf(fact_769_verit__comp__simplify_I29_J,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% verit_comp_simplify(29)
thf(fact_770_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_771_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_772_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_773_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_774_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_775_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_776_mult__cancel__right1,axiom,
! [C2: complex,B3: complex] :
( ( C2
= ( times_times_complex @ B3 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( B3 = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_777_mult__cancel__right1,axiom,
! [C2: real,B3: real] :
( ( C2
= ( times_times_real @ B3 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_778_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_779_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_780_mult__cancel__left1,axiom,
! [C2: complex,B3: complex] :
( ( C2
= ( times_times_complex @ C2 @ B3 ) )
= ( ( C2 = zero_zero_complex )
| ( B3 = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_781_mult__cancel__left1,axiom,
! [C2: real,B3: real] :
( ( C2
= ( times_times_real @ C2 @ B3 ) )
= ( ( C2 = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_782_verit__comp__simplify_I28_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% verit_comp_simplify(28)
thf(fact_783_verit__comp__simplify_I28_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% verit_comp_simplify(28)
thf(fact_784_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_785_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_786_less__numeral__extra_I2_J,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% less_numeral_extra(2)
thf(fact_787_less__numeral__extra_I2_J,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% less_numeral_extra(2)
thf(fact_788_less__1__mult,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_789_less__1__mult,axiom,
! [M: real,N2: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N2 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_790_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_791_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_792_mult__le__one,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_793_mult__le__one,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ B3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_794_mult__right__le__one__le,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X3 @ Y2 ) @ X3 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_795_mult__left__le__one__le,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y2 @ X3 ) @ X3 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_796_mat__diag__one,axiom,
! [N2: nat] :
( ( mat_diag_complex @ N2
@ ^ [X2: nat] : one_one_complex )
= ( one_mat_complex @ N2 ) ) ).
% mat_diag_one
thf(fact_797_mat__diag__one,axiom,
! [N2: nat] :
( ( mat_diag_nat @ N2
@ ^ [X2: nat] : one_one_nat )
= ( one_mat_nat @ N2 ) ) ).
% mat_diag_one
thf(fact_798_mat__diag__one,axiom,
! [N2: nat] :
( ( mat_diag_real @ N2
@ ^ [X2: nat] : one_one_real )
= ( one_mat_real @ N2 ) ) ).
% mat_diag_one
thf(fact_799_density__operator__def,axiom,
( comple5220265106149225959erator
= ( ^ [A2: mat_complex] :
( ( complex_positive @ A2 )
& ( ( comple3184165445352484367omplex @ A2 )
= one_one_complex ) ) ) ) ).
% density_operator_def
thf(fact_800_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_801_mult__less__cancel__right1,axiom,
! [C2: real,B3: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ B3 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B3 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_802_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_803_mult__less__cancel__left1,axiom,
! [C2: real,B3: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ C2 @ B3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B3 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_804_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_805_mult__le__cancel__right1,axiom,
! [C2: real,B3: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ B3 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B3 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_806_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_807_mult__le__cancel__left1,axiom,
! [C2: real,B3: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ C2 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B3 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_808_field__le__mult__one__interval,axiom,
! [X3: real,Y2: real] :
( ! [Z3: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ Z3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z3 @ X3 ) @ Y2 ) ) )
=> ( ord_less_eq_real @ X3 @ Y2 ) ) ).
% field_le_mult_one_interval
thf(fact_809_mult__eq__1,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ A3 @ one_one_complex )
=> ( ( ord_less_eq_complex @ B3 @ one_one_complex )
=> ( ( ( times_times_complex @ A3 @ B3 )
= one_one_complex )
= ( ( A3 = one_one_complex )
& ( B3 = one_one_complex ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_810_mult__eq__1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ B3 @ one_one_nat )
=> ( ( ( times_times_nat @ A3 @ B3 )
= one_one_nat )
= ( ( A3 = one_one_nat )
& ( B3 = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_811_mult__eq__1,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ B3 @ one_one_real )
=> ( ( ( times_times_real @ A3 @ B3 )
= one_one_real )
= ( ( A3 = one_one_real )
& ( B3 = one_one_real ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_812_mult__if__delta,axiom,
! [P: $o,Q3: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q3 )
= Q3 ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q3 )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_813_mult__if__delta,axiom,
! [P: $o,Q3: complex] :
( ( P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q3 )
= Q3 ) )
& ( ~ P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q3 )
= zero_zero_complex ) ) ) ).
% mult_if_delta
thf(fact_814_mult__if__delta,axiom,
! [P: $o,Q3: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q3 )
= Q3 ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q3 )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_815_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_816_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_817_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N2 ) )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_818_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_819_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_820_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M
= ( times_times_nat @ M @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_821_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X5 ) ).
% linordered_field_no_lb
thf(fact_822_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_823_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X4: nat > real] :
( ( P @ X4 )
=> ( P @ ( F @ X4 ) ) )
=> ( ! [X4: nat > real] :
( ( P @ X4 )
=> ! [I: nat] :
( ( Q @ I )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I ) )
& ( ord_less_eq_real @ ( X4 @ I ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I4: nat] : ( ord_less_eq_nat @ ( L2 @ X5 @ I4 ) @ one_one_nat )
& ! [X5: nat > real,I4: nat] :
( ( ( P @ X5 )
& ( Q @ I4 )
& ( ( X5 @ I4 )
= zero_zero_real ) )
=> ( ( L2 @ X5 @ I4 )
= zero_zero_nat ) )
& ! [X5: nat > real,I4: nat] :
( ( ( P @ X5 )
& ( Q @ I4 )
& ( ( X5 @ I4 )
= one_one_real ) )
=> ( ( L2 @ X5 @ I4 )
= one_one_nat ) )
& ! [X5: nat > real,I4: nat] :
( ( ( P @ X5 )
& ( Q @ I4 )
& ( ( L2 @ X5 @ I4 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I4 ) @ ( F @ X5 @ I4 ) ) )
& ! [X5: nat > real,I4: nat] :
( ( ( P @ X5 )
& ( Q @ I4 )
& ( ( L2 @ X5 @ I4 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X5 @ I4 ) @ ( X5 @ I4 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_824_col__mat__plus,axiom,
! [Nr: nat,Nc: nat,M1: list_list_complex,M22: list_list_complex,I3: nat,Pl: complex > complex > complex] :
( ( matrix_mat_complex @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_complex @ Nr @ Nc @ M22 )
=> ( ( ord_less_nat @ I3 @ Nc )
=> ( ( matrix_col_complex @ ( matrix6097015163314587732omplex @ Pl @ M1 @ M22 ) @ I3 )
= ( matrix6198229848702844640omplex @ Pl @ ( matrix_col_complex @ M1 @ I3 ) @ ( matrix_col_complex @ M22 @ I3 ) ) ) ) ) ) ).
% col_mat_plus
thf(fact_825_class__field_Ozero__not__one,axiom,
zero_zero_complex != one_one_complex ).
% class_field.zero_not_one
thf(fact_826_class__field_Ozero__not__one,axiom,
zero_zero_real != one_one_real ).
% class_field.zero_not_one
thf(fact_827_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_828_complete__real,axiom,
! [S2: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S2 )
=> ( ? [Z4: real] :
! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z4 ) )
=> ? [Y3: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S2 )
=> ( ord_less_eq_real @ X5 @ Y3 ) )
& ! [Z4: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z4 ) )
=> ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_829_class__cring_Ofactors__equal,axiom,
! [A3: complex,B3: complex,C2: complex,D3: complex] :
( ( A3 = B3 )
=> ( ( C2 = D3 )
=> ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B3 @ D3 ) ) ) ) ).
% class_cring.factors_equal
thf(fact_830_class__cring_Ofactors__equal,axiom,
! [A3: real,B3: real,C2: real,D3: real] :
( ( A3 = B3 )
=> ( ( C2 = D3 )
=> ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B3 @ D3 ) ) ) ) ).
% class_cring.factors_equal
thf(fact_831_row__mat__plus,axiom,
! [Nr: nat,Nc: nat,M1: list_list_complex,M22: list_list_complex,I3: nat,Pl: complex > complex > complex] :
( ( matrix_mat_complex @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_complex @ Nr @ Nc @ M22 )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( matrix_row_complex @ ( matrix6097015163314587732omplex @ Pl @ M1 @ M22 ) @ I3 )
= ( matrix6198229848702844640omplex @ Pl @ ( matrix_row_complex @ M1 @ I3 ) @ ( matrix_row_complex @ M22 @ I3 ) ) ) ) ) ) ).
% row_mat_plus
thf(fact_832_matrix__mult__to__times__mat,axiom,
! [A: mat_complex,B2: mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ B2 ) )
=> ( ( ( dim_col_complex @ A )
= ( dim_row_complex @ B2 ) )
=> ( ( times_8009071140041733218omplex @ A @ B2 )
= ( mat_of_cols_list @ ( dim_row_complex @ A ) @ ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ ( matrix1515831402840476169omplex @ ( mat_to_cols_list @ A ) ) @ ( mat_to_cols_list @ A ) @ ( mat_to_cols_list @ B2 ) ) ) ) ) ) ) ).
% matrix_mult_to_times_mat
thf(fact_833_pth__7_I2_J,axiom,
! [X3: complex] :
( ( plus_plus_complex @ X3 @ zero_zero_complex )
= X3 ) ).
% pth_7(2)
thf(fact_834_pth__7_I2_J,axiom,
! [X3: real] :
( ( plus_plus_real @ X3 @ zero_zero_real )
= X3 ) ).
% pth_7(2)
thf(fact_835_pth__7_I1_J,axiom,
! [X3: complex] :
( ( plus_plus_complex @ zero_zero_complex @ X3 )
= X3 ) ).
% pth_7(1)
thf(fact_836_pth__7_I1_J,axiom,
! [X3: real] :
( ( plus_plus_real @ zero_zero_real @ X3 )
= X3 ) ).
% pth_7(1)
thf(fact_837_arithmetic__simps_I50_J,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% arithmetic_simps(50)
thf(fact_838_arithmetic__simps_I50_J,axiom,
! [A3: complex] :
( ( plus_plus_complex @ A3 @ zero_zero_complex )
= A3 ) ).
% arithmetic_simps(50)
thf(fact_839_arithmetic__simps_I50_J,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% arithmetic_simps(50)
thf(fact_840_arithmetic__simps_I49_J,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% arithmetic_simps(49)
thf(fact_841_arithmetic__simps_I49_J,axiom,
! [A3: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A3 )
= A3 ) ).
% arithmetic_simps(49)
thf(fact_842_arithmetic__simps_I49_J,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% arithmetic_simps(49)
thf(fact_843_nat__arith_Orule0,axiom,
! [A3: nat] :
( A3
= ( plus_plus_nat @ A3 @ zero_zero_nat ) ) ).
% nat_arith.rule0
thf(fact_844_nat__arith_Orule0,axiom,
! [A3: complex] :
( A3
= ( plus_plus_complex @ A3 @ zero_zero_complex ) ) ).
% nat_arith.rule0
thf(fact_845_nat__arith_Orule0,axiom,
! [A3: real] :
( A3
= ( plus_plus_real @ A3 @ zero_zero_real ) ) ).
% nat_arith.rule0
thf(fact_846_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y2 ) )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_847_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y2: nat] :
( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_848_add__cancel__right__right,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( plus_plus_nat @ A3 @ B3 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_849_add__cancel__right__right,axiom,
! [A3: complex,B3: complex] :
( ( A3
= ( plus_plus_complex @ A3 @ B3 ) )
= ( B3 = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_850_add__cancel__right__right,axiom,
! [A3: real,B3: real] :
( ( A3
= ( plus_plus_real @ A3 @ B3 ) )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_851_add__cancel__right__left,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ A3 ) )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_852_add__cancel__right__left,axiom,
! [A3: complex,B3: complex] :
( ( A3
= ( plus_plus_complex @ B3 @ A3 ) )
= ( B3 = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_853_add__cancel__right__left,axiom,
! [A3: real,B3: real] :
( ( A3
= ( plus_plus_real @ B3 @ A3 ) )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_854_add__cancel__left__right,axiom,
! [A3: nat,B3: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_855_add__cancel__left__right,axiom,
! [A3: complex,B3: complex] :
( ( ( plus_plus_complex @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_856_add__cancel__left__right,axiom,
! [A3: real,B3: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= A3 )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_857_add__cancel__left__left,axiom,
! [B3: nat,A3: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_858_add__cancel__left__left,axiom,
! [B3: complex,A3: complex] :
( ( ( plus_plus_complex @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_859_add__cancel__left__left,axiom,
! [B3: real,A3: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= A3 )
= ( B3 = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_860_double__zero__sym,axiom,
! [A3: real] :
( ( zero_zero_real
= ( plus_plus_real @ A3 @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_861_double__zero,axiom,
! [A3: real] :
( ( ( plus_plus_real @ A3 @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% double_zero
thf(fact_862_add_Ogroup__left__neutral,axiom,
! [A3: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_863_add_Ogroup__left__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_864_comm__monoid__add__class_Oadd__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_865_comm__monoid__add__class_Oadd__0,axiom,
! [A3: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_866_comm__monoid__add__class_Oadd__0,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_867_verit__sum__simplify,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% verit_sum_simplify
thf(fact_868_verit__sum__simplify,axiom,
! [A3: complex] :
( ( plus_plus_complex @ A3 @ zero_zero_complex )
= A3 ) ).
% verit_sum_simplify
thf(fact_869_verit__sum__simplify,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% verit_sum_simplify
thf(fact_870_add__le__imp__le__right,axiom,
! [A3: complex,C2: complex,B3: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A3 @ C2 ) @ ( plus_plus_complex @ B3 @ C2 ) )
=> ( ord_less_eq_complex @ A3 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_871_add__le__imp__le__right,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_872_add__le__imp__le__right,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B3 @ C2 ) )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_873_add__le__cancel__right,axiom,
! [A3: complex,C2: complex,B3: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A3 @ C2 ) @ ( plus_plus_complex @ B3 @ C2 ) )
= ( ord_less_eq_complex @ A3 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_874_add__le__cancel__right,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) )
= ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_875_add__le__cancel__right,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B3 @ C2 ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_876_add__le__imp__le__left,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ C2 @ A3 ) @ ( plus_plus_complex @ C2 @ B3 ) )
=> ( ord_less_eq_complex @ A3 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_877_add__le__imp__le__left,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_878_add__le__imp__le__left,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A3 ) @ ( plus_plus_real @ C2 @ B3 ) )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_879_add__le__cancel__left,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ C2 @ A3 ) @ ( plus_plus_complex @ C2 @ B3 ) )
= ( ord_less_eq_complex @ A3 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_880_add__le__cancel__left,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) )
= ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_881_add__le__cancel__left,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A3 ) @ ( plus_plus_real @ C2 @ B3 ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_882_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C5: nat] :
( B4
= ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_883_add__right__mono,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A3 @ C2 ) @ ( plus_plus_complex @ B3 @ C2 ) ) ) ).
% add_right_mono
thf(fact_884_add__right__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).
% add_right_mono
thf(fact_885_add__right__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B3 @ C2 ) ) ) ).
% add_right_mono
thf(fact_886_less__eqE,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ~ ! [C4: nat] :
( B3
!= ( plus_plus_nat @ A3 @ C4 ) ) ) ).
% less_eqE
thf(fact_887_add__left__mono,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ C2 @ A3 ) @ ( plus_plus_complex @ C2 @ B3 ) ) ) ).
% add_left_mono
thf(fact_888_add__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) ) ) ).
% add_left_mono
thf(fact_889_add__left__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A3 ) @ ( plus_plus_real @ C2 @ B3 ) ) ) ).
% add_left_mono
thf(fact_890_add__mono,axiom,
! [A3: complex,B3: complex,C2: complex,D3: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ C2 @ D3 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A3 @ C2 ) @ ( plus_plus_complex @ B3 @ D3 ) ) ) ) ).
% add_mono
thf(fact_891_add__mono,axiom,
! [A3: nat,B3: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ D3 ) ) ) ) ).
% add_mono
thf(fact_892_add__mono,axiom,
! [A3: real,B3: real,C2: real,D3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C2 @ D3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B3 @ D3 ) ) ) ) ).
% add_mono
thf(fact_893_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I3 @ J )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_894_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_895_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I3 @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_896_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( I3 = J )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_897_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_898_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( I3 = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_899_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I3 @ J )
& ( K = L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_900_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_901_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I3 @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_902_mat__mult__assoc__n,axiom,
! [N2: nat,M1: list_list_nat,M22: list_list_nat,M32: list_list_nat] :
( ( matrix_mat_nat @ N2 @ N2 @ M1 )
=> ( ( matrix_mat_nat @ N2 @ N2 @ M22 )
=> ( ( matrix_mat_nat @ N2 @ N2 @ M32 )
=> ( ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ N2 @ ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ N2 @ M1 @ M22 ) @ M32 )
= ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ N2 @ M1 @ ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ N2 @ M22 @ M32 ) ) ) ) ) ) ).
% mat_mult_assoc_n
thf(fact_903_mat__mult__assoc__n,axiom,
! [N2: nat,M1: list_list_complex,M22: list_list_complex,M32: list_list_complex] :
( ( matrix_mat_complex @ N2 @ N2 @ M1 )
=> ( ( matrix_mat_complex @ N2 @ N2 @ M22 )
=> ( ( matrix_mat_complex @ N2 @ N2 @ M32 )
=> ( ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ N2 @ ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ N2 @ M1 @ M22 ) @ M32 )
= ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ N2 @ M1 @ ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ N2 @ M22 @ M32 ) ) ) ) ) ) ).
% mat_mult_assoc_n
thf(fact_904_mat__mult__assoc__n,axiom,
! [N2: nat,M1: list_list_real,M22: list_list_real,M32: list_list_real] :
( ( matrix_mat_real @ N2 @ N2 @ M1 )
=> ( ( matrix_mat_real @ N2 @ N2 @ M22 )
=> ( ( matrix_mat_real @ N2 @ N2 @ M32 )
=> ( ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ N2 @ ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ N2 @ M1 @ M22 ) @ M32 )
= ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ N2 @ M1 @ ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ N2 @ M22 @ M32 ) ) ) ) ) ) ).
% mat_mult_assoc_n
thf(fact_905_mat__mult__assoc,axiom,
! [Nr: nat,N1: nat,M1: list_list_nat,N22: nat,M22: list_list_nat,Nc: nat,M32: list_list_nat] :
( ( matrix_mat_nat @ Nr @ N1 @ M1 )
=> ( ( matrix_mat_nat @ N1 @ N22 @ M22 )
=> ( ( matrix_mat_nat @ N22 @ Nc @ M32 )
=> ( ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ Nr @ ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ Nr @ M1 @ M22 ) @ M32 )
= ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ Nr @ M1 @ ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ N1 @ M22 @ M32 ) ) ) ) ) ) ).
% mat_mult_assoc
thf(fact_906_mat__mult__assoc,axiom,
! [Nr: nat,N1: nat,M1: list_list_complex,N22: nat,M22: list_list_complex,Nc: nat,M32: list_list_complex] :
( ( matrix_mat_complex @ Nr @ N1 @ M1 )
=> ( ( matrix_mat_complex @ N1 @ N22 @ M22 )
=> ( ( matrix_mat_complex @ N22 @ Nc @ M32 )
=> ( ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ Nr @ ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ Nr @ M1 @ M22 ) @ M32 )
= ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ Nr @ M1 @ ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ N1 @ M22 @ M32 ) ) ) ) ) ) ).
% mat_mult_assoc
thf(fact_907_mat__mult__assoc,axiom,
! [Nr: nat,N1: nat,M1: list_list_real,N22: nat,M22: list_list_real,Nc: nat,M32: list_list_real] :
( ( matrix_mat_real @ Nr @ N1 @ M1 )
=> ( ( matrix_mat_real @ N1 @ N22 @ M22 )
=> ( ( matrix_mat_real @ N22 @ Nc @ M32 )
=> ( ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ Nr @ ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ Nr @ M1 @ M22 ) @ M32 )
= ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ Nr @ M1 @ ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ N1 @ M22 @ M32 ) ) ) ) ) ) ).
% mat_mult_assoc
thf(fact_908_mult__hom_Ohom__add,axiom,
! [C2: nat,X3: nat,Y2: nat] :
( ( times_times_nat @ C2 @ ( plus_plus_nat @ X3 @ Y2 ) )
= ( plus_plus_nat @ ( times_times_nat @ C2 @ X3 ) @ ( times_times_nat @ C2 @ Y2 ) ) ) ).
% mult_hom.hom_add
thf(fact_909_mult__hom_Ohom__add,axiom,
! [C2: complex,X3: complex,Y2: complex] :
( ( times_times_complex @ C2 @ ( plus_plus_complex @ X3 @ Y2 ) )
= ( plus_plus_complex @ ( times_times_complex @ C2 @ X3 ) @ ( times_times_complex @ C2 @ Y2 ) ) ) ).
% mult_hom.hom_add
thf(fact_910_mult__hom_Ohom__add,axiom,
! [C2: real,X3: real,Y2: real] :
( ( times_times_real @ C2 @ ( plus_plus_real @ X3 @ Y2 ) )
= ( plus_plus_real @ ( times_times_real @ C2 @ X3 ) @ ( times_times_real @ C2 @ Y2 ) ) ) ).
% mult_hom.hom_add
thf(fact_911_combine__common__factor,axiom,
! [A3: nat,E2: nat,B3: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A3 @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B3 @ E2 ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_912_combine__common__factor,axiom,
! [A3: complex,E2: complex,B3: complex,C2: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A3 @ E2 ) @ ( plus_plus_complex @ ( times_times_complex @ B3 @ E2 ) @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A3 @ B3 ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_913_combine__common__factor,axiom,
! [A3: real,E2: real,B3: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_914_distrib__right,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% distrib_right
thf(fact_915_distrib__right,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A3 @ B3 ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B3 @ C2 ) ) ) ).
% distrib_right
thf(fact_916_distrib__right,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ).
% distrib_right
thf(fact_917_distrib__left,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( times_times_nat @ A3 @ ( plus_plus_nat @ B3 @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ B3 ) @ ( times_times_nat @ A3 @ C2 ) ) ) ).
% distrib_left
thf(fact_918_distrib__left,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( times_times_complex @ A3 @ ( plus_plus_complex @ B3 @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ B3 ) @ ( times_times_complex @ A3 @ C2 ) ) ) ).
% distrib_left
thf(fact_919_distrib__left,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% distrib_left
thf(fact_920_comm__semiring__class_Odistrib,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_921_comm__semiring__class_Odistrib,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A3 @ B3 ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B3 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_922_comm__semiring__class_Odistrib,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_923_ring__class_Oring__distribs_I1_J,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( times_times_complex @ A3 @ ( plus_plus_complex @ B3 @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ B3 ) @ ( times_times_complex @ A3 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_924_ring__class_Oring__distribs_I1_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_925_ring__class_Oring__distribs_I2_J,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A3 @ B3 ) @ C2 )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B3 @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_926_ring__class_Oring__distribs_I2_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_927_vector__space__over__itself_Ovector__space__assms_I2_J,axiom,
! [A3: complex,B3: complex,X3: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A3 @ B3 ) @ X3 )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ X3 ) @ ( times_times_complex @ B3 @ X3 ) ) ) ).
% vector_space_over_itself.vector_space_assms(2)
thf(fact_928_vector__space__over__itself_Ovector__space__assms_I2_J,axiom,
! [A3: real,B3: real,X3: real] :
( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ X3 )
= ( plus_plus_real @ ( times_times_real @ A3 @ X3 ) @ ( times_times_real @ B3 @ X3 ) ) ) ).
% vector_space_over_itself.vector_space_assms(2)
thf(fact_929_vector__space__over__itself_Ovector__space__assms_I1_J,axiom,
! [A3: complex,X3: complex,Y2: complex] :
( ( times_times_complex @ A3 @ ( plus_plus_complex @ X3 @ Y2 ) )
= ( plus_plus_complex @ ( times_times_complex @ A3 @ X3 ) @ ( times_times_complex @ A3 @ Y2 ) ) ) ).
% vector_space_over_itself.vector_space_assms(1)
thf(fact_930_vector__space__over__itself_Ovector__space__assms_I1_J,axiom,
! [A3: real,X3: real,Y2: real] :
( ( times_times_real @ A3 @ ( plus_plus_real @ X3 @ Y2 ) )
= ( plus_plus_real @ ( times_times_real @ A3 @ X3 ) @ ( times_times_real @ A3 @ Y2 ) ) ) ).
% vector_space_over_itself.vector_space_assms(1)
thf(fact_931_trace__add__linear,axiom,
! [A: mat_real,N2: nat,B2: mat_real] :
( ( member_mat_real @ A @ ( carrier_mat_real @ N2 @ N2 ) )
=> ( ( member_mat_real @ B2 @ ( carrier_mat_real @ N2 @ N2 ) )
=> ( ( complex_trace_real @ ( plus_plus_mat_real @ A @ B2 ) )
= ( plus_plus_real @ ( complex_trace_real @ A ) @ ( complex_trace_real @ B2 ) ) ) ) ) ).
% trace_add_linear
thf(fact_932_trace__add__linear,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple3184165445352484367omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) )
= ( plus_plus_complex @ ( comple3184165445352484367omplex @ A ) @ ( comple3184165445352484367omplex @ B2 ) ) ) ) ) ).
% trace_add_linear
thf(fact_933_Groups_Oadd__ac_I3_J,axiom,
! [B3: complex,A3: complex,C2: complex] :
( ( plus_plus_complex @ B3 @ ( plus_plus_complex @ A3 @ C2 ) )
= ( plus_plus_complex @ A3 @ ( plus_plus_complex @ B3 @ C2 ) ) ) ).
% Groups.add_ac(3)
thf(fact_934_Groups_Oadd__ac_I3_J,axiom,
! [B3: real,A3: real,C2: real] :
( ( plus_plus_real @ B3 @ ( plus_plus_real @ A3 @ C2 ) )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) ) ) ).
% Groups.add_ac(3)
thf(fact_935_Groups_Oadd__ac_I3_J,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A3 @ C2 ) )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).
% Groups.add_ac(3)
thf(fact_936_Groups_Oadd__ac_I2_J,axiom,
( plus_plus_complex
= ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ B4 @ A4 ) ) ) ).
% Groups.add_ac(2)
thf(fact_937_Groups_Oadd__ac_I2_J,axiom,
( plus_plus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% Groups.add_ac(2)
thf(fact_938_Groups_Oadd__ac_I2_J,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% Groups.add_ac(2)
thf(fact_939_Groups_Oadd__ac_I1_J,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A3 @ B3 ) @ C2 )
= ( plus_plus_complex @ A3 @ ( plus_plus_complex @ B3 @ C2 ) ) ) ).
% Groups.add_ac(1)
thf(fact_940_Groups_Oadd__ac_I1_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) ) ) ).
% Groups.add_ac(1)
thf(fact_941_Groups_Oadd__ac_I1_J,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C2 )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).
% Groups.add_ac(1)
thf(fact_942_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A3 @ B3 ) @ C2 )
= ( plus_plus_complex @ A3 @ ( plus_plus_complex @ B3 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_943_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_944_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C2 )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_945_is__num__normalize_I1_J,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A3 @ B3 ) @ C2 )
= ( plus_plus_complex @ A3 @ ( plus_plus_complex @ B3 @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_946_is__num__normalize_I1_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C2 )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_947_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( I3 = J )
& ( K = L ) )
=> ( ( plus_plus_complex @ I3 @ K )
= ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_948_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( I3 = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I3 @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_949_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I3 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_950_group__cancel_Oadd1,axiom,
! [A: complex,K: complex,A3: complex,B3: complex] :
( ( A
= ( plus_plus_complex @ K @ A3 ) )
=> ( ( plus_plus_complex @ A @ B3 )
= ( plus_plus_complex @ K @ ( plus_plus_complex @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_951_group__cancel_Oadd1,axiom,
! [A: real,K: real,A3: real,B3: real] :
( ( A
= ( plus_plus_real @ K @ A3 ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_952_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A3: nat,B3: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_953_group__cancel_Oadd2,axiom,
! [B2: complex,K: complex,B3: complex,A3: complex] :
( ( B2
= ( plus_plus_complex @ K @ B3 ) )
=> ( ( plus_plus_complex @ A3 @ B2 )
= ( plus_plus_complex @ K @ ( plus_plus_complex @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_954_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B3: real,A3: real] :
( ( B2
= ( plus_plus_real @ K @ B3 ) )
=> ( ( plus_plus_real @ A3 @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_955_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B3: nat,A3: nat] :
( ( B2
= ( plus_plus_nat @ K @ B3 ) )
=> ( ( plus_plus_nat @ A3 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_956_add_Oleft__cancel,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ( plus_plus_complex @ A3 @ B3 )
= ( plus_plus_complex @ A3 @ C2 ) )
= ( B3 = C2 ) ) ).
% add.left_cancel
thf(fact_957_add_Oleft__cancel,axiom,
! [A3: real,B3: real,C2: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= ( plus_plus_real @ A3 @ C2 ) )
= ( B3 = C2 ) ) ).
% add.left_cancel
thf(fact_958_add_Oright__cancel,axiom,
! [B3: complex,A3: complex,C2: complex] :
( ( ( plus_plus_complex @ B3 @ A3 )
= ( plus_plus_complex @ C2 @ A3 ) )
= ( B3 = C2 ) ) ).
% add.right_cancel
thf(fact_959_add_Oright__cancel,axiom,
! [B3: real,A3: real,C2: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= ( plus_plus_real @ C2 @ A3 ) )
= ( B3 = C2 ) ) ).
% add.right_cancel
thf(fact_960_add__left__cancel,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ( plus_plus_complex @ A3 @ B3 )
= ( plus_plus_complex @ A3 @ C2 ) )
= ( B3 = C2 ) ) ).
% add_left_cancel
thf(fact_961_add__left__cancel,axiom,
! [A3: real,B3: real,C2: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= ( plus_plus_real @ A3 @ C2 ) )
= ( B3 = C2 ) ) ).
% add_left_cancel
thf(fact_962_add__left__cancel,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= ( plus_plus_nat @ A3 @ C2 ) )
= ( B3 = C2 ) ) ).
% add_left_cancel
thf(fact_963_add__left__imp__eq,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ( plus_plus_complex @ A3 @ B3 )
= ( plus_plus_complex @ A3 @ C2 ) )
=> ( B3 = C2 ) ) ).
% add_left_imp_eq
thf(fact_964_add__left__imp__eq,axiom,
! [A3: real,B3: real,C2: real] :
( ( ( plus_plus_real @ A3 @ B3 )
= ( plus_plus_real @ A3 @ C2 ) )
=> ( B3 = C2 ) ) ).
% add_left_imp_eq
thf(fact_965_add__left__imp__eq,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= ( plus_plus_nat @ A3 @ C2 ) )
=> ( B3 = C2 ) ) ).
% add_left_imp_eq
thf(fact_966_add__right__cancel,axiom,
! [B3: complex,A3: complex,C2: complex] :
( ( ( plus_plus_complex @ B3 @ A3 )
= ( plus_plus_complex @ C2 @ A3 ) )
= ( B3 = C2 ) ) ).
% add_right_cancel
thf(fact_967_add__right__cancel,axiom,
! [B3: real,A3: real,C2: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= ( plus_plus_real @ C2 @ A3 ) )
= ( B3 = C2 ) ) ).
% add_right_cancel
thf(fact_968_add__right__cancel,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= ( plus_plus_nat @ C2 @ A3 ) )
= ( B3 = C2 ) ) ).
% add_right_cancel
thf(fact_969_add__right__imp__eq,axiom,
! [B3: complex,A3: complex,C2: complex] :
( ( ( plus_plus_complex @ B3 @ A3 )
= ( plus_plus_complex @ C2 @ A3 ) )
=> ( B3 = C2 ) ) ).
% add_right_imp_eq
thf(fact_970_add__right__imp__eq,axiom,
! [B3: real,A3: real,C2: real] :
( ( ( plus_plus_real @ B3 @ A3 )
= ( plus_plus_real @ C2 @ A3 ) )
=> ( B3 = C2 ) ) ).
% add_right_imp_eq
thf(fact_971_add__right__imp__eq,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= ( plus_plus_nat @ C2 @ A3 ) )
=> ( B3 = C2 ) ) ).
% add_right_imp_eq
thf(fact_972_add__less__imp__less__right,axiom,
! [A3: complex,C2: complex,B3: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A3 @ C2 ) @ ( plus_plus_complex @ B3 @ C2 ) )
=> ( ord_less_complex @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_973_add__less__imp__less__right,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) )
=> ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_974_add__less__imp__less__right,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B3 @ C2 ) )
=> ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_975_add__less__imp__less__left,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ C2 @ A3 ) @ ( plus_plus_complex @ C2 @ B3 ) )
=> ( ord_less_complex @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_976_add__less__imp__less__left,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) )
=> ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_977_add__less__imp__less__left,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A3 ) @ ( plus_plus_real @ C2 @ B3 ) )
=> ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_978_add__strict__right__mono,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ord_less_complex @ A3 @ B3 )
=> ( ord_less_complex @ ( plus_plus_complex @ A3 @ C2 ) @ ( plus_plus_complex @ B3 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_979_add__strict__right__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_980_add__strict__right__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B3 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_981_add__less__cancel__right,axiom,
! [A3: complex,C2: complex,B3: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A3 @ C2 ) @ ( plus_plus_complex @ B3 @ C2 ) )
= ( ord_less_complex @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_982_add__less__cancel__right,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_983_add__less__cancel__right,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B3 @ C2 ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_984_add__strict__left__mono,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ord_less_complex @ A3 @ B3 )
=> ( ord_less_complex @ ( plus_plus_complex @ C2 @ A3 ) @ ( plus_plus_complex @ C2 @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_985_add__strict__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_986_add__strict__left__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( plus_plus_real @ C2 @ A3 ) @ ( plus_plus_real @ C2 @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_987_add__less__cancel__left,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ C2 @ A3 ) @ ( plus_plus_complex @ C2 @ B3 ) )
= ( ord_less_complex @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_988_add__less__cancel__left,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_989_add__less__cancel__left,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A3 ) @ ( plus_plus_real @ C2 @ B3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_990_add__strict__mono,axiom,
! [A3: complex,B3: complex,C2: complex,D3: complex] :
( ( ord_less_complex @ A3 @ B3 )
=> ( ( ord_less_complex @ C2 @ D3 )
=> ( ord_less_complex @ ( plus_plus_complex @ A3 @ C2 ) @ ( plus_plus_complex @ B3 @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_991_add__strict__mono,axiom,
! [A3: nat,B3: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_992_add__strict__mono,axiom,
! [A3: real,B3: real,C2: real,D3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C2 @ D3 )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B3 @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_993_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_complex @ I3 @ J )
& ( K = L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_994_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_995_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I3 @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_996_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( I3 = J )
& ( ord_less_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_997_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_998_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( I3 = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_999_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_complex @ I3 @ J )
& ( ord_less_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1000_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1001_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I3 @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1002_mat__mult__plus__distrib__left,axiom,
! [Nr: nat,Nc: nat,M1: list_list_nat,M22: list_list_nat,Ncc: nat,M32: list_list_nat] :
( ( matrix_mat_nat @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_nat @ Nr @ Nc @ M22 )
=> ( ( matrix_mat_nat @ Nc @ Ncc @ M32 )
=> ( ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ Nr @ ( matrix_mat_plusI_nat @ plus_plus_nat @ M1 @ M22 ) @ M32 )
= ( matrix_mat_plusI_nat @ plus_plus_nat @ ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ Nr @ M1 @ M32 ) @ ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ Nr @ M22 @ M32 ) ) ) ) ) ) ).
% mat_mult_plus_distrib_left
thf(fact_1003_mat__mult__plus__distrib__left,axiom,
! [Nr: nat,Nc: nat,M1: list_list_complex,M22: list_list_complex,Ncc: nat,M32: list_list_complex] :
( ( matrix_mat_complex @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_complex @ Nr @ Nc @ M22 )
=> ( ( matrix_mat_complex @ Nc @ Ncc @ M32 )
=> ( ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ Nr @ ( matrix6097015163314587732omplex @ plus_plus_complex @ M1 @ M22 ) @ M32 )
= ( matrix6097015163314587732omplex @ plus_plus_complex @ ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ Nr @ M1 @ M32 ) @ ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ Nr @ M22 @ M32 ) ) ) ) ) ) ).
% mat_mult_plus_distrib_left
thf(fact_1004_mat__mult__plus__distrib__left,axiom,
! [Nr: nat,Nc: nat,M1: list_list_real,M22: list_list_real,Ncc: nat,M32: list_list_real] :
( ( matrix_mat_real @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_real @ Nr @ Nc @ M22 )
=> ( ( matrix_mat_real @ Nc @ Ncc @ M32 )
=> ( ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ Nr @ ( matrix4160047064883189586I_real @ plus_plus_real @ M1 @ M22 ) @ M32 )
= ( matrix4160047064883189586I_real @ plus_plus_real @ ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ Nr @ M1 @ M32 ) @ ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ Nr @ M22 @ M32 ) ) ) ) ) ) ).
% mat_mult_plus_distrib_left
thf(fact_1005_mat__mult__plus__distrib__right,axiom,
! [Nr: nat,Nc: nat,M1: list_list_nat,Ncc: nat,M22: list_list_nat,M32: list_list_nat] :
( ( matrix_mat_nat @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_nat @ Nc @ Ncc @ M22 )
=> ( ( matrix_mat_nat @ Nc @ Ncc @ M32 )
=> ( ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ Nr @ M1 @ ( matrix_mat_plusI_nat @ plus_plus_nat @ M22 @ M32 ) )
= ( matrix_mat_plusI_nat @ plus_plus_nat @ ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ Nr @ M1 @ M22 ) @ ( matrix_mat_multI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ Nr @ M1 @ M32 ) ) ) ) ) ) ).
% mat_mult_plus_distrib_right
thf(fact_1006_mat__mult__plus__distrib__right,axiom,
! [Nr: nat,Nc: nat,M1: list_list_complex,Ncc: nat,M22: list_list_complex,M32: list_list_complex] :
( ( matrix_mat_complex @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_complex @ Nc @ Ncc @ M22 )
=> ( ( matrix_mat_complex @ Nc @ Ncc @ M32 )
=> ( ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ Nr @ M1 @ ( matrix6097015163314587732omplex @ plus_plus_complex @ M22 @ M32 ) )
= ( matrix6097015163314587732omplex @ plus_plus_complex @ ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ Nr @ M1 @ M22 ) @ ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ Nr @ M1 @ M32 ) ) ) ) ) ) ).
% mat_mult_plus_distrib_right
thf(fact_1007_mat__mult__plus__distrib__right,axiom,
! [Nr: nat,Nc: nat,M1: list_list_real,Ncc: nat,M22: list_list_real,M32: list_list_real] :
( ( matrix_mat_real @ Nr @ Nc @ M1 )
=> ( ( matrix_mat_real @ Nc @ Ncc @ M22 )
=> ( ( matrix_mat_real @ Nc @ Ncc @ M32 )
=> ( ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ Nr @ M1 @ ( matrix4160047064883189586I_real @ plus_plus_real @ M22 @ M32 ) )
= ( matrix4160047064883189586I_real @ plus_plus_real @ ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ Nr @ M1 @ M22 ) @ ( matrix6810070775276521660I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ Nr @ M1 @ M32 ) ) ) ) ) ) ).
% mat_mult_plus_distrib_right
thf(fact_1008_zero__compare__simps_I3_J,axiom,
! [A3: complex,B3: complex,C2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ B3 @ C2 )
=> ( ord_less_eq_complex @ B3 @ ( plus_plus_complex @ A3 @ C2 ) ) ) ) ).
% zero_compare_simps(3)
thf(fact_1009_zero__compare__simps_I3_J,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% zero_compare_simps(3)
thf(fact_1010_zero__compare__simps_I3_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A3 @ C2 ) ) ) ) ).
% zero_compare_simps(3)
thf(fact_1011_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A3 @ A3 ) )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1012_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ A3 ) @ zero_zero_real )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1013_le__add__same__cancel2,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ ( plus_plus_complex @ B3 @ A3 ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_1014_le__add__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_1015_le__add__same__cancel2,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ ( plus_plus_real @ B3 @ A3 ) )
= ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_1016_le__add__same__cancel1,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ ( plus_plus_complex @ A3 @ B3 ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_1017_le__add__same__cancel1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_1018_le__add__same__cancel1,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ ( plus_plus_real @ A3 @ B3 ) )
= ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_1019_add__le__same__cancel2,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A3 @ B3 ) @ B3 )
= ( ord_less_eq_complex @ A3 @ zero_zero_complex ) ) ).
% add_le_same_cancel2
thf(fact_1020_add__le__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1021_add__le__same__cancel2,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B3 ) @ B3 )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_1022_add__le__same__cancel1,axiom,
! [B3: complex,A3: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ B3 @ A3 ) @ B3 )
= ( ord_less_eq_complex @ A3 @ zero_zero_complex ) ) ).
% add_le_same_cancel1
thf(fact_1023_add__le__same__cancel1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1024_add__le__same__cancel1,axiom,
! [B3: real,A3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B3 @ A3 ) @ B3 )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_1025_add__nonpos__eq__0__iff,axiom,
! [X3: complex,Y2: complex] :
( ( ord_less_eq_complex @ X3 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ Y2 @ zero_zero_complex )
=> ( ( ( plus_plus_complex @ X3 @ Y2 )
= zero_zero_complex )
= ( ( X3 = zero_zero_complex )
& ( Y2 = zero_zero_complex ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1026_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1027_add__nonpos__eq__0__iff,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X3 @ Y2 )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1028_add__nonneg__eq__0__iff,axiom,
! [X3: complex,Y2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ X3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ Y2 )
=> ( ( ( plus_plus_complex @ X3 @ Y2 )
= zero_zero_complex )
= ( ( X3 = zero_zero_complex )
& ( Y2 = zero_zero_complex ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1029_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1030_add__nonneg__eq__0__iff,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( plus_plus_real @ X3 @ Y2 )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1031_add__nonpos__nonpos,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B3 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A3 @ B3 ) @ zero_zero_complex ) ) ) ).
% add_nonpos_nonpos
thf(fact_1032_add__nonpos__nonpos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1033_add__nonpos__nonpos,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_1034_add__nonneg__nonneg,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B3 )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( plus_plus_complex @ A3 @ B3 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1035_add__nonneg__nonneg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1036_add__nonneg__nonneg,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1037_add__increasing2,axiom,
! [C2: complex,B3: complex,A3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ( ord_less_eq_complex @ B3 @ A3 )
=> ( ord_less_eq_complex @ B3 @ ( plus_plus_complex @ A3 @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1038_add__increasing2,axiom,
! [C2: nat,B3: nat,A3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1039_add__increasing2,axiom,
! [C2: real,B3: real,A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ B3 @ A3 )
=> ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A3 @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1040_add__decreasing2,axiom,
! [C2: complex,A3: complex,B3: complex] :
( ( ord_less_eq_complex @ C2 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A3 @ C2 ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_1041_add__decreasing2,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_1042_add__decreasing2,axiom,
! [C2: real,A3: real,B3: real] :
( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C2 ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_1043_add__decreasing,axiom,
! [A3: complex,C2: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ C2 @ B3 )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A3 @ C2 ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_1044_add__decreasing,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_1045_add__decreasing,axiom,
! [A3: real,C2: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ C2 @ B3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C2 ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_1046_mat__to__cols__list__times__mat,axiom,
! [A: mat_complex,B2: mat_complex] :
( ( ( dim_col_complex @ A )
= ( dim_row_complex @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_col_complex @ A ) )
=> ( ( mat_to_cols_list @ ( times_8009071140041733218omplex @ A @ B2 ) )
= ( matrix2876711754638949054omplex @ zero_zero_complex @ plus_plus_complex @ times_times_complex @ ( matrix1515831402840476169omplex @ ( mat_to_cols_list @ A ) ) @ ( mat_to_cols_list @ A ) @ ( mat_to_cols_list @ B2 ) ) ) ) ) ).
% mat_to_cols_list_times_mat
thf(fact_1047_sum__le__prod1,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ B3 @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B3 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ) ).
% sum_le_prod1
thf(fact_1048_add__eq__self__zero,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= M )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1049_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1050_add__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1051_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1052_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1053_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N: nat] :
? [K5: nat] :
( N
= ( plus_plus_nat @ M3 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1054_trans__le__add2,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1055_trans__le__add1,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1056_add__le__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1057_add__le__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1058_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1059_add__leD2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1060_add__leD1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% add_leD1
thf(fact_1061_le__add2,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% le_add2
thf(fact_1062_le__add1,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% le_add1
thf(fact_1063_add__leE,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1064_left__add__mult__distrib,axiom,
! [I3: nat,U4: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I3 @ J ) @ U4 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1065_nat__distrib_I1_J,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% nat_distrib(1)
thf(fact_1066_add__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1067_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1068_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1069_trans__less__add2,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1070_trans__less__add1,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1071_add__less__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1072_not__add__less2,axiom,
! [J: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_1073_not__add__less1,axiom,
! [I3: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ I3 ) ).
% not_add_less1
thf(fact_1074_add__less__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1075_add__lessD1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
=> ( ord_less_nat @ I3 @ K ) ) ).
% add_lessD1
thf(fact_1076_mat__assoc__test_I13_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( plus_p8323303612493835998omplex @ A @ B2 )
= ( plus_p8323303612493835998omplex @ B2 @ A ) ) ) ) ) ) ).
% mat_assoc_test(13)
thf(fact_1077_mat__assoc__test_I14_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ C )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ C @ B2 ) @ A ) ) ) ) ) ) ).
% mat_assoc_test(14)
thf(fact_1078_mat__assoc__test_I15_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ ( plus_p8323303612493835998omplex @ C @ D ) )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ A @ C ) @ ( plus_p8323303612493835998omplex @ B2 @ D ) ) ) ) ) ) ) ).
% mat_assoc_test(15)
thf(fact_1079_less__imp__add__positive,axiom,
! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I3 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1080_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1081_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M7: nat,N3: nat] :
( ( ord_less_nat @ M7 @ N3 )
=> ( ord_less_nat @ ( F @ M7 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1082_mat__assoc__test_I7_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( times_8009071140041733218omplex @ ( plus_p8323303612493835998omplex @ A @ B2 ) @ ( plus_p8323303612493835998omplex @ B2 @ C ) )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A @ B2 ) @ ( times_8009071140041733218omplex @ B2 @ B2 ) ) @ ( times_8009071140041733218omplex @ A @ C ) ) @ ( times_8009071140041733218omplex @ B2 @ C ) ) ) ) ) ) ) ).
% mat_assoc_test(7)
thf(fact_1083_Complex__Matrix_Opositive__add,axiom,
! [A: mat_complex,B2: mat_complex,N2: nat] :
( ( complex_positive @ A )
=> ( ( complex_positive @ B2 )
=> ( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( complex_positive @ ( plus_p8323303612493835998omplex @ A @ B2 ) ) ) ) ) ) ).
% Complex_Matrix.positive_add
thf(fact_1084_lowner__le__add,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( complex_lowner_le @ A @ B2 )
=> ( ( complex_lowner_le @ C @ D )
=> ( complex_lowner_le @ ( plus_p8323303612493835998omplex @ A @ C ) @ ( plus_p8323303612493835998omplex @ B2 @ D ) ) ) ) ) ) ) ) ).
% lowner_le_add
thf(fact_1085_kuhn__lemma,axiom,
! [P5: nat,N2: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P5 )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P5 ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( ( ( Label @ X4 @ I )
= zero_zero_nat )
| ( ( Label @ X4 @ I )
= one_one_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P5 ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( ( ( X4 @ I )
= zero_zero_nat )
=> ( ( Label @ X4 @ I )
= zero_zero_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P5 ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( ( ( X4 @ I )
= P5 )
=> ( ( Label @ X4 @ I )
= one_one_nat ) ) ) )
=> ~ ! [Q4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_nat @ ( Q4 @ I4 ) @ P5 ) )
=> ~ ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ? [R3: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q4 @ J4 ) @ ( R3 @ J4 ) )
& ( ord_less_eq_nat @ ( R3 @ J4 ) @ ( plus_plus_nat @ ( Q4 @ J4 ) @ one_one_nat ) ) ) )
& ? [S3: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q4 @ J4 ) @ ( S3 @ J4 ) )
& ( ord_less_eq_nat @ ( S3 @ J4 ) @ ( plus_plus_nat @ ( Q4 @ J4 ) @ one_one_nat ) ) ) )
& ( ( Label @ R3 @ I4 )
!= ( Label @ S3 @ I4 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1086_mat__assoc__test_I12_J,axiom,
! [A: mat_complex,N2: nat,B2: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( comple3184165445352484367omplex @ ( plus_p8323303612493835998omplex @ A @ ( times_8009071140041733218omplex @ B2 @ C ) ) )
= ( plus_plus_complex @ ( comple3184165445352484367omplex @ A ) @ ( comple3184165445352484367omplex @ ( times_8009071140041733218omplex @ C @ B2 ) ) ) ) ) ) ) ) ).
% mat_assoc_test(12)
thf(fact_1087_square__bound__lemma,axiom,
! [X3: real] : ( ord_less_real @ X3 @ ( times_times_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( plus_plus_real @ one_one_real @ X3 ) ) ) ).
% square_bound_lemma
thf(fact_1088_not__real__square__gt__zero,axiom,
! [X3: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X3 @ X3 ) ) )
= ( X3 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1089_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A6: nat,B7: nat] :
( ( P @ A6 @ B7 )
= ( P @ B7 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ zero_zero_nat )
=> ( ! [A6: nat,B7: nat] :
( ( P @ A6 @ B7 )
=> ( P @ A6 @ ( plus_plus_nat @ A6 @ B7 ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Euclid_induct
thf(fact_1090_inverts__mat__sym,axiom,
! [A: mat_complex,B2: mat_complex] :
( ( inverts_mat_complex @ A @ B2 )
=> ( ( ( dim_row_complex @ B2 )
= ( dim_col_complex @ A ) )
=> ( ( square_mat_complex @ B2 )
=> ( inverts_mat_complex @ B2 @ A ) ) ) ) ).
% inverts_mat_sym
thf(fact_1091_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [M7: nat] : ( P @ M7 @ zero_zero_nat )
=> ( ! [M7: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo_nat @ M7 @ N3 ) )
=> ( P @ M7 @ N3 ) ) )
=> ( P @ M @ N2 ) ) ) ).
% gcd_nat_induct
thf(fact_1092_bezw_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [X4: nat,Y3: nat] :
( ( ( Y3 != zero_zero_nat )
=> ( P @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) )
=> ( ( ( Y3 != zero_zero_nat )
=> ( P @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) )
=> ( ( ( Y3 != zero_zero_nat )
=> ( P @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) )
=> ( P @ X4 @ Y3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ).
% bezw.induct
thf(fact_1093_gcd__nat_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [X4: nat,Y3: nat] :
( ( ( Y3 != zero_zero_nat )
=> ( P @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) )
=> ( P @ X4 @ Y3 ) )
=> ( P @ A0 @ A1 ) ) ).
% gcd_nat.induct
thf(fact_1094_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1095_verit__le__mono__div,axiom,
! [A: nat,B2: nat,N2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A @ N2 )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B2 @ N2 )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B2 @ N2 ) ) ) ) ).
% verit_le_mono_div
thf(fact_1096_div__mod__decomp,axiom,
! [A: nat,N2: nat] :
( A
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A @ N2 ) ) ) ).
% div_mod_decomp
thf(fact_1097_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y2: nat] :
( ( if_nat @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y2: nat] :
( ( if_nat @ $true @ X3 @ Y2 )
= X3 ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X3: real,Y2: real] :
( ( if_real @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X3: real,Y2: real] :
( ( if_real @ $true @ X3 @ Y2 )
= X3 ) ).
thf(help_If_3_1_If_001t__Complex__Ocomplex_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X3: complex,Y2: complex] :
( ( if_complex @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X3: complex,Y2: complex] :
( ( if_complex @ $true @ X3 @ Y2 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [Bl: list_mat_complex,Ul: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Ul )
= ( size_s5969786470865220249omplex @ al ) )
& ! [I: nat] :
( ~ ( ord_less_nat @ I @ ( size_s5969786470865220249omplex @ al ) )
| ( ( member_mat_complex @ ( nth_mat_complex @ Ul @ I ) @ ( carrier_mat_complex @ ( dim_row_complex @ ( nth_mat_complex @ al @ I ) ) @ ( dim_col_complex @ ( nth_mat_complex @ al @ I ) ) ) )
& ( comple6660659447773130958omplex @ ( nth_mat_complex @ Ul @ I ) )
& ( member_mat_complex @ ( nth_mat_complex @ Bl @ I ) @ ( carrier_mat_complex @ ( dim_row_complex @ ( nth_mat_complex @ al @ I ) ) @ ( dim_col_complex @ ( nth_mat_complex @ al @ I ) ) ) ) ) )
& ( spectr5409772854192057952omplex @ ( diag_b9145358668110806138omplex @ al ) @ ( diag_b9145358668110806138omplex @ Bl ) @ ( diag_b9145358668110806138omplex @ Ul ) ) ) ).
%------------------------------------------------------------------------------