TPTP Problem File: SLH0119^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% 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_03510_140942__19708020_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1513 ( 580 unt; 358 typ; 0 def)
% Number of atoms : 3204 (1684 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 10940 ( 349 ~; 72 |; 281 &;8639 @)
% ( 0 <=>;1599 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 55 ( 54 usr)
% Number of type conns : 852 ( 852 >; 0 *; 0 +; 0 <<)
% Number of symbols : 307 ( 304 usr; 66 con; 0-4 aty)
% Number of variables : 3145 ( 48 ^;2930 !; 167 ?;3145 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:39:04.913
%------------------------------------------------------------------------------
% Could-be-implicit typings (54)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_Pr8195022564563857607omplex: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_Su1178597446371505947omplex: $tType ).
thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Complex__Ocomplex_Mt__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
finite279839453003621671l_num1: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_Pr6450005376003605217omplex: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Ounit_J_J,type,
set_Pr7172745005698865197t_unit: $tType ).
thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Nat__Onat_Mt__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
finite1289000397740218697l_num1: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_Su328059466923093813omplex: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Ounit_J_J,type,
set_Su1050799096618353793t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_Pr8061060647479028738omplex: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Nat__Onat_J_J,type,
set_Pr4190393606172363740ex_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
set_Pr5094982260447487303t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_Su7239676323109711534omplex: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Nat__Onat_J_J,type,
set_Su3369009281803046536ex_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
set_Su4110612849109743515t_unit: $tType ).
thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Complex__Ocomplex_Mt__Numeral____Type__Onum1_J,type,
finite3194102281137443150l_num1: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
set_Pr1763845938948868674it_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
set_Pr4334478416066269672t_unit: $tType ).
thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Nat__Onat_Mt__Numeral____Type__Onum1_J,type,
finite2525469894391432876l_num1: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
set_Su4968945780807083758it_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
set_Su7539578257924484756t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_mat_mat_complex: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_list_mat_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_set_mat_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Sum_sum_nat_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
mat_mat_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
set_set_Product_unit: $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__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
set_list_complex: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
list_set_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__Matrix__Omat_It__Nat__Onat_J_J,type,
list_mat_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Nat__Onat_J_J,type,
set_mat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
set_Product_unit: $tType ).
thf(ty_n_t__Matrix__Omat_It__Complex__Ocomplex_J,type,
mat_complex: $tType ).
thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Matrix__Omat_It__Real__Oreal_J,type,
mat_real: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_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__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $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 (304)
thf(sy_c_Cartesian__Space_Ovector_001t__Complex__Ocomplex_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
cartes2146103711486359127l_num1: list_complex > finite279839453003621671l_num1 ).
thf(sy_c_Cartesian__Space_Ovector_001t__Complex__Ocomplex_001t__Numeral____Type__Onum1,type,
cartes7240268600962231432l_num1: list_complex > finite3194102281137443150l_num1 ).
thf(sy_c_Cartesian__Space_Ovector_001t__Nat__Onat_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
cartes7700031802712742009l_num1: list_nat > finite1289000397740218697l_num1 ).
thf(sy_c_Cartesian__Space_Ovector_001t__Nat__Onat_001t__Numeral____Type__Onum1,type,
cartes6052806112279933926l_num1: list_nat > finite2525469894391432876l_num1 ).
thf(sy_c_Commuting__Hermitian_Odiag__compat_001t__Complex__Ocomplex,type,
commut5261563022830629508omplex: mat_complex > list_nat > $o ).
thf(sy_c_Commuting__Hermitian_Odiag__diff_001t__Complex__Ocomplex,type,
commut4502369927624756007omplex: mat_complex > list_nat > $o ).
thf(sy_c_Commuting__Hermitian_Odiag__diff_001t__Nat__Onat,type,
commut2433368883435016265ff_nat: mat_nat > list_nat > $o ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001t__Complex__Ocomplex,type,
commut93809757773076895omplex: list_complex > list_nat ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
commut5736191610077499254omplex: list_mat_complex > list_nat ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001t__Nat__Onat,type,
commut2436974278740741825ps_nat: list_nat > list_nat ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001t__Real__Oreal,type,
commut8680161604938074397s_real: list_real > list_nat ).
thf(sy_c_Commuting__Hermitian_Oextract__subdiags_001t__Complex__Ocomplex,type,
commut6900707758132580272omplex: mat_complex > list_nat > list_mat_complex ).
thf(sy_c_Commuting__Hermitian_Oextract__subdiags_001t__Nat__Onat,type,
commut2742254738641004242gs_nat: mat_nat > list_nat > list_mat_nat ).
thf(sy_c_Commuting__Hermitian_Olst__diff_001t__Complex__Ocomplex,type,
commut1410864796179263225omplex: list_complex > list_nat > $o ).
thf(sy_c_Commuting__Hermitian_Olst__diff_001t__Nat__Onat,type,
commut7647841724617136155ff_nat: list_nat > list_nat > $o ).
thf(sy_c_Commuting__Hermitian_Oper__diag_001t__Complex__Ocomplex,type,
commut4119912100034661455omplex: mat_complex > ( nat > nat ) > mat_complex ).
thf(sy_c_Commuting__Hermitian_Oper__diag_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
commut3385207333667201222omplex: mat_mat_complex > ( nat > nat ) > mat_mat_complex ).
thf(sy_c_Commuting__Hermitian_Oper__diag_001t__Nat__Onat,type,
commut5604902300900073841ag_nat: mat_nat > ( nat > nat ) > mat_nat ).
thf(sy_c_Commuting__Hermitian_Oper__diag_001t__Real__Oreal,type,
commut7814478499001091021g_real: mat_real > ( nat > nat ) > mat_real ).
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Complex__Matrix_Ohermitian_001t__Complex__Ocomplex,type,
comple8306762464034002205omplex: mat_complex > $o ).
thf(sy_c_Complex__Matrix_Ounitary_001t__Complex__Ocomplex,type,
comple6660659447773130958omplex: mat_complex > $o ).
thf(sy_c_Determinant_Omat__delete_001t__Complex__Ocomplex,type,
mat_delete_complex: mat_complex > nat > nat > mat_complex ).
thf(sy_c_Determinant_Opermutation__insert_001t__Nat__Onat,type,
permut3695043542826343943rt_nat: nat > nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Determinant_Opermutation__insert_001t__Real__Oreal,type,
permut4060954620988167523t_real: real > nat > ( real > nat ) > real > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
finite32325147380013007omplex: set_mat_complex > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
finite7047982916621727056omplex: set_mat_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
finite7366070217807234576omplex: set_Pr8195022564563857607omplex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Nat__Onat_J,type,
finite5159396851285225149ex_nat: set_Pr4190393606172363740ex_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Ounit_J,type,
finite4703860625191999094t_unit: set_Pr7172745005698865197t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
finite3477794401636876771omplex: set_Pr8061060647479028738omplex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
finite5113082511001691337t_unit: set_Pr4334478416066269672t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
finite7222479058111421226omplex: set_Pr6450005376003605217omplex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
finite5187522816498166307it_nat: set_Pr1763845938948868674it_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J,type,
finite6816719414181127824t_unit: set_Pr5094982260447487303t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Ounit,type,
finite4290736615968046902t_unit: set_Product_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
finite1349200545324696496omplex: set_set_mat_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Ounit_J,type,
finite1772178364199683094t_unit: set_set_Product_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
finite6995816559193130876omplex: set_Su1178597446371505947omplex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Nat__Onat_J,type,
finite3494777059389363281ex_nat: set_Su3369009281803046536ex_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Ounit_J,type,
finite6535610298543002082t_unit: set_Su1050799096618353793t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
finite1813174609741014903omplex: set_Su7239676323109711534omplex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6187706683773761046at_nat: set_Sum_sum_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
finite4327512606132785245t_unit: set_Su7539578257924484756t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
finite9054228731462424214omplex: set_Su328059466923093813omplex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
finite4401952911629260215it_nat: set_Su4968945780807083758it_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J,type,
finite3146551501593861116t_unit: set_Su4110612849109743515t_unit > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
minus_2412168080157227406omplex: mat_complex > mat_complex > mat_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
minus_8760755521168068590omplex: set_mat_complex > set_mat_complex > set_mat_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Ounit_J,type,
minus_6452836326544984404t_unit: set_Product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
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_Oone__class_Oone_001t__Set__Oset_It__Nat__Onat_J,type,
one_one_set_nat: set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Real__Oreal_J,type,
one_one_set_real: set_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__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_Otimes__class_Otimes_001t__Set__Oset_It__Complex__Ocomplex_J,type,
times_6048082448287401577omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
times_6731331324747250370omplex: set_mat_complex > set_mat_complex > set_mat_complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Real__Oreal_J,type,
zero_zero_set_real: set_real ).
thf(sy_c_If_001t__Complex__Ocomplex,type,
if_complex: $o > complex > complex > complex ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
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_Lattices__Big_Oord__class_Oarg__min__on_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Nat__Onat,type,
lattic8691243167872488990ex_nat: ( mat_complex > nat ) > set_mat_complex > mat_complex ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Real__Oreal,type,
lattic2495253371305538042x_real: ( mat_complex > real ) > set_mat_complex > mat_complex ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).
thf(sy_c_Linear__Algebra__Complements_Oprojector_001t__Complex__Ocomplex,type,
linear5633924348262549461omplex: mat_complex > $o ).
thf(sy_c_List_Oappend_001t__Complex__Ocomplex,type,
append_complex: list_complex > list_complex > list_complex ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Complex__Ocomplex_J,type,
append_list_complex: list_list_complex > list_list_complex > list_list_complex ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
append_mat_complex: list_mat_complex > list_mat_complex > list_mat_complex ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Real__Oreal,type,
append_real: list_real > list_real > list_real ).
thf(sy_c_List_Oconcat_001t__Complex__Ocomplex,type,
concat_complex: list_list_complex > list_complex ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Oconcat_001t__Real__Oreal,type,
concat_real: list_list_real > list_real ).
thf(sy_c_List_Ogen__length_001t__Complex__Ocomplex,type,
gen_length_complex: nat > list_complex > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Olist_OCons_001t__Complex__Ocomplex,type,
cons_complex: complex > list_complex > list_complex ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Complex__Ocomplex_J,type,
cons_list_complex: list_complex > list_list_complex > list_list_complex ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
cons_mat_complex: mat_complex > list_mat_complex > list_mat_complex ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
cons_real: real > list_real > list_real ).
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__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
nil_mat_complex: list_mat_complex ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Real__Oreal,type,
nil_real: list_real ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Complex__Ocomplex_J,type,
nil_set_complex: list_set_complex ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Nat__Onat_J,type,
nil_set_nat: list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
map_complex_complex: ( complex > complex ) > list_complex > list_complex ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Nat__Onat,type,
map_complex_nat: ( complex > nat ) > list_complex > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Real__Oreal,type,
map_complex_real: ( complex > real ) > list_complex > list_real ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Complex__Ocomplex_J_001t__List__Olist_It__Complex__Ocomplex_J,type,
map_li2870275437539113154omplex: ( list_complex > list_complex ) > list_list_complex > list_list_complex ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Complex__Ocomplex_J_001t__List__Olist_It__Real__Oreal_J,type,
map_li971590449312185664t_real: ( list_complex > list_real ) > list_list_complex > list_list_real ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
map_ma6165852935686130436omplex: ( mat_complex > mat_complex ) > list_mat_complex > list_mat_complex ).
thf(sy_c_List_Olist_Omap_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Nat__Onat,type,
map_mat_complex_nat: ( mat_complex > nat ) > list_mat_complex > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Real__Oreal,type,
map_mat_complex_real: ( mat_complex > real ) > list_mat_complex > list_real ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Complex__Ocomplex,type,
map_nat_complex: ( nat > complex ) > list_nat > list_complex ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
map_nat_mat_complex: ( nat > mat_complex ) > list_nat > list_mat_complex ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Real__Oreal,type,
map_nat_real: ( nat > real ) > list_nat > list_real ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
map_real_mat_complex: ( real > mat_complex ) > list_real > list_mat_complex ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
map_real_nat: ( real > nat ) > list_real > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
map_real_real: ( real > real ) > list_real > list_real ).
thf(sy_c_List_Olistset_001t__Complex__Ocomplex,type,
listset_complex: list_set_complex > set_list_complex ).
thf(sy_c_List_Olistset_001t__Nat__Onat,type,
listset_nat: list_set_nat > set_list_nat ).
thf(sy_c_List_Omap__tailrec_001t__Complex__Ocomplex_001t__Real__Oreal,type,
map_ta5686879364588479338x_real: ( complex > real ) > list_complex > list_real ).
thf(sy_c_List_Omap__tailrec__rev_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
map_ta3224203966507818835omplex: ( complex > complex ) > list_complex > list_complex > list_complex ).
thf(sy_c_List_Omap__tailrec__rev_001t__Complex__Ocomplex_001t__Nat__Onat,type,
map_ta4351949837888814197ex_nat: ( complex > nat ) > list_complex > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001t__Complex__Ocomplex_001t__Real__Oreal,type,
map_ta6589641090014469329x_real: ( complex > real ) > list_complex > list_real > list_real ).
thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__Complex__Ocomplex,type,
map_ta732166513454076533omplex: ( nat > complex ) > list_nat > list_complex > list_complex ).
thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__Nat__Onat,type,
map_ta7164188454487880599at_nat: ( nat > nat ) > list_nat > list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
maps_complex_complex: ( complex > list_complex ) > list_complex > list_complex ).
thf(sy_c_List_Omaps_001t__Complex__Ocomplex_001t__Nat__Onat,type,
maps_complex_nat: ( complex > list_nat ) > list_complex > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Complex__Ocomplex,type,
maps_nat_complex: ( nat > list_complex ) > list_nat > list_complex ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Omember_001t__Complex__Ocomplex,type,
member_complex: list_complex > complex > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_On__lists_001t__Complex__Ocomplex,type,
n_lists_complex: nat > list_complex > list_list_complex ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
nth_complex: list_complex > nat > 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_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Real__Oreal,type,
nth_real: list_real > nat > real ).
thf(sy_c_List_Oproduct__lists_001t__Complex__Ocomplex,type,
produc7545014605101902079omplex: list_list_complex > list_list_complex ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orev_001t__Complex__Ocomplex,type,
rev_complex: list_complex > list_complex ).
thf(sy_c_List_Orev_001t__List__Olist_It__Nat__Onat_J,type,
rev_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orev_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
rev_mat_complex: list_mat_complex > list_mat_complex ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orev_001t__Real__Oreal,type,
rev_real: list_real > list_real ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Real__Oreal,type,
sorted_wrt_real: ( real > real > $o ) > list_real > $o ).
thf(sy_c_List_Osubseqs_001t__Complex__Ocomplex,type,
subseqs_complex: list_complex > list_list_complex ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Complex__Ocomplex,type,
carrier_mat_complex: nat > nat > set_mat_complex ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
carrie8442657464762054641omplex: nat > nat > set_mat_mat_complex ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Nat__Onat,type,
carrier_mat_nat: nat > nat > set_mat_nat ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Real__Oreal,type,
carrier_mat_real: nat > nat > set_mat_real ).
thf(sy_c_Matrix_Odiag__block__mat_001t__Complex__Ocomplex,type,
diag_b9145358668110806138omplex: list_mat_complex > mat_complex ).
thf(sy_c_Matrix_Odiag__block__mat_001t__Nat__Onat,type,
diag_block_mat_nat: list_mat_nat > mat_nat ).
thf(sy_c_Matrix_Odiag__mat_001t__Complex__Ocomplex,type,
diag_mat_complex: mat_complex > list_complex ).
thf(sy_c_Matrix_Odiag__mat_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
diag_mat_mat_complex: mat_mat_complex > list_mat_complex ).
thf(sy_c_Matrix_Odiag__mat_001t__Nat__Onat,type,
diag_mat_nat: mat_nat > list_nat ).
thf(sy_c_Matrix_Odiag__mat_001t__Real__Oreal,type,
diag_mat_real: mat_real > list_real ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Complex__Ocomplex,type,
diagonal_mat_complex: mat_complex > $o ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Nat__Onat,type,
diagonal_mat_nat: mat_nat > $o ).
thf(sy_c_Matrix_Oone__mat_001t__Complex__Ocomplex,type,
one_mat_complex: nat > mat_complex ).
thf(sy_c_Matrix_Osmult__mat_001t__Complex__Ocomplex,type,
smult_mat_complex: complex > mat_complex > mat_complex ).
thf(sy_c_Matrix_Osmult__mat_001t__Nat__Onat,type,
smult_mat_nat: nat > mat_nat > mat_nat ).
thf(sy_c_Matrix_Osmult__mat_001t__Real__Oreal,type,
smult_mat_real: real > mat_real > mat_real ).
thf(sy_c_Matrix_Oundef__vec_001t__Complex__Ocomplex,type,
undef_vec_complex: nat > complex ).
thf(sy_c_Matrix_Oundef__vec_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
undef_2495355514574404529omplex: nat > mat_complex ).
thf(sy_c_Matrix_Oundef__vec_001t__Nat__Onat,type,
undef_vec_nat: nat > nat ).
thf(sy_c_Matrix_Oundef__vec_001t__Real__Oreal,type,
undef_vec_real: nat > real ).
thf(sy_c_Missing__Unsorted_Omax__list__non__empty_001t__Nat__Onat,type,
missin53001312869816611ty_nat: list_nat > nat ).
thf(sy_c_Missing__VectorSpace_Ofind__indices_001t__Complex__Ocomplex,type,
missin8834916005246747252omplex: complex > list_complex > list_nat ).
thf(sy_c_Missing__VectorSpace_Ofind__indices_001t__Nat__Onat,type,
missin5050847376130023830es_nat: nat > list_nat > list_nat ).
thf(sy_c_Modules_Omodule_Ospan_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
span_complex_complex: ( complex > complex > complex ) > set_complex > set_complex ).
thf(sy_c_Modules_Omodule_Ospan_001t__Real__Oreal_001t__Real__Oreal,type,
span_real_real: ( real > real > real ) > set_real > set_real ).
thf(sy_c_Multiset__Permutations_Opermutations__of__list__impl_001t__Complex__Ocomplex,type,
multis6086220127510270003omplex: list_complex > list_list_complex ).
thf(sy_c_Multiset__Permutations_Opermutations__of__list__impl_001t__Nat__Onat,type,
multis1338830684155981141pl_nat: list_nat > list_list_nat ).
thf(sy_c_Multiset__Permutations_Opermutations__of__list__impl__aux_001t__Complex__Ocomplex,type,
multis1292119725739966671omplex: list_complex > list_complex > list_list_complex ).
thf(sy_c_Multiset__Permutations_Opermutations__of__list__impl__aux_001t__Nat__Onat,type,
multis6640212516482799089ux_nat: list_nat > list_nat > list_list_nat ).
thf(sy_c_Multiset__Permutations_Opermutations__of__set_001t__Complex__Ocomplex,type,
multis1932168107469466731omplex: set_complex > set_list_complex ).
thf(sy_c_Multiset__Permutations_Opermutations__of__set_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
multis2234121214477010346omplex: set_mat_complex > set_list_mat_complex ).
thf(sy_c_Multiset__Permutations_Opermutations__of__set_001t__Nat__Onat,type,
multis1655833086286526861et_nat: set_nat > set_list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
size_s5969786470865220249omplex: list_mat_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
size_size_list_real: list_real > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Matrix__Omat_It__Complex__Ocomplex_J_M_Eo_J,type,
bot_bo2514468519737825834plex_o: mat_complex > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
bot_bot_set_complex: set_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
bot_bo6492010485567502472omplex: set_list_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
bot_bo6377478972893813113omplex: set_list_mat_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
bot_bo7165004461764951667omplex: set_mat_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Ounit_J,type,
bot_bo3957492148770167129t_unit: set_Product_unit ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
ord_less_mat_complex: mat_complex > mat_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_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
ord_le5598786136212072115omplex: set_mat_complex > set_mat_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $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__Matrix__Omat_It__Complex__Ocomplex_J,type,
ord_le1403324449407493959omplex: mat_complex > mat_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__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $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__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Complex__Ocomplex_J,type,
top_top_set_complex: set_complex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
top_to1861530291043981143omplex: set_mat_complex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
top_to6470415304537177623omplex: set_Pr8195022564563857607omplex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Nat__Onat_J_J,type,
top_to3326070779871765132ex_nat: set_Pr4190393606172363740ex_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Ounit_J_J,type,
top_to843579388925295997t_unit: set_Pr7172745005698865197t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
top_to7196737821178430130omplex: set_Pr8061060647479028738omplex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
top_to8544742955230171288t_unit: set_Pr4334478416066269672t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
top_to120839759230036017omplex: set_Pr6450005376003605217omplex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
top_to5974110478112770290it_nat: set_Pr1763845938948868674it_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
top_to1835807148980544151t_unit: set_Pr5094982260447487303t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
top_to5556459383890298295omplex: set_set_mat_complex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
top_to1767297665138865437t_unit: set_set_Product_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
top_to442280769018103243omplex: set_Su1178597446371505947omplex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Nat__Onat_J_J,type,
top_to2112016286549842904ex_nat: set_Su3369009281803046536ex_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Ounit_J_J,type,
top_to1024881433771703089t_unit: set_Su1050799096618353793t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
top_to5982683327856507902omplex: set_Su7239676323109711534omplex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
top_to5465250082899874788t_unit: set_Su7539578257924484756t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
top_to302141804076443109omplex: set_Su328059466923093813omplex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
top_to2894617605782473790it_nat: set_Su4968945780807083758it_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
top_to2771918933716375115t_unit: set_Su4110612849109743515t_unit ).
thf(sy_c_Polynomial_Oplus__coeffs_001t__Complex__Ocomplex,type,
plus_coeffs_complex: list_complex > list_complex > list_complex ).
thf(sy_c_Polynomial_Oplus__coeffs_001t__Nat__Onat,type,
plus_coeffs_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Projective__Measurements_Odiag__elems_001t__Complex__Ocomplex,type,
projec2809893096078145286omplex: mat_complex > set_complex ).
thf(sy_c_Projective__Measurements_Odiag__elems_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
projec1765981369499306831omplex: mat_mat_complex > set_mat_complex ).
thf(sy_c_Projective__Measurements_Odiag__elems_001t__Nat__Onat,type,
projec8639844951311350312ms_nat: mat_nat > set_nat ).
thf(sy_c_Schur__Decomposition_Ocorthogonal__mat_001t__Complex__Ocomplex,type,
schur_549222400177443379omplex: mat_complex > $o ).
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__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
insert_complex: complex > set_complex > set_complex ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Complex__Ocomplex_J,type,
insert_list_complex: list_complex > set_list_complex > set_list_complex ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
insert8600743036574769149omplex: list_mat_complex > set_list_mat_complex > set_list_mat_complex ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
insert_mat_complex: mat_complex > set_mat_complex > set_mat_complex ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Ois__empty_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
is_empty_mat_complex: set_mat_complex > $o ).
thf(sy_c_Set_Ois__singleton_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
is_sin1068006998250866843omplex: set_mat_complex > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Oremove_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
remove_mat_complex: mat_complex > set_mat_complex > set_mat_complex ).
thf(sy_c_Set_Othe__elem_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
the_elem_mat_complex: set_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_Sublist_Olongest__common__prefix_001t__Complex__Ocomplex,type,
longes3069803181527244196omplex: list_complex > list_complex > list_complex ).
thf(sy_c_Sublist_Olongest__common__prefix_001t__Nat__Onat,type,
longes266370323089874118ix_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Sublist_Oprefix_001t__Complex__Ocomplex,type,
prefix_complex: list_complex > list_complex > $o ).
thf(sy_c_Sublist_Oprefix_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
prefix_mat_complex: list_mat_complex > list_mat_complex > $o ).
thf(sy_c_Sublist_Oprefix_001t__Nat__Onat,type,
prefix_nat: list_nat > list_nat > $o ).
thf(sy_c_Sublist_Oprefix_001t__Real__Oreal,type,
prefix_real: list_real > list_real > $o ).
thf(sy_c_Sublist_Oprefixes_001t__Complex__Ocomplex,type,
prefixes_complex: list_complex > list_list_complex ).
thf(sy_c_Sublist_Oprefixes_001t__Nat__Onat,type,
prefixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osublists_001t__Complex__Ocomplex,type,
sublists_complex: list_complex > list_list_complex ).
thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
sublists_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__Complex__Ocomplex,type,
suffixes_complex: list_complex > list_list_complex ).
thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
suffixes_nat: list_nat > list_list_nat ).
thf(sy_c_Utility_Omax__list,type,
max_list: list_nat > nat ).
thf(sy_c_Vector__Spaces_Ofinite__dimensional__vector__space_Odimension_001t__Real__Oreal,type,
vector5117482691322076262n_real: set_real > nat ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex2: complex > set_complex > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
member_mat_complex: mat_complex > set_mat_complex > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
member7752848204589936667omplex: mat_mat_complex > set_mat_mat_complex > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Nat__Onat_J,type,
member_mat_nat: mat_nat > set_mat_nat > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Real__Oreal_J,type,
member_mat_real: mat_real > set_mat_real > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_Af,type,
af: set_mat_complex ).
thf(sy_v_Afa____,type,
afa: set_mat_complex ).
thf(sy_v_Afp____,type,
afp: set_mat_complex ).
thf(sy_v_Ap____,type,
ap: mat_complex ).
thf(sy_v_Bs____,type,
bs: mat_complex ).
thf(sy_v_Us____,type,
us: mat_complex ).
thf(sy_v_eqcl____,type,
eqcl: list_nat ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_ia____,type,
ia: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_na____,type,
na: nat ).
% Relevant facts (1145)
thf(fact_0_ubprops_I6_J,axiom,
( ( commut93809757773076895omplex @ ( diag_mat_complex @ bs ) )
!= nil_nat ) ).
% ubprops(6)
thf(fact_1__092_060open_062diag__diff_ABs_Aeqcl_092_060close_062,axiom,
commut4502369927624756007omplex @ bs @ eqcl ).
% \<open>diag_diff Bs eqcl\<close>
thf(fact_2_eqcl__def,axiom,
( eqcl
= ( commut93809757773076895omplex @ ( diag_mat_complex @ bs ) ) ) ).
% eqcl_def
thf(fact_3_eq__comps_Osimps_I1_J,axiom,
( ( commut2436974278740741825ps_nat @ nil_nat )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_4_eq__comps_Osimps_I1_J,axiom,
( ( commut93809757773076895omplex @ nil_complex )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_5_eq__comps__empty__if,axiom,
! [L: list_nat] :
( ( ( commut2436974278740741825ps_nat @ L )
= nil_nat )
=> ( L = nil_nat ) ) ).
% eq_comps_empty_if
thf(fact_6_eq__comps__empty__if,axiom,
! [L: list_complex] :
( ( ( commut93809757773076895omplex @ L )
= nil_nat )
=> ( L = nil_complex ) ) ).
% eq_comps_empty_if
thf(fact_7_eq__comps__not__empty,axiom,
! [L: list_nat] :
( ( L != nil_nat )
=> ( ( commut2436974278740741825ps_nat @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_8_eq__comps__not__empty,axiom,
! [L: list_complex] :
( ( L != nil_complex )
=> ( ( commut93809757773076895omplex @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_9_lst__diff_Osimps_I1_J,axiom,
! [L: list_nat] :
( ( commut7647841724617136155ff_nat @ L @ nil_nat )
= ( L = nil_nat ) ) ).
% lst_diff.simps(1)
thf(fact_10_lst__diff_Osimps_I1_J,axiom,
! [L: list_complex] :
( ( commut1410864796179263225omplex @ L @ nil_nat )
= ( L = nil_complex ) ) ).
% lst_diff.simps(1)
thf(fact_11_find__indices__Nil,axiom,
! [X: nat] :
( ( missin5050847376130023830es_nat @ X @ nil_nat )
= nil_nat ) ).
% find_indices_Nil
thf(fact_12_find__indices__Nil,axiom,
! [X: complex] :
( ( missin8834916005246747252omplex @ X @ nil_complex )
= nil_nat ) ).
% find_indices_Nil
thf(fact_13_eq__comps_Ocases,axiom,
! [X: list_complex] :
( ( X != nil_complex )
=> ( ! [X2: complex] :
( X
!= ( cons_complex @ X2 @ nil_complex ) )
=> ~ ! [X2: complex,Y: complex,L2: list_complex] :
( X
!= ( cons_complex @ X2 @ ( cons_complex @ Y @ L2 ) ) ) ) ) ).
% eq_comps.cases
thf(fact_14_eq__comps_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ~ ! [X2: nat,Y: nat,L2: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ Y @ L2 ) ) ) ) ) ).
% eq_comps.cases
thf(fact_15_eq__comps_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Y: nat,L2: list_nat] :
( ( P @ ( cons_nat @ Y @ L2 ) )
=> ( P @ ( cons_nat @ X2 @ ( cons_nat @ Y @ L2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_16_eq__comps_Oinduct,axiom,
! [P: list_complex > $o,A0: list_complex] :
( ( P @ nil_complex )
=> ( ! [X2: complex] : ( P @ ( cons_complex @ X2 @ nil_complex ) )
=> ( ! [X2: complex,Y: complex,L2: list_complex] :
( ( P @ ( cons_complex @ Y @ L2 ) )
=> ( P @ ( cons_complex @ X2 @ ( cons_complex @ Y @ L2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_17_member__rec_I2_J,axiom,
! [Y2: nat] :
~ ( member_nat @ nil_nat @ Y2 ) ).
% member_rec(2)
thf(fact_18_member__rec_I2_J,axiom,
! [Y2: complex] :
~ ( member_complex @ nil_complex @ Y2 ) ).
% member_rec(2)
thf(fact_19_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_20_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_complex @ N @ nil_complex )
= N ) ).
% gen_length_code(1)
thf(fact_21_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_22_maps__simps_I2_J,axiom,
! [F: nat > list_complex] :
( ( maps_nat_complex @ F @ nil_nat )
= nil_complex ) ).
% maps_simps(2)
thf(fact_23_maps__simps_I2_J,axiom,
! [F: complex > list_nat] :
( ( maps_complex_nat @ F @ nil_complex )
= nil_nat ) ).
% maps_simps(2)
thf(fact_24_maps__simps_I2_J,axiom,
! [F: complex > list_complex] :
( ( maps_complex_complex @ F @ nil_complex )
= nil_complex ) ).
% maps_simps(2)
thf(fact_25_ubprops_I2_J,axiom,
diagonal_mat_complex @ bs ).
% ubprops(2)
thf(fact_26_ubprops_I1_J,axiom,
member_mat_complex @ bs @ ( carrier_mat_complex @ na @ na ) ).
% ubprops(1)
thf(fact_27_ubprops_I7_J,axiom,
( ( diag_mat_complex @ bs )
!= nil_complex ) ).
% ubprops(7)
thf(fact_28_ubprops_I5_J,axiom,
commut4502369927624756007omplex @ bs @ ( commut93809757773076895omplex @ ( diag_mat_complex @ bs ) ) ).
% ubprops(5)
thf(fact_29_list_Osimps_I1_J,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.simps(1)
thf(fact_30_list_Osimps_I1_J,axiom,
! [X21: complex,X22: list_complex,Y21: complex,Y22: list_complex] :
( ( ( cons_complex @ X21 @ X22 )
= ( cons_complex @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.simps(1)
thf(fact_31_member__rec_I1_J,axiom,
! [X: nat,Xs: list_nat,Y2: nat] :
( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_nat @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_32_member__rec_I1_J,axiom,
! [X: complex,Xs: list_complex,Y2: complex] :
( ( member_complex @ ( cons_complex @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_complex @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_33_List_Otranspose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X2: nat,Xs2: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_34_List_Otranspose_Ocases,axiom,
! [X: list_list_complex] :
( ( X != nil_list_complex )
=> ( ! [Xss: list_list_complex] :
( X
!= ( cons_list_complex @ nil_complex @ Xss ) )
=> ~ ! [X2: complex,Xs2: list_complex,Xss: list_list_complex] :
( X
!= ( cons_list_complex @ ( cons_complex @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_35_not__Cons__self,axiom,
! [Xs: list_nat,X: nat] :
( Xs
!= ( cons_nat @ X @ Xs ) ) ).
% not_Cons_self
thf(fact_36_not__Cons__self,axiom,
! [Xs: list_complex,X: complex] :
( Xs
!= ( cons_complex @ X @ Xs ) ) ).
% not_Cons_self
thf(fact_37_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_38_list__nonempty__induct,axiom,
! [Xs: list_complex,P: list_complex > $o] :
( ( Xs != nil_complex )
=> ( ! [X2: complex] : ( P @ ( cons_complex @ X2 @ nil_complex ) )
=> ( ! [X2: complex,Xs2: list_complex] :
( ( Xs2 != nil_complex )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_39_induct__list012,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Y: nat,Zs: list_nat] :
( ( P @ Zs )
=> ( ( P @ ( cons_nat @ Y @ Zs ) )
=> ( P @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_40_induct__list012,axiom,
! [P: list_complex > $o,Xs: list_complex] :
( ( P @ nil_complex )
=> ( ! [X2: complex] : ( P @ ( cons_complex @ X2 @ nil_complex ) )
=> ( ! [X2: complex,Y: complex,Zs: list_complex] :
( ( P @ Zs )
=> ( ( P @ ( cons_complex @ Y @ Zs ) )
=> ( P @ ( cons_complex @ X2 @ ( cons_complex @ Y @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_41_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X2 @ Xs2 ) @ nil_nat )
=> ( ! [Y: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y @ Ys2 ) )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_42_list__induct2_H,axiom,
! [P: list_nat > list_complex > $o,Xs: list_nat,Ys: list_complex] :
( ( P @ nil_nat @ nil_complex )
=> ( ! [X2: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X2 @ Xs2 ) @ nil_complex )
=> ( ! [Y: complex,Ys2: list_complex] : ( P @ nil_nat @ ( cons_complex @ Y @ Ys2 ) )
=> ( ! [X2: nat,Xs2: list_nat,Y: complex,Ys2: list_complex] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_43_list__induct2_H,axiom,
! [P: list_complex > list_nat > $o,Xs: list_complex,Ys: list_nat] :
( ( P @ nil_complex @ nil_nat )
=> ( ! [X2: complex,Xs2: list_complex] : ( P @ ( cons_complex @ X2 @ Xs2 ) @ nil_nat )
=> ( ! [Y: nat,Ys2: list_nat] : ( P @ nil_complex @ ( cons_nat @ Y @ Ys2 ) )
=> ( ! [X2: complex,Xs2: list_complex,Y: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_44_list__induct2_H,axiom,
! [P: list_complex > list_complex > $o,Xs: list_complex,Ys: list_complex] :
( ( P @ nil_complex @ nil_complex )
=> ( ! [X2: complex,Xs2: list_complex] : ( P @ ( cons_complex @ X2 @ Xs2 ) @ nil_complex )
=> ( ! [Y: complex,Ys2: list_complex] : ( P @ nil_complex @ ( cons_complex @ Y @ Ys2 ) )
=> ( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_45_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y3: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_46_neq__Nil__conv,axiom,
! [Xs: list_complex] :
( ( Xs != nil_complex )
= ( ? [Y3: complex,Ys3: list_complex] :
( Xs
= ( cons_complex @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_47_map__tailrec__rev_Oinduct,axiom,
! [P: ( nat > nat ) > list_nat > list_nat > $o,A0: nat > nat,A1: list_nat,A2: list_nat] :
( ! [F2: nat > nat,X_1: list_nat] : ( P @ F2 @ nil_nat @ X_1 )
=> ( ! [F2: nat > nat,A: nat,As: list_nat,Bs: list_nat] :
( ( P @ F2 @ As @ ( cons_nat @ ( F2 @ A ) @ Bs ) )
=> ( P @ F2 @ ( cons_nat @ A @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_48_map__tailrec__rev_Oinduct,axiom,
! [P: ( complex > nat ) > list_complex > list_nat > $o,A0: complex > nat,A1: list_complex,A2: list_nat] :
( ! [F2: complex > nat,X_1: list_nat] : ( P @ F2 @ nil_complex @ X_1 )
=> ( ! [F2: complex > nat,A: complex,As: list_complex,Bs: list_nat] :
( ( P @ F2 @ As @ ( cons_nat @ ( F2 @ A ) @ Bs ) )
=> ( P @ F2 @ ( cons_complex @ A @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_49_map__tailrec__rev_Oinduct,axiom,
! [P: ( nat > complex ) > list_nat > list_complex > $o,A0: nat > complex,A1: list_nat,A2: list_complex] :
( ! [F2: nat > complex,X_1: list_complex] : ( P @ F2 @ nil_nat @ X_1 )
=> ( ! [F2: nat > complex,A: nat,As: list_nat,Bs: list_complex] :
( ( P @ F2 @ As @ ( cons_complex @ ( F2 @ A ) @ Bs ) )
=> ( P @ F2 @ ( cons_nat @ A @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_50_map__tailrec__rev_Oinduct,axiom,
! [P: ( complex > complex ) > list_complex > list_complex > $o,A0: complex > complex,A1: list_complex,A2: list_complex] :
( ! [F2: complex > complex,X_1: list_complex] : ( P @ F2 @ nil_complex @ X_1 )
=> ( ! [F2: complex > complex,A: complex,As: list_complex,Bs: list_complex] :
( ( P @ F2 @ As @ ( cons_complex @ ( F2 @ A ) @ Bs ) )
=> ( P @ F2 @ ( cons_complex @ A @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_51_successively_Oinduct,axiom,
! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
( ! [P2: nat > nat > $o] : ( P @ P2 @ nil_nat )
=> ( ! [P2: nat > nat > $o,X2: nat] : ( P @ P2 @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [P2: nat > nat > $o,X2: nat,Y: nat,Xs2: list_nat] :
( ( P @ P2 @ ( cons_nat @ Y @ Xs2 ) )
=> ( P @ P2 @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_52_successively_Oinduct,axiom,
! [P: ( complex > complex > $o ) > list_complex > $o,A0: complex > complex > $o,A1: list_complex] :
( ! [P2: complex > complex > $o] : ( P @ P2 @ nil_complex )
=> ( ! [P2: complex > complex > $o,X2: complex] : ( P @ P2 @ ( cons_complex @ X2 @ nil_complex ) )
=> ( ! [P2: complex > complex > $o,X2: complex,Y: complex,Xs2: list_complex] :
( ( P @ P2 @ ( cons_complex @ Y @ Xs2 ) )
=> ( P @ P2 @ ( cons_complex @ X2 @ ( cons_complex @ Y @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_53_remdups__adj_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Y: nat,Xs2: list_nat] :
( ( ( X2 = Y )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) ) )
=> ( ( ( X2 != Y )
=> ( P @ ( cons_nat @ Y @ Xs2 ) ) )
=> ( P @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_54_remdups__adj_Oinduct,axiom,
! [P: list_complex > $o,A0: list_complex] :
( ( P @ nil_complex )
=> ( ! [X2: complex] : ( P @ ( cons_complex @ X2 @ nil_complex ) )
=> ( ! [X2: complex,Y: complex,Xs2: list_complex] :
( ( ( X2 = Y )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) ) )
=> ( ( ( X2 != Y )
=> ( P @ ( cons_complex @ Y @ Xs2 ) ) )
=> ( P @ ( cons_complex @ X2 @ ( cons_complex @ Y @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_55_sorted__wrt_Oinduct,axiom,
! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
( ! [P2: nat > nat > $o] : ( P @ P2 @ nil_nat )
=> ( ! [P2: nat > nat > $o,X2: nat,Ys2: list_nat] :
( ( P @ P2 @ Ys2 )
=> ( P @ P2 @ ( cons_nat @ X2 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_56_sorted__wrt_Oinduct,axiom,
! [P: ( complex > complex > $o ) > list_complex > $o,A0: complex > complex > $o,A1: list_complex] :
( ! [P2: complex > complex > $o] : ( P @ P2 @ nil_complex )
=> ( ! [P2: complex > complex > $o,X2: complex,Ys2: list_complex] :
( ( P @ P2 @ Ys2 )
=> ( P @ P2 @ ( cons_complex @ X2 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_57_shuffles_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [Xs2: list_nat] : ( P @ Xs2 @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( P @ Xs2 @ ( cons_nat @ Y @ Ys2 ) )
=> ( ( P @ ( cons_nat @ X2 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_58_shuffles_Oinduct,axiom,
! [P: list_complex > list_complex > $o,A0: list_complex,A1: list_complex] :
( ! [X_1: list_complex] : ( P @ nil_complex @ X_1 )
=> ( ! [Xs2: list_complex] : ( P @ Xs2 @ nil_complex )
=> ( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex] :
( ( P @ Xs2 @ ( cons_complex @ Y @ Ys2 ) )
=> ( ( P @ ( cons_complex @ X2 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_59_min__list_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ! [X2: nat,Xs2: list_nat] :
( ! [X212: nat,X222: list_nat] :
( ( Xs2
= ( cons_nat @ X212 @ X222 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) ) )
=> ( ( P @ nil_nat )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_60_min__list_Oinduct,axiom,
! [P: list_complex > $o,A0: list_complex] :
( ! [X2: complex,Xs2: list_complex] :
( ! [X212: complex,X222: list_complex] :
( ( Xs2
= ( cons_complex @ X212 @ X222 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) ) )
=> ( ( P @ nil_complex )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_61_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs2 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_62_min__list_Ocases,axiom,
! [X: list_complex] :
( ! [X2: complex,Xs2: list_complex] :
( X
!= ( cons_complex @ X2 @ Xs2 ) )
=> ( X = nil_complex ) ) ).
% min_list.cases
thf(fact_63_splice_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [X2: nat,Xs2: list_nat,Ys2: list_nat] :
( ( P @ Ys2 @ Xs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_64_splice_Oinduct,axiom,
! [P: list_complex > list_complex > $o,A0: list_complex,A1: list_complex] :
( ! [X_1: list_complex] : ( P @ nil_complex @ X_1 )
=> ( ! [X2: complex,Xs2: list_complex,Ys2: list_complex] :
( ( P @ Ys2 @ Xs2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_65_list_Oinducts,axiom,
! [P: list_nat > $o,List: list_nat] :
( ( P @ nil_nat )
=> ( ! [X1: nat,X23: list_nat] :
( ( P @ X23 )
=> ( P @ ( cons_nat @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_66_list_Oinducts,axiom,
! [P: list_complex > $o,List: list_complex] :
( ( P @ nil_complex )
=> ( ! [X1: complex,X23: list_complex] :
( ( P @ X23 )
=> ( P @ ( cons_complex @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_67_list_Oexhaust,axiom,
! [Y2: list_nat] :
( ( Y2 != nil_nat )
=> ~ ! [X213: nat,X223: list_nat] :
( Y2
!= ( cons_nat @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_68_list_Oexhaust,axiom,
! [Y2: list_complex] :
( ( Y2 != nil_complex )
=> ~ ! [X213: complex,X223: list_complex] :
( Y2
!= ( cons_complex @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_69_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_70_list_OdiscI,axiom,
! [List: list_complex,X21: complex,X22: list_complex] :
( ( List
= ( cons_complex @ X21 @ X22 ) )
=> ( List != nil_complex ) ) ).
% list.discI
thf(fact_71_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_72_list_Odistinct_I1_J,axiom,
! [X21: complex,X22: list_complex] :
( nil_complex
!= ( cons_complex @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_73_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_74_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat2 @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A4: set_mat_complex] :
( ( collect_mat_complex
@ ^ [X3: mat_complex] : ( member_mat_complex @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_77_lst__diff__imp__diag__diff,axiom,
! [D: mat_nat,N: nat,M: list_nat] :
( ( member_mat_nat @ D @ ( carrier_mat_nat @ N @ N ) )
=> ( ( commut7647841724617136155ff_nat @ ( diag_mat_nat @ D ) @ M )
=> ( commut2433368883435016265ff_nat @ D @ M ) ) ) ).
% lst_diff_imp_diag_diff
thf(fact_78_lst__diff__imp__diag__diff,axiom,
! [D: mat_complex,N: nat,M: list_nat] :
( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( commut1410864796179263225omplex @ ( diag_mat_complex @ D ) @ M )
=> ( commut4502369927624756007omplex @ D @ M ) ) ) ).
% lst_diff_imp_diag_diff
thf(fact_79_max__list__non__empty_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,V: nat,Va: list_nat] :
( ( P @ ( cons_nat @ V @ Va ) )
=> ( P @ ( cons_nat @ X2 @ ( cons_nat @ V @ Va ) ) ) )
=> ( ( P @ nil_nat )
=> ( P @ A0 ) ) ) ) ).
% max_list_non_empty.induct
thf(fact_80_max__list__non__empty_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,V: nat,Va: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ V @ Va ) ) )
=> ( X = nil_nat ) ) ) ).
% max_list_non_empty.cases
thf(fact_81_minus__poly__rev__list_Oinduct,axiom,
! [P: list_complex > list_complex > $o,A0: list_complex,A1: list_complex] :
( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) ) )
=> ( ! [Xs2: list_complex] : ( P @ Xs2 @ nil_complex )
=> ( ! [Y: complex,Ys2: list_complex] : ( P @ nil_complex @ ( cons_complex @ Y @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% minus_poly_rev_list.induct
thf(fact_82_longest__common__prefix_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( ( X2 = Y )
=> ( P @ Xs2 @ Ys2 ) )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [Uu: list_nat] : ( P @ Uu @ nil_nat )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_83_longest__common__prefix_Oinduct,axiom,
! [P: list_complex > list_complex > $o,A0: list_complex,A1: list_complex] :
( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex] :
( ( ( X2 = Y )
=> ( P @ Xs2 @ Ys2 ) )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) ) )
=> ( ! [X_1: list_complex] : ( P @ nil_complex @ X_1 )
=> ( ! [Uu: list_complex] : ( P @ Uu @ nil_complex )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_84_plus__coeffs_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [Xs2: list_nat] : ( P @ Xs2 @ nil_nat )
=> ( ! [V: nat,Va: list_nat] : ( P @ nil_nat @ ( cons_nat @ V @ Va ) )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% plus_coeffs.induct
thf(fact_85_plus__coeffs_Oinduct,axiom,
! [P: list_complex > list_complex > $o,A0: list_complex,A1: list_complex] :
( ! [Xs2: list_complex] : ( P @ Xs2 @ nil_complex )
=> ( ! [V: complex,Va: list_complex] : ( P @ nil_complex @ ( cons_complex @ V @ Va ) )
=> ( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% plus_coeffs.induct
thf(fact_86_eq__comps_Osimps_I2_J,axiom,
! [X: complex] :
( ( commut93809757773076895omplex @ ( cons_complex @ X @ nil_complex ) )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% eq_comps.simps(2)
thf(fact_87_eq__comps_Osimps_I2_J,axiom,
! [X: nat] :
( ( commut2436974278740741825ps_nat @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% eq_comps.simps(2)
thf(fact_88_map__tailrec__rev_Oelims,axiom,
! [X: nat > nat,Xa: list_nat,Xb: list_nat,Y2: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_nat )
=> ( Y2 != Xb ) )
=> ~ ! [A: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A @ As ) )
=> ( Y2
!= ( map_ta7164188454487880599at_nat @ X @ As @ ( cons_nat @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_89_map__tailrec__rev_Oelims,axiom,
! [X: nat > complex,Xa: list_nat,Xb: list_complex,Y2: list_complex] :
( ( ( map_ta732166513454076533omplex @ X @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_nat )
=> ( Y2 != Xb ) )
=> ~ ! [A: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A @ As ) )
=> ( Y2
!= ( map_ta732166513454076533omplex @ X @ As @ ( cons_complex @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_90_map__tailrec__rev_Oelims,axiom,
! [X: complex > nat,Xa: list_complex,Xb: list_nat,Y2: list_nat] :
( ( ( map_ta4351949837888814197ex_nat @ X @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_complex )
=> ( Y2 != Xb ) )
=> ~ ! [A: complex,As: list_complex] :
( ( Xa
= ( cons_complex @ A @ As ) )
=> ( Y2
!= ( map_ta4351949837888814197ex_nat @ X @ As @ ( cons_nat @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_91_map__tailrec__rev_Oelims,axiom,
! [X: complex > complex,Xa: list_complex,Xb: list_complex,Y2: list_complex] :
( ( ( map_ta3224203966507818835omplex @ X @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_complex )
=> ( Y2 != Xb ) )
=> ~ ! [A: complex,As: list_complex] :
( ( Xa
= ( cons_complex @ A @ As ) )
=> ( Y2
!= ( map_ta3224203966507818835omplex @ X @ As @ ( cons_complex @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_92_list__encode_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) ) )
=> ( P @ A0 ) ) ) ).
% list_encode.induct
thf(fact_93_list__encode_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X2: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs2 ) ) ) ).
% list_encode.cases
thf(fact_94_ubprops_I4_J,axiom,
member_mat_complex @ us @ ( carrier_mat_complex @ na @ na ) ).
% ubprops(4)
thf(fact_95_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > nat,A3: nat,As2: list_nat,Bs2: list_nat] :
( ( map_ta7164188454487880599at_nat @ F @ ( cons_nat @ A3 @ As2 ) @ Bs2 )
= ( map_ta7164188454487880599at_nat @ F @ As2 @ ( cons_nat @ ( F @ A3 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_96_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > complex,A3: nat,As2: list_nat,Bs2: list_complex] :
( ( map_ta732166513454076533omplex @ F @ ( cons_nat @ A3 @ As2 ) @ Bs2 )
= ( map_ta732166513454076533omplex @ F @ As2 @ ( cons_complex @ ( F @ A3 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_97_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: complex > nat,A3: complex,As2: list_complex,Bs2: list_nat] :
( ( map_ta4351949837888814197ex_nat @ F @ ( cons_complex @ A3 @ As2 ) @ Bs2 )
= ( map_ta4351949837888814197ex_nat @ F @ As2 @ ( cons_nat @ ( F @ A3 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_98_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: complex > complex,A3: complex,As2: list_complex,Bs2: list_complex] :
( ( map_ta3224203966507818835omplex @ F @ ( cons_complex @ A3 @ As2 ) @ Bs2 )
= ( map_ta3224203966507818835omplex @ F @ As2 @ ( cons_complex @ ( F @ A3 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_99__092_060open_062Ap_A_092_060in_062_Acarrier__mat_An_An_092_060close_062,axiom,
member_mat_complex @ ap @ ( carrier_mat_complex @ na @ na ) ).
% \<open>Ap \<in> carrier_mat n n\<close>
thf(fact_100_Suc_Oprems_I4_J,axiom,
! [A4: mat_complex] :
( ( member_mat_complex @ A4 @ afa )
=> ( member_mat_complex @ A4 @ ( carrier_mat_complex @ na @ na ) ) ) ).
% Suc.prems(4)
thf(fact_101_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_102_sublists_Osimps_I1_J,axiom,
( ( sublists_complex @ nil_complex )
= ( cons_list_complex @ nil_complex @ nil_list_complex ) ) ).
% sublists.simps(1)
thf(fact_103_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_104_product__lists_Osimps_I1_J,axiom,
( ( produc7545014605101902079omplex @ nil_list_complex )
= ( cons_list_complex @ nil_complex @ nil_list_complex ) ) ).
% product_lists.simps(1)
thf(fact_105_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_106_subseqs_Osimps_I1_J,axiom,
( ( subseqs_complex @ nil_complex )
= ( cons_list_complex @ nil_complex @ nil_list_complex ) ) ).
% subseqs.simps(1)
thf(fact_107_max__list__non__empty_Osimps_I1_J,axiom,
! [X: nat] :
( ( missin53001312869816611ty_nat @ ( cons_nat @ X @ nil_nat ) )
= X ) ).
% max_list_non_empty.simps(1)
thf(fact_108_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_109_suffixes_Osimps_I1_J,axiom,
( ( suffixes_complex @ nil_complex )
= ( cons_list_complex @ nil_complex @ nil_list_complex ) ) ).
% suffixes.simps(1)
thf(fact_110_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_111_prefixes_Osimps_I1_J,axiom,
( ( prefixes_complex @ nil_complex )
= ( cons_list_complex @ nil_complex @ nil_list_complex ) ) ).
% prefixes.simps(1)
thf(fact_112_plus__coeffs_Osimps_I2_J,axiom,
! [V2: nat,Va2: list_nat] :
( ( plus_coeffs_nat @ nil_nat @ ( cons_nat @ V2 @ Va2 ) )
= ( cons_nat @ V2 @ Va2 ) ) ).
% plus_coeffs.simps(2)
thf(fact_113_plus__coeffs_Osimps_I2_J,axiom,
! [V2: complex,Va2: list_complex] :
( ( plus_coeffs_complex @ nil_complex @ ( cons_complex @ V2 @ Va2 ) )
= ( cons_complex @ V2 @ Va2 ) ) ).
% plus_coeffs.simps(2)
thf(fact_114_ubprops_I3_J,axiom,
comple6660659447773130958omplex @ us ).
% ubprops(3)
thf(fact_115__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Ap_O_AAp_A_092_060in_062_AAf_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Ap: mat_complex] :
~ ( member_mat_complex @ Ap @ afa ) ).
% \<open>\<And>thesis. (\<And>Ap. Ap \<in> Af \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_116__092_060open_062_092_060exists_062A_O_AA_A_092_060in_062_AAf_092_060close_062,axiom,
? [A5: mat_complex] : ( member_mat_complex @ A5 @ afa ) ).
% \<open>\<exists>A. A \<in> Af\<close>
thf(fact_117__092_060open_062Ap_A_092_060in_062_AAf_092_060close_062,axiom,
member_mat_complex @ ap @ afa ).
% \<open>Ap \<in> Af\<close>
thf(fact_118_Suc_I4_J,axiom,
afa != bot_bo7165004461764951667omplex ).
% Suc(4)
thf(fact_119_Suc_I2_J,axiom,
finite7047982916621727056omplex @ afa ).
% Suc(2)
thf(fact_120_plus__coeffs_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( plus_coeffs_nat @ Xs @ nil_nat )
= Xs ) ).
% plus_coeffs.simps(1)
thf(fact_121_plus__coeffs_Osimps_I1_J,axiom,
! [Xs: list_complex] :
( ( plus_coeffs_complex @ Xs @ nil_complex )
= Xs ) ).
% plus_coeffs.simps(1)
thf(fact_122_Suc_I8_J,axiom,
! [A4: mat_complex,B: mat_complex] :
( ( member_mat_complex @ A4 @ afa )
=> ( ( member_mat_complex @ B @ afa )
=> ( ( times_8009071140041733218omplex @ A4 @ B )
= ( times_8009071140041733218omplex @ B @ A4 ) ) ) ) ).
% Suc(8)
thf(fact_123_suffixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( suffixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs2 ) )
& ( Xs
= ( suffixes_nat @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_124_suffixes__eq__snoc,axiom,
! [Ys: list_complex,Xs: list_list_complex,X: list_complex] :
( ( ( suffixes_complex @ Ys )
= ( append_list_complex @ Xs @ ( cons_list_complex @ X @ nil_list_complex ) ) )
= ( ( ( ( Ys = nil_complex )
& ( Xs = nil_list_complex ) )
| ? [Z: complex,Zs2: list_complex] :
( ( Ys
= ( cons_complex @ Z @ Zs2 ) )
& ( Xs
= ( suffixes_complex @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_125__092_060open_062hermitian_AAp_092_060close_062,axiom,
comple8306762464034002205omplex @ ap ).
% \<open>hermitian Ap\<close>
thf(fact_126_Suc_I7_J,axiom,
! [A4: mat_complex] :
( ( member_mat_complex @ A4 @ afa )
=> ( comple8306762464034002205omplex @ A4 ) ) ).
% Suc(7)
thf(fact_127_suffixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_128_suffixes_Osimps_I2_J,axiom,
! [X: complex,Xs: list_complex] :
( ( suffixes_complex @ ( cons_complex @ X @ Xs ) )
= ( append_list_complex @ ( suffixes_complex @ Xs ) @ ( cons_list_complex @ ( cons_complex @ X @ Xs ) @ nil_list_complex ) ) ) ).
% suffixes.simps(2)
thf(fact_129_prefixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( prefixes_nat @ ( cons_nat @ X @ Xs ) )
= ( cons_list_nat @ nil_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_130_prefixes_Osimps_I2_J,axiom,
! [X: complex,Xs: list_complex] :
( ( prefixes_complex @ ( cons_complex @ X @ Xs ) )
= ( cons_list_complex @ nil_complex @ ( map_li2870275437539113154omplex @ ( cons_complex @ X ) @ ( prefixes_complex @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_131_longest__common__prefix_Osimps_I1_J,axiom,
! [X: nat,Y2: nat,Xs: list_nat,Ys: list_nat] :
( ( ( X = Y2 )
=> ( ( longes266370323089874118ix_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= ( cons_nat @ X @ ( longes266370323089874118ix_nat @ Xs @ Ys ) ) ) )
& ( ( X != Y2 )
=> ( ( longes266370323089874118ix_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= nil_nat ) ) ) ).
% longest_common_prefix.simps(1)
thf(fact_132_longest__common__prefix_Osimps_I1_J,axiom,
! [X: complex,Y2: complex,Xs: list_complex,Ys: list_complex] :
( ( ( X = Y2 )
=> ( ( longes3069803181527244196omplex @ ( cons_complex @ X @ Xs ) @ ( cons_complex @ Y2 @ Ys ) )
= ( cons_complex @ X @ ( longes3069803181527244196omplex @ Xs @ Ys ) ) ) )
& ( ( X != Y2 )
=> ( ( longes3069803181527244196omplex @ ( cons_complex @ X @ Xs ) @ ( cons_complex @ Y2 @ Ys ) )
= nil_complex ) ) ) ).
% longest_common_prefix.simps(1)
thf(fact_133_longest__common__prefix_Oelims,axiom,
! [X: list_nat,Xa: list_nat,Y2: list_nat] :
( ( ( longes266370323089874118ix_nat @ X @ Xa )
= Y2 )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ! [Y: nat,Ys2: list_nat] :
( ( Xa
= ( cons_nat @ Y @ Ys2 ) )
=> ~ ( ( ( X2 = Y )
=> ( Y2
= ( cons_nat @ X2 @ ( longes266370323089874118ix_nat @ Xs2 @ Ys2 ) ) ) )
& ( ( X2 != Y )
=> ( Y2 = nil_nat ) ) ) ) )
=> ( ( ( X = nil_nat )
=> ( Y2 != nil_nat ) )
=> ~ ( ( Xa = nil_nat )
=> ( Y2 != nil_nat ) ) ) ) ) ).
% longest_common_prefix.elims
thf(fact_134_longest__common__prefix_Oelims,axiom,
! [X: list_complex,Xa: list_complex,Y2: list_complex] :
( ( ( longes3069803181527244196omplex @ X @ Xa )
= Y2 )
=> ( ! [X2: complex,Xs2: list_complex] :
( ( X
= ( cons_complex @ X2 @ Xs2 ) )
=> ! [Y: complex,Ys2: list_complex] :
( ( Xa
= ( cons_complex @ Y @ Ys2 ) )
=> ~ ( ( ( X2 = Y )
=> ( Y2
= ( cons_complex @ X2 @ ( longes3069803181527244196omplex @ Xs2 @ Ys2 ) ) ) )
& ( ( X2 != Y )
=> ( Y2 = nil_complex ) ) ) ) )
=> ( ( ( X = nil_complex )
=> ( Y2 != nil_complex ) )
=> ~ ( ( Xa = nil_complex )
=> ( Y2 != nil_complex ) ) ) ) ) ).
% longest_common_prefix.elims
thf(fact_135_assms_I1_J,axiom,
finite7047982916621727056omplex @ af ).
% assms(1)
thf(fact_136_assms_I2_J,axiom,
af != bot_bo7165004461764951667omplex ).
% assms(2)
thf(fact_137_map__append,axiom,
! [F: complex > real,Xs: list_complex,Ys: list_complex] :
( ( map_complex_real @ F @ ( append_complex @ Xs @ Ys ) )
= ( append_real @ ( map_complex_real @ F @ Xs ) @ ( map_complex_real @ F @ Ys ) ) ) ).
% map_append
thf(fact_138_append__eq__map__conv,axiom,
! [Ys: list_real,Zs3: list_real,F: complex > real,Xs: list_complex] :
( ( ( append_real @ Ys @ Zs3 )
= ( map_complex_real @ F @ Xs ) )
= ( ? [Us: list_complex,Vs: list_complex] :
( ( Xs
= ( append_complex @ Us @ Vs ) )
& ( Ys
= ( map_complex_real @ F @ Us ) )
& ( Zs3
= ( map_complex_real @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_139_map__eq__append__conv,axiom,
! [F: complex > real,Xs: list_complex,Ys: list_real,Zs3: list_real] :
( ( ( map_complex_real @ F @ Xs )
= ( append_real @ Ys @ Zs3 ) )
= ( ? [Us: list_complex,Vs: list_complex] :
( ( Xs
= ( append_complex @ Us @ Vs ) )
& ( Ys
= ( map_complex_real @ F @ Us ) )
& ( Zs3
= ( map_complex_real @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_140_infinite__imp__elem,axiom,
! [A4: set_mat_complex] :
( ~ ( finite7047982916621727056omplex @ A4 )
=> ? [X2: mat_complex] : ( member_mat_complex @ X2 @ A4 ) ) ).
% infinite_imp_elem
thf(fact_141_infinite__imp__elem,axiom,
! [A4: set_nat] :
( ~ ( finite_finite_nat @ A4 )
=> ? [X2: nat] : ( member_nat2 @ X2 @ A4 ) ) ).
% infinite_imp_elem
thf(fact_142_map__eq__Cons__conv,axiom,
! [F: complex > real,Xs: list_complex,Y2: real,Ys: list_real] :
( ( ( map_complex_real @ F @ Xs )
= ( cons_real @ Y2 @ Ys ) )
= ( ? [Z: complex,Zs2: list_complex] :
( ( Xs
= ( cons_complex @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y2 )
& ( ( map_complex_real @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_143_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( ? [Z: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y2 )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_144_map__eq__Cons__conv,axiom,
! [F: complex > nat,Xs: list_complex,Y2: nat,Ys: list_nat] :
( ( ( map_complex_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( ? [Z: complex,Zs2: list_complex] :
( ( Xs
= ( cons_complex @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y2 )
& ( ( map_complex_nat @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_145_map__eq__Cons__conv,axiom,
! [F: nat > complex,Xs: list_nat,Y2: complex,Ys: list_complex] :
( ( ( map_nat_complex @ F @ Xs )
= ( cons_complex @ Y2 @ Ys ) )
= ( ? [Z: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y2 )
& ( ( map_nat_complex @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_146_map__eq__Cons__conv,axiom,
! [F: complex > complex,Xs: list_complex,Y2: complex,Ys: list_complex] :
( ( ( map_complex_complex @ F @ Xs )
= ( cons_complex @ Y2 @ Ys ) )
= ( ? [Z: complex,Zs2: list_complex] :
( ( Xs
= ( cons_complex @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y2 )
& ( ( map_complex_complex @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_147_Cons__eq__map__conv,axiom,
! [X: real,Xs: list_real,F: complex > real,Ys: list_complex] :
( ( ( cons_real @ X @ Xs )
= ( map_complex_real @ F @ Ys ) )
= ( ? [Z: complex,Zs2: list_complex] :
( ( Ys
= ( cons_complex @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_complex_real @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_148_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_149_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: complex > nat,Ys: list_complex] :
( ( ( cons_nat @ X @ Xs )
= ( map_complex_nat @ F @ Ys ) )
= ( ? [Z: complex,Zs2: list_complex] :
( ( Ys
= ( cons_complex @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_complex_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_150_Cons__eq__map__conv,axiom,
! [X: complex,Xs: list_complex,F: nat > complex,Ys: list_nat] :
( ( ( cons_complex @ X @ Xs )
= ( map_nat_complex @ F @ Ys ) )
= ( ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_nat_complex @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_151_Cons__eq__map__conv,axiom,
! [X: complex,Xs: list_complex,F: complex > complex,Ys: list_complex] :
( ( ( cons_complex @ X @ Xs )
= ( map_complex_complex @ F @ Ys ) )
= ( ? [Z: complex,Zs2: list_complex] :
( ( Ys
= ( cons_complex @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_complex_complex @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_152_map__eq__Cons__D,axiom,
! [F: complex > real,Xs: list_complex,Y2: real,Ys: list_real] :
( ( ( map_complex_real @ F @ Xs )
= ( cons_real @ Y2 @ Ys ) )
=> ? [Z2: complex,Zs: list_complex] :
( ( Xs
= ( cons_complex @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_complex_real @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_153_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z2: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_nat_nat @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_154_map__eq__Cons__D,axiom,
! [F: complex > nat,Xs: list_complex,Y2: nat,Ys: list_nat] :
( ( ( map_complex_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z2: complex,Zs: list_complex] :
( ( Xs
= ( cons_complex @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_complex_nat @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_155_map__eq__Cons__D,axiom,
! [F: nat > complex,Xs: list_nat,Y2: complex,Ys: list_complex] :
( ( ( map_nat_complex @ F @ Xs )
= ( cons_complex @ Y2 @ Ys ) )
=> ? [Z2: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_nat_complex @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_156_map__eq__Cons__D,axiom,
! [F: complex > complex,Xs: list_complex,Y2: complex,Ys: list_complex] :
( ( ( map_complex_complex @ F @ Xs )
= ( cons_complex @ Y2 @ Ys ) )
=> ? [Z2: complex,Zs: list_complex] :
( ( Xs
= ( cons_complex @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_complex_complex @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_157_Cons__eq__map__D,axiom,
! [X: real,Xs: list_real,F: complex > real,Ys: list_complex] :
( ( ( cons_real @ X @ Xs )
= ( map_complex_real @ F @ Ys ) )
=> ? [Z2: complex,Zs: list_complex] :
( ( Ys
= ( cons_complex @ Z2 @ Zs ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_complex_real @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_158_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z2: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_159_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: complex > nat,Ys: list_complex] :
( ( ( cons_nat @ X @ Xs )
= ( map_complex_nat @ F @ Ys ) )
=> ? [Z2: complex,Zs: list_complex] :
( ( Ys
= ( cons_complex @ Z2 @ Zs ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_complex_nat @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_160_Cons__eq__map__D,axiom,
! [X: complex,Xs: list_complex,F: nat > complex,Ys: list_nat] :
( ( ( cons_complex @ X @ Xs )
= ( map_nat_complex @ F @ Ys ) )
=> ? [Z2: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_complex @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_161_Cons__eq__map__D,axiom,
! [X: complex,Xs: list_complex,F: complex > complex,Ys: list_complex] :
( ( ( cons_complex @ X @ Xs )
= ( map_complex_complex @ F @ Ys ) )
=> ? [Z2: complex,Zs: list_complex] :
( ( Ys
= ( cons_complex @ Z2 @ Zs ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_complex_complex @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_162_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_163_list_Osimps_I9_J,axiom,
! [F: nat > complex,X21: nat,X22: list_nat] :
( ( map_nat_complex @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_complex @ ( F @ X21 ) @ ( map_nat_complex @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_164_list_Osimps_I9_J,axiom,
! [F: complex > real,X21: complex,X22: list_complex] :
( ( map_complex_real @ F @ ( cons_complex @ X21 @ X22 ) )
= ( cons_real @ ( F @ X21 ) @ ( map_complex_real @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_165_list_Osimps_I9_J,axiom,
! [F: complex > nat,X21: complex,X22: list_complex] :
( ( map_complex_nat @ F @ ( cons_complex @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_complex_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_166_list_Osimps_I9_J,axiom,
! [F: complex > complex,X21: complex,X22: list_complex] :
( ( map_complex_complex @ F @ ( cons_complex @ X21 @ X22 ) )
= ( cons_complex @ ( F @ X21 ) @ ( map_complex_complex @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_167_list_Omap_I1_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.map(1)
thf(fact_168_list_Omap_I1_J,axiom,
! [F: nat > complex] :
( ( map_nat_complex @ F @ nil_nat )
= nil_complex ) ).
% list.map(1)
thf(fact_169_list_Omap_I1_J,axiom,
! [F: complex > nat] :
( ( map_complex_nat @ F @ nil_complex )
= nil_nat ) ).
% list.map(1)
thf(fact_170_list_Omap_I1_J,axiom,
! [F: complex > complex] :
( ( map_complex_complex @ F @ nil_complex )
= nil_complex ) ).
% list.map(1)
thf(fact_171_list_Omap_I1_J,axiom,
! [F: complex > real] :
( ( map_complex_real @ F @ nil_complex )
= nil_real ) ).
% list.map(1)
thf(fact_172_list_Omap__disc__iff,axiom,
! [F: nat > nat,A3: list_nat] :
( ( ( map_nat_nat @ F @ A3 )
= nil_nat )
= ( A3 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_173_list_Omap__disc__iff,axiom,
! [F: complex > nat,A3: list_complex] :
( ( ( map_complex_nat @ F @ A3 )
= nil_nat )
= ( A3 = nil_complex ) ) ).
% list.map_disc_iff
thf(fact_174_list_Omap__disc__iff,axiom,
! [F: nat > complex,A3: list_nat] :
( ( ( map_nat_complex @ F @ A3 )
= nil_complex )
= ( A3 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_175_list_Omap__disc__iff,axiom,
! [F: complex > complex,A3: list_complex] :
( ( ( map_complex_complex @ F @ A3 )
= nil_complex )
= ( A3 = nil_complex ) ) ).
% list.map_disc_iff
thf(fact_176_list_Omap__disc__iff,axiom,
! [F: complex > real,A3: list_complex] :
( ( ( map_complex_real @ F @ A3 )
= nil_real )
= ( A3 = nil_complex ) ) ).
% list.map_disc_iff
thf(fact_177_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_178_Nil__is__map__conv,axiom,
! [F: complex > nat,Xs: list_complex] :
( ( nil_nat
= ( map_complex_nat @ F @ Xs ) )
= ( Xs = nil_complex ) ) ).
% Nil_is_map_conv
thf(fact_179_Nil__is__map__conv,axiom,
! [F: nat > complex,Xs: list_nat] :
( ( nil_complex
= ( map_nat_complex @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_180_Nil__is__map__conv,axiom,
! [F: complex > complex,Xs: list_complex] :
( ( nil_complex
= ( map_complex_complex @ F @ Xs ) )
= ( Xs = nil_complex ) ) ).
% Nil_is_map_conv
thf(fact_181_Nil__is__map__conv,axiom,
! [F: complex > real,Xs: list_complex] :
( ( nil_real
= ( map_complex_real @ F @ Xs ) )
= ( Xs = nil_complex ) ) ).
% Nil_is_map_conv
thf(fact_182_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_183_map__is__Nil__conv,axiom,
! [F: complex > nat,Xs: list_complex] :
( ( ( map_complex_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_complex ) ) ).
% map_is_Nil_conv
thf(fact_184_map__is__Nil__conv,axiom,
! [F: nat > complex,Xs: list_nat] :
( ( ( map_nat_complex @ F @ Xs )
= nil_complex )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_185_map__is__Nil__conv,axiom,
! [F: complex > complex,Xs: list_complex] :
( ( ( map_complex_complex @ F @ Xs )
= nil_complex )
= ( Xs = nil_complex ) ) ).
% map_is_Nil_conv
thf(fact_186_map__is__Nil__conv,axiom,
! [F: complex > real,Xs: list_complex] :
( ( ( map_complex_real @ F @ Xs )
= nil_real )
= ( Xs = nil_complex ) ) ).
% map_is_Nil_conv
thf(fact_187_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs3: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs3 ) )
=> ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs3 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_188_Cons__eq__appendI,axiom,
! [X: complex,Xs1: list_complex,Ys: list_complex,Xs: list_complex,Zs3: list_complex] :
( ( ( cons_complex @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_complex @ Xs1 @ Zs3 ) )
=> ( ( cons_complex @ X @ Xs )
= ( append_complex @ Ys @ Zs3 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_189_append__Cons,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
= ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_190_append__Cons,axiom,
! [X: complex,Xs: list_complex,Ys: list_complex] :
( ( append_complex @ ( cons_complex @ X @ Xs ) @ Ys )
= ( cons_complex @ X @ ( append_complex @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_191_append_Osimps_I1_J,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append.simps(1)
thf(fact_192_append_Osimps_I1_J,axiom,
! [Ys: list_complex] :
( ( append_complex @ nil_complex @ Ys )
= Ys ) ).
% append.simps(1)
thf(fact_193_append_Oleft__neutral,axiom,
! [A3: list_nat] :
( ( append_nat @ nil_nat @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_194_append_Oleft__neutral,axiom,
! [A3: list_complex] :
( ( append_complex @ nil_complex @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_195_append_Oright__neutral,axiom,
! [A3: list_nat] :
( ( append_nat @ A3 @ nil_nat )
= A3 ) ).
% append.right_neutral
thf(fact_196_append_Oright__neutral,axiom,
! [A3: list_complex] :
( ( append_complex @ A3 @ nil_complex )
= A3 ) ).
% append.right_neutral
thf(fact_197_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_198_append__Nil2,axiom,
! [Xs: list_complex] :
( ( append_complex @ Xs @ nil_complex )
= Xs ) ).
% append_Nil2
thf(fact_199_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_200_eq__Nil__appendI,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( Xs = Ys )
=> ( Xs
= ( append_complex @ nil_complex @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_201_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_202_append__self__conv,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( ( append_complex @ Xs @ Ys )
= Xs )
= ( Ys = nil_complex ) ) ).
% append_self_conv
thf(fact_203_self__append__conv,axiom,
! [Y2: list_nat,Ys: list_nat] :
( ( Y2
= ( append_nat @ Y2 @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_204_self__append__conv,axiom,
! [Y2: list_complex,Ys: list_complex] :
( ( Y2
= ( append_complex @ Y2 @ Ys ) )
= ( Ys = nil_complex ) ) ).
% self_append_conv
thf(fact_205_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_206_append__self__conv2,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( ( append_complex @ Xs @ Ys )
= Ys )
= ( Xs = nil_complex ) ) ).
% append_self_conv2
thf(fact_207_self__append__conv2,axiom,
! [Y2: list_nat,Xs: list_nat] :
( ( Y2
= ( append_nat @ Xs @ Y2 ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_208_self__append__conv2,axiom,
! [Y2: list_complex,Xs: list_complex] :
( ( Y2
= ( append_complex @ Xs @ Y2 ) )
= ( Xs = nil_complex ) ) ).
% self_append_conv2
thf(fact_209_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_210_Nil__is__append__conv,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( nil_complex
= ( append_complex @ Xs @ Ys ) )
= ( ( Xs = nil_complex )
& ( Ys = nil_complex ) ) ) ).
% Nil_is_append_conv
thf(fact_211_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_212_append__is__Nil__conv,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( ( append_complex @ Xs @ Ys )
= nil_complex )
= ( ( Xs = nil_complex )
& ( Ys = nil_complex ) ) ) ).
% append_is_Nil_conv
thf(fact_213_sublists_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( sublists_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( sublists_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_214_sublists_Osimps_I2_J,axiom,
! [X: complex,Xs: list_complex] :
( ( sublists_complex @ ( cons_complex @ X @ Xs ) )
= ( append_list_complex @ ( sublists_complex @ Xs ) @ ( map_li2870275437539113154omplex @ ( cons_complex @ X ) @ ( prefixes_complex @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_215_longest__common__prefix_Osimps_I3_J,axiom,
! [Uu2: list_nat] :
( ( longes266370323089874118ix_nat @ Uu2 @ nil_nat )
= nil_nat ) ).
% longest_common_prefix.simps(3)
thf(fact_216_longest__common__prefix_Osimps_I3_J,axiom,
! [Uu2: list_complex] :
( ( longes3069803181527244196omplex @ Uu2 @ nil_complex )
= nil_complex ) ).
% longest_common_prefix.simps(3)
thf(fact_217_longest__common__prefix_Osimps_I2_J,axiom,
! [Uv: list_nat] :
( ( longes266370323089874118ix_nat @ nil_nat @ Uv )
= nil_nat ) ).
% longest_common_prefix.simps(2)
thf(fact_218_longest__common__prefix_Osimps_I2_J,axiom,
! [Uv: list_complex] :
( ( longes3069803181527244196omplex @ nil_complex @ Uv )
= nil_complex ) ).
% longest_common_prefix.simps(2)
thf(fact_219_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_220_rev__nonempty__induct,axiom,
! [Xs: list_complex,P: list_complex > $o] :
( ( Xs != nil_complex )
=> ( ! [X2: complex] : ( P @ ( cons_complex @ X2 @ nil_complex ) )
=> ( ! [X2: complex,Xs2: list_complex] :
( ( Xs2 != nil_complex )
=> ( ( P @ Xs2 )
=> ( P @ ( append_complex @ Xs2 @ ( cons_complex @ X2 @ nil_complex ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_221_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs3: list_nat,X: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs3 )
= ( cons_nat @ X @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs3
= ( cons_nat @ X @ Xs ) ) )
| ? [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs3 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_222_append__eq__Cons__conv,axiom,
! [Ys: list_complex,Zs3: list_complex,X: complex,Xs: list_complex] :
( ( ( append_complex @ Ys @ Zs3 )
= ( cons_complex @ X @ Xs ) )
= ( ( ( Ys = nil_complex )
& ( Zs3
= ( cons_complex @ X @ Xs ) ) )
| ? [Ys4: list_complex] :
( ( Ys
= ( cons_complex @ X @ Ys4 ) )
& ( ( append_complex @ Ys4 @ Zs3 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_223_Cons__eq__append__conv,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs3 ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X @ Xs )
= Zs3 ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_nat @ Ys4 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_224_Cons__eq__append__conv,axiom,
! [X: complex,Xs: list_complex,Ys: list_complex,Zs3: list_complex] :
( ( ( cons_complex @ X @ Xs )
= ( append_complex @ Ys @ Zs3 ) )
= ( ( ( Ys = nil_complex )
& ( ( cons_complex @ X @ Xs )
= Zs3 ) )
| ? [Ys4: list_complex] :
( ( ( cons_complex @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_complex @ Ys4 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_225_append1__eq__conv,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y2: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_226_append1__eq__conv,axiom,
! [Xs: list_complex,X: complex,Ys: list_complex,Y2: complex] :
( ( ( append_complex @ Xs @ ( cons_complex @ X @ nil_complex ) )
= ( append_complex @ Ys @ ( cons_complex @ Y2 @ nil_complex ) ) )
= ( ( Xs = Ys )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_227_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys2: list_nat,Y: nat] :
( Xs
!= ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_228_rev__exhaust,axiom,
! [Xs: list_complex] :
( ( Xs != nil_complex )
=> ~ ! [Ys2: list_complex,Y: complex] :
( Xs
!= ( append_complex @ Ys2 @ ( cons_complex @ Y @ nil_complex ) ) ) ) ).
% rev_exhaust
thf(fact_229_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_230_rev__induct,axiom,
! [P: list_complex > $o,Xs: list_complex] :
( ( P @ nil_complex )
=> ( ! [X2: complex,Xs2: list_complex] :
( ( P @ Xs2 )
=> ( P @ ( append_complex @ Xs2 @ ( cons_complex @ X2 @ nil_complex ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_231_prefixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( prefixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( append_nat @ Zs2 @ ( cons_nat @ Z @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_232_prefixes__eq__snoc,axiom,
! [Ys: list_complex,Xs: list_list_complex,X: list_complex] :
( ( ( prefixes_complex @ Ys )
= ( append_list_complex @ Xs @ ( cons_list_complex @ X @ nil_list_complex ) ) )
= ( ( ( ( Ys = nil_complex )
& ( Xs = nil_list_complex ) )
| ? [Z: complex,Zs2: list_complex] :
( ( Ys
= ( append_complex @ Zs2 @ ( cons_complex @ Z @ nil_complex ) ) )
& ( Xs
= ( prefixes_complex @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_233_prefixes__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_234_prefixes__snoc,axiom,
! [Xs: list_complex,X: complex] :
( ( prefixes_complex @ ( append_complex @ Xs @ ( cons_complex @ X @ nil_complex ) ) )
= ( append_list_complex @ ( prefixes_complex @ Xs ) @ ( cons_list_complex @ ( append_complex @ Xs @ ( cons_complex @ X @ nil_complex ) ) @ nil_list_complex ) ) ) ).
% prefixes_snoc
thf(fact_235__092_060open_062_092_060forall_062A_092_060in_062Afp_O_Ahermitian_AA_A_092_060and_062_AA_A_092_060in_062_Acarrier__mat_An_An_092_060close_062,axiom,
! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ afp )
=> ( ( comple8306762464034002205omplex @ X4 )
& ( member_mat_complex @ X4 @ ( carrier_mat_complex @ na @ na ) ) ) ) ).
% \<open>\<forall>A\<in>Afp. hermitian A \<and> A \<in> carrier_mat n n\<close>
thf(fact_236__092_060open_062_092_060forall_062A_092_060in_062Afp_O_AAp_A_K_AA_A_061_AA_A_K_AAp_092_060close_062,axiom,
! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ afp )
=> ( ( times_8009071140041733218omplex @ ap @ X4 )
= ( times_8009071140041733218omplex @ X4 @ ap ) ) ) ).
% \<open>\<forall>A\<in>Afp. Ap * A = A * Ap\<close>
thf(fact_237__092_060open_062finite_AAfp_092_060close_062,axiom,
finite7047982916621727056omplex @ afp ).
% \<open>finite Afp\<close>
thf(fact_238__092_060open_062_092_060forall_062A_092_060in_062Afp_O_A_092_060forall_062B_092_060in_062Afp_O_AA_A_K_AB_A_061_AB_A_K_AA_092_060close_062,axiom,
! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ afp )
=> ! [Xa2: mat_complex] :
( ( member_mat_complex @ Xa2 @ afp )
=> ( ( times_8009071140041733218omplex @ X4 @ Xa2 )
= ( times_8009071140041733218omplex @ Xa2 @ X4 ) ) ) ) ).
% \<open>\<forall>A\<in>Afp. \<forall>B\<in>Afp. A * B = B * A\<close>
thf(fact_239_diagonal__mat__times__diag,axiom,
! [A4: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( diagonal_mat_complex @ A4 )
=> ( ( diagonal_mat_complex @ B )
=> ( diagonal_mat_complex @ ( times_8009071140041733218omplex @ A4 @ B ) ) ) ) ) ) ).
% diagonal_mat_times_diag
thf(fact_240_diagonal__mat__sq__diag,axiom,
! [B: mat_complex,N: nat] :
( ( diagonal_mat_complex @ B )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( diagonal_mat_complex @ ( times_8009071140041733218omplex @ B @ B ) ) ) ) ).
% diagonal_mat_sq_diag
thf(fact_241_diagonal__mat__commute,axiom,
! [A4: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( diagonal_mat_complex @ A4 )
=> ( ( diagonal_mat_complex @ B )
=> ( ( times_8009071140041733218omplex @ A4 @ B )
= ( times_8009071140041733218omplex @ B @ A4 ) ) ) ) ) ) ).
% diagonal_mat_commute
thf(fact_242_unitary__times__unitary,axiom,
! [P: mat_complex,N: nat,Q: mat_complex] :
( ( member_mat_complex @ P @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ Q @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ P )
=> ( ( comple6660659447773130958omplex @ Q )
=> ( comple6660659447773130958omplex @ ( times_8009071140041733218omplex @ P @ Q ) ) ) ) ) ) ).
% unitary_times_unitary
thf(fact_243_assms_I5_J,axiom,
! [A4: mat_complex] :
( ( member_mat_complex @ A4 @ af )
=> ( comple8306762464034002205omplex @ A4 ) ) ).
% assms(5)
thf(fact_244_assms_I6_J,axiom,
! [A4: mat_complex,B: mat_complex] :
( ( member_mat_complex @ A4 @ af )
=> ( ( member_mat_complex @ B @ af )
=> ( ( times_8009071140041733218omplex @ A4 @ B )
= ( times_8009071140041733218omplex @ B @ A4 ) ) ) ) ).
% assms(6)
thf(fact_245_assms_I3_J,axiom,
! [A4: mat_complex] :
( ( member_mat_complex @ A4 @ af )
=> ( member_mat_complex @ A4 @ ( carrier_mat_complex @ n @ n ) ) ) ).
% assms(3)
thf(fact_246_mat__assoc__test_I1_J,axiom,
! [A4: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A4 @ B ) @ ( times_8009071140041733218omplex @ C @ D ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A4 @ B ) @ C ) @ D ) ) ) ) ) ) ).
% mat_assoc_test(1)
thf(fact_247_hermitian__square__hermitian,axiom,
! [A4: mat_complex] :
( ( comple8306762464034002205omplex @ A4 )
=> ( comple8306762464034002205omplex @ ( times_8009071140041733218omplex @ A4 @ A4 ) ) ) ).
% hermitian_square_hermitian
thf(fact_248_assoc__mult__mat,axiom,
! [A4: mat_complex,N_1: nat,N_2: nat,B: mat_complex,N_3: nat,C: mat_complex,N_4: nat] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N_1 @ N_2 ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N_2 @ N_3 ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N_3 @ N_4 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A4 @ B ) @ C )
= ( times_8009071140041733218omplex @ A4 @ ( times_8009071140041733218omplex @ B @ C ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_249_mult__carrier__mat,axiom,
! [A4: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A4 @ B ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_250_find__indices__snoc,axiom,
! [X: complex,Ys: list_complex,Y2: complex] :
( ( missin8834916005246747252omplex @ X @ ( append_complex @ Ys @ ( cons_complex @ Y2 @ nil_complex ) ) )
= ( append_nat @ ( missin8834916005246747252omplex @ X @ Ys ) @ ( if_list_nat @ ( X = Y2 ) @ ( cons_nat @ ( size_s3451745648224563538omplex @ Ys ) @ nil_nat ) @ nil_nat ) ) ) ).
% find_indices_snoc
thf(fact_251_find__indices__snoc,axiom,
! [X: nat,Ys: list_nat,Y2: nat] :
( ( missin5050847376130023830es_nat @ X @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( append_nat @ ( missin5050847376130023830es_nat @ X @ Ys ) @ ( if_list_nat @ ( X = Y2 ) @ ( cons_nat @ ( size_size_list_nat @ Ys ) @ nil_nat ) @ nil_nat ) ) ) ).
% find_indices_snoc
thf(fact_252_finite_Ointros_I1_J,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.intros(1)
thf(fact_253_finite_Ointros_I1_J,axiom,
finite7047982916621727056omplex @ bot_bo7165004461764951667omplex ).
% finite.intros(1)
thf(fact_254_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_255_infinite__imp__nonempty,axiom,
! [S: set_mat_complex] :
( ~ ( finite7047982916621727056omplex @ S )
=> ( S != bot_bo7165004461764951667omplex ) ) ).
% infinite_imp_nonempty
thf(fact_256_finite__transitivity__chain,axiom,
! [A4: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [X2: nat] :
~ ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y: nat,Z2: nat] :
( ( R @ X2 @ Y )
=> ( ( R @ Y @ Z2 )
=> ( R @ X2 @ Z2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A4 )
=> ? [Y4: nat] :
( ( member_nat2 @ Y4 @ A4 )
& ( R @ X2 @ Y4 ) ) )
=> ( A4 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_257_finite__transitivity__chain,axiom,
! [A4: set_mat_complex,R: mat_complex > mat_complex > $o] :
( ( finite7047982916621727056omplex @ A4 )
=> ( ! [X2: mat_complex] :
~ ( R @ X2 @ X2 )
=> ( ! [X2: mat_complex,Y: mat_complex,Z2: mat_complex] :
( ( R @ X2 @ Y )
=> ( ( R @ Y @ Z2 )
=> ( R @ X2 @ Z2 ) ) )
=> ( ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ A4 )
=> ? [Y4: mat_complex] :
( ( member_mat_complex @ Y4 @ A4 )
& ( R @ X2 @ Y4 ) ) )
=> ( A4 = bot_bo7165004461764951667omplex ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_258_more__arith__simps_I6_J,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% more_arith_simps(6)
thf(fact_259_more__arith__simps_I6_J,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% more_arith_simps(6)
thf(fact_260_more__arith__simps_I6_J,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% more_arith_simps(6)
thf(fact_261_more__arith__simps_I5_J,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% more_arith_simps(5)
thf(fact_262_more__arith__simps_I5_J,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% more_arith_simps(5)
thf(fact_263_more__arith__simps_I5_J,axiom,
! [A3: complex] :
( ( times_times_complex @ one_one_complex @ A3 )
= A3 ) ).
% more_arith_simps(5)
thf(fact_264_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_265_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_266_length__map,axiom,
! [F: complex > real,Xs: list_complex] :
( ( size_size_list_real @ ( map_complex_real @ F @ Xs ) )
= ( size_s3451745648224563538omplex @ Xs ) ) ).
% length_map
thf(fact_267_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_268_map__eq__imp__length__eq,axiom,
! [F: complex > real,Xs: list_complex,G: complex > real,Ys: list_complex] :
( ( ( map_complex_real @ F @ Xs )
= ( map_complex_real @ G @ Ys ) )
=> ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_269_map__eq__imp__length__eq,axiom,
! [F: complex > real,Xs: list_complex,G: nat > real,Ys: list_nat] :
( ( ( map_complex_real @ F @ Xs )
= ( map_nat_real @ G @ Ys ) )
=> ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_270_map__eq__imp__length__eq,axiom,
! [F: nat > real,Xs: list_nat,G: complex > real,Ys: list_complex] :
( ( ( map_nat_real @ F @ Xs )
= ( map_complex_real @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_271_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us2: list_nat,Vs2: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us2 )
= ( size_size_list_nat @ Vs2 ) ) )
=> ( ( ( append_nat @ Xs @ Us2 )
= ( append_nat @ Ys @ Vs2 ) )
= ( ( Xs = Ys )
& ( Us2 = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_272_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_complex,Ws: list_complex,P: list_complex > list_complex > list_complex > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs3 )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_complex @ nil_complex )
=> ( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex,Z2: complex,Zs: list_complex,W: complex,Ws2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) @ ( cons_complex @ Z2 @ Zs ) @ ( cons_complex @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_273_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_complex,Ws: list_nat,P: list_complex > list_complex > list_complex > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_complex @ nil_nat )
=> ( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex,Z2: complex,Zs: list_complex,W: nat,Ws2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) @ ( cons_complex @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_274_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_nat,Ws: list_complex,P: list_complex > list_complex > list_nat > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_nat @ nil_complex )
=> ( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex,Z2: nat,Zs: list_nat,W: complex,Ws2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_complex @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_275_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_nat,Ws: list_nat,P: list_complex > list_complex > list_nat > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_nat @ nil_nat )
=> ( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex,Z2: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_276_list__induct4,axiom,
! [Xs: list_complex,Ys: list_nat,Zs3: list_complex,Ws: list_complex,P: list_complex > list_nat > list_complex > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs3 )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_complex @ nil_complex )
=> ( ! [X2: complex,Xs2: list_complex,Y: nat,Ys2: list_nat,Z2: complex,Zs: list_complex,W: complex,Ws2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_complex @ Z2 @ Zs ) @ ( cons_complex @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_277_list__induct4,axiom,
! [Xs: list_complex,Ys: list_nat,Zs3: list_complex,Ws: list_nat,P: list_complex > list_nat > list_complex > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_complex @ nil_nat )
=> ( ! [X2: complex,Xs2: list_complex,Y: nat,Ys2: list_nat,Z2: complex,Zs: list_complex,W: nat,Ws2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_complex @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_278_list__induct4,axiom,
! [Xs: list_complex,Ys: list_nat,Zs3: list_nat,Ws: list_complex,P: list_complex > list_nat > list_nat > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_nat @ nil_complex )
=> ( ! [X2: complex,Xs2: list_complex,Y: nat,Ys2: list_nat,Z2: nat,Zs: list_nat,W: complex,Ws2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_complex @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_279_list__induct4,axiom,
! [Xs: list_complex,Ys: list_nat,Zs3: list_nat,Ws: list_nat,P: list_complex > list_nat > list_nat > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: complex,Xs2: list_complex,Y: nat,Ys2: list_nat,Z2: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_280_list__induct4,axiom,
! [Xs: list_nat,Ys: list_complex,Zs3: list_complex,Ws: list_complex,P: list_nat > list_complex > list_complex > list_complex > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs3 )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_nat @ nil_complex @ nil_complex @ nil_complex )
=> ( ! [X2: nat,Xs2: list_nat,Y: complex,Ys2: list_complex,Z2: complex,Zs: list_complex,W: complex,Ws2: list_complex] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) @ ( cons_complex @ Z2 @ Zs ) @ ( cons_complex @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_281_list__induct4,axiom,
! [Xs: list_nat,Ys: list_complex,Zs3: list_complex,Ws: list_nat,P: list_nat > list_complex > list_complex > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_complex @ nil_complex @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: complex,Ys2: list_complex,Z2: complex,Zs: list_complex,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) @ ( cons_complex @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_282_list__induct3,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_complex,P: list_complex > list_complex > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_complex )
=> ( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex,Z2: complex,Zs: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) @ ( cons_complex @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_283_list__induct3,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_nat,P: list_complex > list_complex > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_nat )
=> ( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex,Z2: nat,Zs: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_284_list__induct3,axiom,
! [Xs: list_complex,Ys: list_nat,Zs3: list_complex,P: list_complex > list_nat > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_complex )
=> ( ! [X2: complex,Xs2: list_complex,Y: nat,Ys2: list_nat,Z2: complex,Zs: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_complex @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_285_list__induct3,axiom,
! [Xs: list_complex,Ys: list_nat,Zs3: list_nat,P: list_complex > list_nat > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_nat )
=> ( ! [X2: complex,Xs2: list_complex,Y: nat,Ys2: list_nat,Z2: nat,Zs: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_286_list__induct3,axiom,
! [Xs: list_nat,Ys: list_complex,Zs3: list_complex,P: list_nat > list_complex > list_complex > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_complex @ nil_complex )
=> ( ! [X2: nat,Xs2: list_nat,Y: complex,Ys2: list_complex,Z2: complex,Zs: list_complex] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) @ ( cons_complex @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_287_list__induct3,axiom,
! [Xs: list_nat,Ys: list_complex,Zs3: list_nat,P: list_nat > list_complex > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_complex @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: complex,Ys2: list_complex,Z2: nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_288_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs3: list_complex,P: list_nat > list_nat > list_complex > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_complex )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat,Z2: complex,Zs: list_complex] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_complex @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_289_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs3: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat,Z2: nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_290_list__induct2,axiom,
! [Xs: list_complex,Ys: list_complex,P: list_complex > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( P @ nil_complex @ nil_complex )
=> ( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_291_list__induct2,axiom,
! [Xs: list_complex,Ys: list_nat,P: list_complex > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_complex @ nil_nat )
=> ( ! [X2: complex,Xs2: list_complex,Y: nat,Ys2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_292_list__induct2,axiom,
! [Xs: list_nat,Ys: list_complex,P: list_nat > list_complex > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( P @ nil_nat @ nil_complex )
=> ( ! [X2: nat,Xs2: list_nat,Y: complex,Ys2: list_complex] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_293_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_294_same__length__different,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( Xs != Ys )
=> ( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ? [Pre: list_complex,X2: complex,Xs3: list_complex,Y: complex,Ys5: list_complex] :
( ( X2 != Y )
& ( Xs
= ( append_complex @ Pre @ ( append_complex @ ( cons_complex @ X2 @ nil_complex ) @ Xs3 ) ) )
& ( Ys
= ( append_complex @ Pre @ ( append_complex @ ( cons_complex @ Y @ nil_complex ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_295_same__length__different,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X2: nat,Xs3: list_nat,Y: nat,Ys5: list_nat] :
( ( X2 != Y )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X2 @ nil_nat ) @ Xs3 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_296_more__arith__simps_I11_J,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B2 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% more_arith_simps(11)
thf(fact_297_more__arith__simps_I11_J,axiom,
! [A3: complex,B2: complex,C2: complex] :
( ( times_times_complex @ ( times_times_complex @ A3 @ B2 ) @ C2 )
= ( times_times_complex @ A3 @ ( times_times_complex @ B2 @ C2 ) ) ) ).
% more_arith_simps(11)
thf(fact_298_eq__comps__singleton,axiom,
! [A3: nat,L: list_complex] :
( ( ( cons_nat @ A3 @ nil_nat )
= ( commut93809757773076895omplex @ L ) )
=> ( A3
= ( size_s3451745648224563538omplex @ L ) ) ) ).
% eq_comps_singleton
thf(fact_299_eq__comps__singleton,axiom,
! [A3: nat,L: list_nat] :
( ( ( cons_nat @ A3 @ nil_nat )
= ( commut2436974278740741825ps_nat @ L ) )
=> ( A3
= ( size_size_list_nat @ L ) ) ) ).
% eq_comps_singleton
thf(fact_300_vector__space__over__itself_Oscale__one,axiom,
! [X: real] :
( ( times_times_real @ one_one_real @ X )
= X ) ).
% vector_space_over_itself.scale_one
thf(fact_301_vector__space__over__itself_Oscale__one,axiom,
! [X: complex] :
( ( times_times_complex @ one_one_complex @ X )
= X ) ).
% vector_space_over_itself.scale_one
thf(fact_302_mult_Ocomm__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.comm_neutral
thf(fact_303_mult_Ocomm__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.comm_neutral
thf(fact_304_mult_Ocomm__neutral,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ one_one_complex )
= A3 ) ).
% mult.comm_neutral
thf(fact_305_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_306_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_307_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_308_Groups_Omult__ac_I3_J,axiom,
! [B2: nat,A3: nat,C2: nat] :
( ( times_times_nat @ B2 @ ( times_times_nat @ A3 @ C2 ) )
= ( times_times_nat @ A3 @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% Groups.mult_ac(3)
thf(fact_309_Groups_Omult__ac_I3_J,axiom,
! [B2: complex,A3: complex,C2: complex] :
( ( times_times_complex @ B2 @ ( times_times_complex @ A3 @ C2 ) )
= ( times_times_complex @ A3 @ ( times_times_complex @ B2 @ C2 ) ) ) ).
% Groups.mult_ac(3)
thf(fact_310_Groups_Omult__ac_I2_J,axiom,
( times_times_nat
= ( ^ [A6: nat,B3: nat] : ( times_times_nat @ B3 @ A6 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_311_Groups_Omult__ac_I2_J,axiom,
( times_times_complex
= ( ^ [A6: complex,B3: complex] : ( times_times_complex @ B3 @ A6 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_312_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B2 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_313_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: complex,B2: complex,C2: complex] :
( ( times_times_complex @ ( times_times_complex @ A3 @ B2 ) @ C2 )
= ( times_times_complex @ A3 @ ( times_times_complex @ B2 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_314_vector__space__over__itself_Ovector__space__assms_I3_J,axiom,
! [A3: complex,B2: complex,X: complex] :
( ( times_times_complex @ A3 @ ( times_times_complex @ B2 @ X ) )
= ( times_times_complex @ ( times_times_complex @ A3 @ B2 ) @ X ) ) ).
% vector_space_over_itself.vector_space_assms(3)
thf(fact_315_vector__space__over__itself_Oscale__left__commute,axiom,
! [A3: complex,B2: complex,X: complex] :
( ( times_times_complex @ A3 @ ( times_times_complex @ B2 @ X ) )
= ( times_times_complex @ B2 @ ( times_times_complex @ A3 @ X ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_316_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_317_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_318_commute__diag__compat,axiom,
! [D: mat_complex,N: nat,B: mat_complex,L: list_nat] :
( ( diagonal_mat_complex @ D )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ( times_8009071140041733218omplex @ B @ D )
= ( times_8009071140041733218omplex @ D @ B ) )
=> ( ( commut4502369927624756007omplex @ D @ L )
=> ( commut5261563022830629508omplex @ B @ L ) ) ) ) ) ) ).
% commute_diag_compat
thf(fact_319_Afp__def,axiom,
( afp
= ( minus_8760755521168068590omplex @ afa @ ( insert_mat_complex @ ap @ bot_bo7165004461764951667omplex ) ) ) ).
% Afp_def
thf(fact_320_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs3: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs3 ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys = nil_nat )
& ( Zs3 = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs4: list_nat,Xs5: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs4 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs4 ) )
& ( Zs3
= ( append_nat @ Xs5 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_321_concat__eq__append__conv,axiom,
! [Xss2: list_list_complex,Ys: list_complex,Zs3: list_complex] :
( ( ( concat_complex @ Xss2 )
= ( append_complex @ Ys @ Zs3 ) )
= ( ( ( Xss2 = nil_list_complex )
=> ( ( Ys = nil_complex )
& ( Zs3 = nil_complex ) ) )
& ( ( Xss2 != nil_list_complex )
=> ? [Xss1: list_list_complex,Xs4: list_complex,Xs5: list_complex,Xss22: list_list_complex] :
( ( Xss2
= ( append_list_complex @ Xss1 @ ( cons_list_complex @ ( append_complex @ Xs4 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_complex @ ( concat_complex @ Xss1 ) @ Xs4 ) )
& ( Zs3
= ( append_complex @ Xs5 @ ( concat_complex @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_322_forall__vector__1,axiom,
( ( ^ [P3: finite2525469894391432876l_num1 > $o] :
! [X5: finite2525469894391432876l_num1] : ( P3 @ X5 ) )
= ( ^ [P4: finite2525469894391432876l_num1 > $o] :
! [X3: nat] : ( P4 @ ( cartes6052806112279933926l_num1 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) ) ).
% forall_vector_1
thf(fact_323_forall__vector__1,axiom,
( ( ^ [P3: finite3194102281137443150l_num1 > $o] :
! [X5: finite3194102281137443150l_num1] : ( P3 @ X5 ) )
= ( ^ [P4: finite3194102281137443150l_num1 > $o] :
! [X3: complex] : ( P4 @ ( cartes7240268600962231432l_num1 @ ( cons_complex @ X3 @ nil_complex ) ) ) ) ) ).
% forall_vector_1
thf(fact_324_Diff__not__in,axiom,
! [A3: nat,A4: set_nat] :
~ ( member_nat2 @ A3 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ).
% Diff_not_in
thf(fact_325_Diff__not__in,axiom,
! [A3: mat_complex,A4: set_mat_complex] :
~ ( member_mat_complex @ A3 @ ( minus_8760755521168068590omplex @ A4 @ ( insert_mat_complex @ A3 @ bot_bo7165004461764951667omplex ) ) ) ).
% Diff_not_in
thf(fact_326_finite__Diff__insert,axiom,
! [A4: set_nat,A3: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A3 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_327_finite__Diff__insert,axiom,
! [A4: set_mat_complex,A3: mat_complex,B: set_mat_complex] :
( ( finite7047982916621727056omplex @ ( minus_8760755521168068590omplex @ A4 @ ( insert_mat_complex @ A3 @ B ) ) )
= ( finite7047982916621727056omplex @ ( minus_8760755521168068590omplex @ A4 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_328_finite__empty__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P @ A4 )
=> ( ! [A: nat,A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( member_nat2 @ A @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_329_finite__empty__induct,axiom,
! [A4: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ A4 )
=> ( ( P @ A4 )
=> ( ! [A: mat_complex,A5: set_mat_complex] :
( ( finite7047982916621727056omplex @ A5 )
=> ( ( member_mat_complex @ A @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_8760755521168068590omplex @ A5 @ ( insert_mat_complex @ A @ bot_bo7165004461764951667omplex ) ) ) ) ) )
=> ( P @ bot_bo7165004461764951667omplex ) ) ) ) ).
% finite_empty_induct
thf(fact_330_infinite__coinduct,axiom,
! [X6: set_nat > $o,A4: set_nat] :
( ( X6 @ A4 )
=> ( ! [A5: set_nat] :
( ( X6 @ A5 )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ A5 )
& ( ( X6 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_331_infinite__coinduct,axiom,
! [X6: set_mat_complex > $o,A4: set_mat_complex] :
( ( X6 @ A4 )
=> ( ! [A5: set_mat_complex] :
( ( X6 @ A5 )
=> ? [X4: mat_complex] :
( ( member_mat_complex @ X4 @ A5 )
& ( ( X6 @ ( minus_8760755521168068590omplex @ A5 @ ( insert_mat_complex @ X4 @ bot_bo7165004461764951667omplex ) ) )
| ~ ( finite7047982916621727056omplex @ ( minus_8760755521168068590omplex @ A5 @ ( insert_mat_complex @ X4 @ bot_bo7165004461764951667omplex ) ) ) ) ) )
=> ~ ( finite7047982916621727056omplex @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_332_infinite__remove,axiom,
! [S: set_nat,A3: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_333_infinite__remove,axiom,
! [S: set_mat_complex,A3: mat_complex] :
( ~ ( finite7047982916621727056omplex @ S )
=> ~ ( finite7047982916621727056omplex @ ( minus_8760755521168068590omplex @ S @ ( insert_mat_complex @ A3 @ bot_bo7165004461764951667omplex ) ) ) ) ).
% infinite_remove
thf(fact_334_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A3: complex,X: complex,Y2: complex] :
( ( times_times_complex @ A3 @ ( minus_minus_complex @ X @ Y2 ) )
= ( minus_minus_complex @ ( times_times_complex @ A3 @ X ) @ ( times_times_complex @ A3 @ Y2 ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_335_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A3: complex,B2: complex,X: complex] :
( ( times_times_complex @ ( minus_minus_complex @ A3 @ B2 ) @ X )
= ( minus_minus_complex @ ( times_times_complex @ A3 @ X ) @ ( times_times_complex @ B2 @ X ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_336_finite_Ointros_I2_J,axiom,
! [A4: set_mat_complex,A3: mat_complex] :
( ( finite7047982916621727056omplex @ A4 )
=> ( finite7047982916621727056omplex @ ( insert_mat_complex @ A3 @ A4 ) ) ) ).
% finite.intros(2)
thf(fact_337_finite_Ointros_I2_J,axiom,
! [A4: set_nat,A3: nat] :
( ( finite_finite_nat @ A4 )
=> ( finite_finite_nat @ ( insert_nat @ A3 @ A4 ) ) ) ).
% finite.intros(2)
thf(fact_338_finite__insert,axiom,
! [A3: mat_complex,A4: set_mat_complex] :
( ( finite7047982916621727056omplex @ ( insert_mat_complex @ A3 @ A4 ) )
= ( finite7047982916621727056omplex @ A4 ) ) ).
% finite_insert
thf(fact_339_finite__insert,axiom,
! [A3: nat,A4: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A3 @ A4 ) )
= ( finite_finite_nat @ A4 ) ) ).
% finite_insert
thf(fact_340_finite__Diff,axiom,
! [A4: set_nat,B: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B ) ) ) ).
% finite_Diff
thf(fact_341_finite__Diff,axiom,
! [A4: set_mat_complex,B: set_mat_complex] :
( ( finite7047982916621727056omplex @ A4 )
=> ( finite7047982916621727056omplex @ ( minus_8760755521168068590omplex @ A4 @ B ) ) ) ).
% finite_Diff
thf(fact_342_finite__Diff2,axiom,
! [B: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_Diff2
thf(fact_343_finite__Diff2,axiom,
! [B: set_mat_complex,A4: set_mat_complex] :
( ( finite7047982916621727056omplex @ B )
=> ( ( finite7047982916621727056omplex @ ( minus_8760755521168068590omplex @ A4 @ B ) )
= ( finite7047982916621727056omplex @ A4 ) ) ) ).
% finite_Diff2
thf(fact_344_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_345_Diff__infinite__finite,axiom,
! [T: set_mat_complex,S: set_mat_complex] :
( ( finite7047982916621727056omplex @ T )
=> ( ~ ( finite7047982916621727056omplex @ S )
=> ~ ( finite7047982916621727056omplex @ ( minus_8760755521168068590omplex @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_346_infinite__finite__induct,axiom,
! [P: set_nat > $o,A4: set_nat] :
( ! [A5: set_nat] :
( ~ ( finite_finite_nat @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat2 @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_347_infinite__finite__induct,axiom,
! [P: set_mat_complex > $o,A4: set_mat_complex] :
( ! [A5: set_mat_complex] :
( ~ ( finite7047982916621727056omplex @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [X2: mat_complex,F3: set_mat_complex] :
( ( finite7047982916621727056omplex @ F3 )
=> ( ~ ( member_mat_complex @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_mat_complex @ X2 @ F3 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_348_finite__ne__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat2 @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_349_finite__ne__induct,axiom,
! [F4: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ F4 )
=> ( ( F4 != bot_bo7165004461764951667omplex )
=> ( ! [X2: mat_complex] : ( P @ ( insert_mat_complex @ X2 @ bot_bo7165004461764951667omplex ) )
=> ( ! [X2: mat_complex,F3: set_mat_complex] :
( ( finite7047982916621727056omplex @ F3 )
=> ( ( F3 != bot_bo7165004461764951667omplex )
=> ( ~ ( member_mat_complex @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_mat_complex @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_350_finite__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat2 @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_351_finite__induct,axiom,
! [F4: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ F4 )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [X2: mat_complex,F3: set_mat_complex] :
( ( finite7047982916621727056omplex @ F3 )
=> ( ~ ( member_mat_complex @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_mat_complex @ X2 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_352_finite_Oinducts,axiom,
! [X: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ X )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: set_nat,A: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( insert_nat @ A @ A5 ) ) ) )
=> ( P @ X ) ) ) ) ).
% finite.inducts
thf(fact_353_finite_Oinducts,axiom,
! [X: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ X )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [A5: set_mat_complex,A: mat_complex] :
( ( finite7047982916621727056omplex @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( insert_mat_complex @ A @ A5 ) ) ) )
=> ( P @ X ) ) ) ) ).
% finite.inducts
thf(fact_354_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A6: set_nat] :
( ( A6 = bot_bot_set_nat )
| ? [A7: set_nat,B3: nat] :
( ( A6
= ( insert_nat @ B3 @ A7 ) )
& ( finite_finite_nat @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_355_finite_Osimps,axiom,
( finite7047982916621727056omplex
= ( ^ [A6: set_mat_complex] :
( ( A6 = bot_bo7165004461764951667omplex )
| ? [A7: set_mat_complex,B3: mat_complex] :
( ( A6
= ( insert_mat_complex @ B3 @ A7 ) )
& ( finite7047982916621727056omplex @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_356_finite_Ocases,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ~ ! [A5: set_nat] :
( ? [A: nat] :
( A3
= ( insert_nat @ A @ A5 ) )
=> ~ ( finite_finite_nat @ A5 ) ) ) ) ).
% finite.cases
thf(fact_357_finite_Ocases,axiom,
! [A3: set_mat_complex] :
( ( finite7047982916621727056omplex @ A3 )
=> ( ( A3 != bot_bo7165004461764951667omplex )
=> ~ ! [A5: set_mat_complex] :
( ? [A: mat_complex] :
( A3
= ( insert_mat_complex @ A @ A5 ) )
=> ~ ( finite7047982916621727056omplex @ A5 ) ) ) ) ).
% finite.cases
thf(fact_358_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_359_concat_Osimps_I1_J,axiom,
( ( concat_complex @ nil_list_complex )
= nil_complex ) ).
% concat.simps(1)
thf(fact_360_map__concat,axiom,
! [F: complex > real,Xs: list_list_complex] :
( ( map_complex_real @ F @ ( concat_complex @ Xs ) )
= ( concat_real @ ( map_li971590449312185664t_real @ ( map_complex_real @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_361_vector__space__over__itself_Ofinite__Basis,axiom,
finite_finite_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ).
% vector_space_over_itself.finite_Basis
thf(fact_362_permutations__of__set__aux_Oinduct,axiom,
! [P: list_complex > set_complex > $o,A0: list_complex,A1: set_complex] :
( ! [Acc: list_complex,A5: set_complex] :
( ! [X4: complex] :
( ( finite3207457112153483333omplex @ A5 )
=> ( ( A5 != bot_bot_set_complex )
=> ( ( member_complex2 @ X4 @ A5 )
=> ( P @ ( cons_complex @ X4 @ Acc ) @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) ) ) )
=> ( P @ Acc @ A5 ) )
=> ( P @ A0 @ A1 ) ) ).
% permutations_of_set_aux.induct
thf(fact_363_permutations__of__set__aux_Oinduct,axiom,
! [P: list_nat > set_nat > $o,A0: list_nat,A1: set_nat] :
( ! [Acc: list_nat,A5: set_nat] :
( ! [X4: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( member_nat2 @ X4 @ A5 )
=> ( P @ ( cons_nat @ X4 @ Acc ) @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ Acc @ A5 ) )
=> ( P @ A0 @ A1 ) ) ).
% permutations_of_set_aux.induct
thf(fact_364_permutations__of__set__aux_Oinduct,axiom,
! [P: list_mat_complex > set_mat_complex > $o,A0: list_mat_complex,A1: set_mat_complex] :
( ! [Acc: list_mat_complex,A5: set_mat_complex] :
( ! [X4: mat_complex] :
( ( finite7047982916621727056omplex @ A5 )
=> ( ( A5 != bot_bo7165004461764951667omplex )
=> ( ( member_mat_complex @ X4 @ A5 )
=> ( P @ ( cons_mat_complex @ X4 @ Acc ) @ ( minus_8760755521168068590omplex @ A5 @ ( insert_mat_complex @ X4 @ bot_bo7165004461764951667omplex ) ) ) ) ) )
=> ( P @ Acc @ A5 ) )
=> ( P @ A0 @ A1 ) ) ).
% permutations_of_set_aux.induct
thf(fact_365_insert__Diff__single,axiom,
! [A3: mat_complex,A4: set_mat_complex] :
( ( insert_mat_complex @ A3 @ ( minus_8760755521168068590omplex @ A4 @ ( insert_mat_complex @ A3 @ bot_bo7165004461764951667omplex ) ) )
= ( insert_mat_complex @ A3 @ A4 ) ) ).
% insert_Diff_single
thf(fact_366_Diff__insert__absorb,axiom,
! [X: nat,A4: set_nat] :
( ~ ( member_nat2 @ X @ A4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_367_Diff__insert__absorb,axiom,
! [X: mat_complex,A4: set_mat_complex] :
( ~ ( member_mat_complex @ X @ A4 )
=> ( ( minus_8760755521168068590omplex @ ( insert_mat_complex @ X @ A4 ) @ ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_368_Diff__insert2,axiom,
! [A4: set_mat_complex,A3: mat_complex,B: set_mat_complex] :
( ( minus_8760755521168068590omplex @ A4 @ ( insert_mat_complex @ A3 @ B ) )
= ( minus_8760755521168068590omplex @ ( minus_8760755521168068590omplex @ A4 @ ( insert_mat_complex @ A3 @ bot_bo7165004461764951667omplex ) ) @ B ) ) ).
% Diff_insert2
thf(fact_369_insert__Diff,axiom,
! [A3: nat,A4: set_nat] :
( ( member_nat2 @ A3 @ A4 )
=> ( ( insert_nat @ A3 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_370_insert__Diff,axiom,
! [A3: mat_complex,A4: set_mat_complex] :
( ( member_mat_complex @ A3 @ A4 )
=> ( ( insert_mat_complex @ A3 @ ( minus_8760755521168068590omplex @ A4 @ ( insert_mat_complex @ A3 @ bot_bo7165004461764951667omplex ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_371_Diff__insert,axiom,
! [A4: set_mat_complex,A3: mat_complex,B: set_mat_complex] :
( ( minus_8760755521168068590omplex @ A4 @ ( insert_mat_complex @ A3 @ B ) )
= ( minus_8760755521168068590omplex @ ( minus_8760755521168068590omplex @ A4 @ B ) @ ( insert_mat_complex @ A3 @ bot_bo7165004461764951667omplex ) ) ) ).
% Diff_insert
thf(fact_372_set__one,axiom,
( one_one_set_nat
= ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ).
% set_one
thf(fact_373_set__one,axiom,
( one_one_set_real
= ( insert_real @ one_one_real @ bot_bot_set_real ) ) ).
% set_one
thf(fact_374_dimension__eq__1,axiom,
( ( vector5117482691322076262n_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) )
= one_one_nat ) ).
% dimension_eq_1
thf(fact_375_minus__carrier__mat_H,axiom,
! [A4: mat_complex,Nr: nat,Nc: nat,B: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( minus_2412168080157227406omplex @ A4 @ B ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% minus_carrier_mat'
thf(fact_376_minus__carrier__mat,axiom,
! [B: mat_complex,Nr: nat,Nc: nat,A4: mat_complex] :
( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( minus_2412168080157227406omplex @ A4 @ B ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_377_mult__minus__distrib__mat,axiom,
! [A4: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat,C: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A4 @ ( minus_2412168080157227406omplex @ B @ C ) )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A4 @ B ) @ ( times_8009071140041733218omplex @ A4 @ C ) ) ) ) ) ) ).
% mult_minus_distrib_mat
thf(fact_378_minus__mult__distrib__mat,axiom,
! [A4: mat_complex,Nr: nat,N: nat,B: mat_complex,C: mat_complex,Nc: nat] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( minus_2412168080157227406omplex @ A4 @ B ) @ C )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A4 @ C ) @ ( times_8009071140041733218omplex @ B @ C ) ) ) ) ) ) ).
% minus_mult_distrib_mat
thf(fact_379_mat__assoc__test_I9_J,axiom,
! [A4: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A4 @ ( minus_2412168080157227406omplex @ B @ C ) ) @ D )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A4 @ B ) @ D ) @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A4 @ C ) @ D ) ) ) ) ) ) ) ).
% mat_assoc_test(9)
thf(fact_380_set__times__elim,axiom,
! [X: mat_complex,A4: set_mat_complex,B: set_mat_complex] :
( ( member_mat_complex @ X @ ( times_6731331324747250370omplex @ A4 @ B ) )
=> ~ ! [A: mat_complex,B4: mat_complex] :
( ( X
= ( times_8009071140041733218omplex @ A @ B4 ) )
=> ( ( member_mat_complex @ A @ A4 )
=> ~ ( member_mat_complex @ B4 @ B ) ) ) ) ).
% set_times_elim
thf(fact_381_set__times__elim,axiom,
! [X: nat,A4: set_nat,B: set_nat] :
( ( member_nat2 @ X @ ( times_times_set_nat @ A4 @ B ) )
=> ~ ! [A: nat,B4: nat] :
( ( X
= ( times_times_nat @ A @ B4 ) )
=> ( ( member_nat2 @ A @ A4 )
=> ~ ( member_nat2 @ B4 @ B ) ) ) ) ).
% set_times_elim
thf(fact_382_set__times__elim,axiom,
! [X: complex,A4: set_complex,B: set_complex] :
( ( member_complex2 @ X @ ( times_6048082448287401577omplex @ A4 @ B ) )
=> ~ ! [A: complex,B4: complex] :
( ( X
= ( times_times_complex @ A @ B4 ) )
=> ( ( member_complex2 @ A @ A4 )
=> ~ ( member_complex2 @ B4 @ B ) ) ) ) ).
% set_times_elim
thf(fact_383_set__times__intro,axiom,
! [A3: mat_complex,C: set_mat_complex,B2: mat_complex,D: set_mat_complex] :
( ( member_mat_complex @ A3 @ C )
=> ( ( member_mat_complex @ B2 @ D )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A3 @ B2 ) @ ( times_6731331324747250370omplex @ C @ D ) ) ) ) ).
% set_times_intro
thf(fact_384_set__times__intro,axiom,
! [A3: nat,C: set_nat,B2: nat,D: set_nat] :
( ( member_nat2 @ A3 @ C )
=> ( ( member_nat2 @ B2 @ D )
=> ( member_nat2 @ ( times_times_nat @ A3 @ B2 ) @ ( times_times_set_nat @ C @ D ) ) ) ) ).
% set_times_intro
thf(fact_385_set__times__intro,axiom,
! [A3: complex,C: set_complex,B2: complex,D: set_complex] :
( ( member_complex2 @ A3 @ C )
=> ( ( member_complex2 @ B2 @ D )
=> ( member_complex2 @ ( times_times_complex @ A3 @ B2 ) @ ( times_6048082448287401577omplex @ C @ D ) ) ) ) ).
% set_times_intro
thf(fact_386_emptyE,axiom,
! [A3: nat] :
~ ( member_nat2 @ A3 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_387_emptyE,axiom,
! [A3: mat_complex] :
~ ( member_mat_complex @ A3 @ bot_bo7165004461764951667omplex ) ).
% emptyE
thf(fact_388_equals0D,axiom,
! [A4: set_nat,A3: nat] :
( ( A4 = bot_bot_set_nat )
=> ~ ( member_nat2 @ A3 @ A4 ) ) ).
% equals0D
thf(fact_389_equals0D,axiom,
! [A4: set_mat_complex,A3: mat_complex] :
( ( A4 = bot_bo7165004461764951667omplex )
=> ~ ( member_mat_complex @ A3 @ A4 ) ) ).
% equals0D
thf(fact_390_equals0I,axiom,
! [A4: set_nat] :
( ! [Y: nat] :
~ ( member_nat2 @ Y @ A4 )
=> ( A4 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_391_equals0I,axiom,
! [A4: set_mat_complex] :
( ! [Y: mat_complex] :
~ ( member_mat_complex @ Y @ A4 )
=> ( A4 = bot_bo7165004461764951667omplex ) ) ).
% equals0I
thf(fact_392_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat2 @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_393_empty__iff,axiom,
! [C2: mat_complex] :
~ ( member_mat_complex @ C2 @ bot_bo7165004461764951667omplex ) ).
% empty_iff
thf(fact_394_ex__in__conv,axiom,
! [A4: set_nat] :
( ( ? [X3: nat] : ( member_nat2 @ X3 @ A4 ) )
= ( A4 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_395_ex__in__conv,axiom,
! [A4: set_mat_complex] :
( ( ? [X3: mat_complex] : ( member_mat_complex @ X3 @ A4 ) )
= ( A4 != bot_bo7165004461764951667omplex ) ) ).
% ex_in_conv
thf(fact_396_all__not__in__conv,axiom,
! [A4: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat2 @ X3 @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_397_all__not__in__conv,axiom,
! [A4: set_mat_complex] :
( ( ! [X3: mat_complex] :
~ ( member_mat_complex @ X3 @ A4 ) )
= ( A4 = bot_bo7165004461764951667omplex ) ) ).
% all_not_in_conv
thf(fact_398_Collect__empty__eq,axiom,
! [P: mat_complex > $o] :
( ( ( collect_mat_complex @ P )
= bot_bo7165004461764951667omplex )
= ( ! [X3: mat_complex] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_399_empty__Collect__eq,axiom,
! [P: mat_complex > $o] :
( ( bot_bo7165004461764951667omplex
= ( collect_mat_complex @ P ) )
= ( ! [X3: mat_complex] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_400_finite__set__times,axiom,
! [S2: set_mat_complex,T2: set_mat_complex] :
( ( finite7047982916621727056omplex @ S2 )
=> ( ( finite7047982916621727056omplex @ T2 )
=> ( finite7047982916621727056omplex @ ( times_6731331324747250370omplex @ S2 @ T2 ) ) ) ) ).
% finite_set_times
thf(fact_401_finite__set__times,axiom,
! [S2: set_nat,T2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( finite_finite_nat @ ( times_times_set_nat @ S2 @ T2 ) ) ) ) ).
% finite_set_times
thf(fact_402_singletonD,axiom,
! [B2: nat,A3: nat] :
( ( member_nat2 @ B2 @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
=> ( B2 = A3 ) ) ).
% singletonD
thf(fact_403_singletonD,axiom,
! [B2: mat_complex,A3: mat_complex] :
( ( member_mat_complex @ B2 @ ( insert_mat_complex @ A3 @ bot_bo7165004461764951667omplex ) )
=> ( B2 = A3 ) ) ).
% singletonD
thf(fact_404_singletonI,axiom,
! [A3: nat] : ( member_nat2 @ A3 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_405_singletonI,axiom,
! [A3: mat_complex] : ( member_mat_complex @ A3 @ ( insert_mat_complex @ A3 @ bot_bo7165004461764951667omplex ) ) ).
% singletonI
thf(fact_406_singleton__iff,axiom,
! [B2: nat,A3: nat] :
( ( member_nat2 @ B2 @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
= ( B2 = A3 ) ) ).
% singleton_iff
thf(fact_407_singleton__iff,axiom,
! [B2: mat_complex,A3: mat_complex] :
( ( member_mat_complex @ B2 @ ( insert_mat_complex @ A3 @ bot_bo7165004461764951667omplex ) )
= ( B2 = A3 ) ) ).
% singleton_iff
thf(fact_408_doubleton__eq__iff,axiom,
! [A3: mat_complex,B2: mat_complex,C2: mat_complex,D2: mat_complex] :
( ( ( insert_mat_complex @ A3 @ ( insert_mat_complex @ B2 @ bot_bo7165004461764951667omplex ) )
= ( insert_mat_complex @ C2 @ ( insert_mat_complex @ D2 @ bot_bo7165004461764951667omplex ) ) )
= ( ( ( A3 = C2 )
& ( B2 = D2 ) )
| ( ( A3 = D2 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_409_empty__not__insert,axiom,
! [A3: mat_complex,A4: set_mat_complex] :
( bot_bo7165004461764951667omplex
!= ( insert_mat_complex @ A3 @ A4 ) ) ).
% empty_not_insert
thf(fact_410_singleton__inject,axiom,
! [A3: mat_complex,B2: mat_complex] :
( ( ( insert_mat_complex @ A3 @ bot_bo7165004461764951667omplex )
= ( insert_mat_complex @ B2 @ bot_bo7165004461764951667omplex ) )
=> ( A3 = B2 ) ) ).
% singleton_inject
thf(fact_411_Diff__empty,axiom,
! [A4: set_mat_complex] :
( ( minus_8760755521168068590omplex @ A4 @ bot_bo7165004461764951667omplex )
= A4 ) ).
% Diff_empty
thf(fact_412_empty__Diff,axiom,
! [A4: set_mat_complex] :
( ( minus_8760755521168068590omplex @ bot_bo7165004461764951667omplex @ A4 )
= bot_bo7165004461764951667omplex ) ).
% empty_Diff
thf(fact_413_Diff__cancel,axiom,
! [A4: set_mat_complex] :
( ( minus_8760755521168068590omplex @ A4 @ A4 )
= bot_bo7165004461764951667omplex ) ).
% Diff_cancel
thf(fact_414_hermitian__minus,axiom,
! [A4: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple8306762464034002205omplex @ A4 )
=> ( ( comple8306762464034002205omplex @ B )
=> ( comple8306762464034002205omplex @ ( minus_2412168080157227406omplex @ A4 @ B ) ) ) ) ) ) ).
% hermitian_minus
thf(fact_415_cross3__simps_I28_J,axiom,
! [A3: complex,B2: complex,C2: complex] :
( ( times_times_complex @ ( minus_minus_complex @ A3 @ B2 ) @ C2 )
= ( minus_minus_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B2 @ C2 ) ) ) ).
% cross3_simps(28)
thf(fact_416_cross3__simps_I27_J,axiom,
! [A3: complex,B2: complex,C2: complex] :
( ( times_times_complex @ A3 @ ( minus_minus_complex @ B2 @ C2 ) )
= ( minus_minus_complex @ ( times_times_complex @ A3 @ B2 ) @ ( times_times_complex @ A3 @ C2 ) ) ) ).
% cross3_simps(27)
thf(fact_417_cross3__simps_I26_J,axiom,
! [B2: nat,C2: nat,A3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B2 @ C2 ) @ A3 )
= ( minus_minus_nat @ ( times_times_nat @ B2 @ A3 ) @ ( times_times_nat @ C2 @ A3 ) ) ) ).
% cross3_simps(26)
thf(fact_418_cross3__simps_I26_J,axiom,
! [B2: complex,C2: complex,A3: complex] :
( ( times_times_complex @ ( minus_minus_complex @ B2 @ C2 ) @ A3 )
= ( minus_minus_complex @ ( times_times_complex @ B2 @ A3 ) @ ( times_times_complex @ C2 @ A3 ) ) ) ).
% cross3_simps(26)
thf(fact_419_cross3__simps_I25_J,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( times_times_nat @ A3 @ ( minus_minus_nat @ B2 @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A3 @ B2 ) @ ( times_times_nat @ A3 @ C2 ) ) ) ).
% cross3_simps(25)
thf(fact_420_cross3__simps_I25_J,axiom,
! [A3: complex,B2: complex,C2: complex] :
( ( times_times_complex @ A3 @ ( minus_minus_complex @ B2 @ C2 ) )
= ( minus_minus_complex @ ( times_times_complex @ A3 @ B2 ) @ ( times_times_complex @ A3 @ C2 ) ) ) ).
% cross3_simps(25)
thf(fact_421_inf__period_I2_J,axiom,
! [P: complex > $o,D: complex,Q: complex > $o] :
( ! [X2: complex,K: complex] :
( ( P @ X2 )
= ( P @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K @ D ) ) ) )
=> ( ! [X2: complex,K: complex] :
( ( Q @ X2 )
= ( Q @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K @ D ) ) ) )
=> ! [X4: complex,K2: complex] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K2 @ D ) ) )
| ( Q @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K2 @ D ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_422_inf__period_I1_J,axiom,
! [P: complex > $o,D: complex,Q: complex > $o] :
( ! [X2: complex,K: complex] :
( ( P @ X2 )
= ( P @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K @ D ) ) ) )
=> ( ! [X2: complex,K: complex] :
( ( Q @ X2 )
= ( Q @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K @ D ) ) ) )
=> ! [X4: complex,K2: complex] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K2 @ D ) ) )
& ( Q @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K2 @ D ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_423_projector__def,axiom,
( linear5633924348262549461omplex
= ( ^ [M2: mat_complex] :
( ( comple8306762464034002205omplex @ M2 )
& ( ( times_8009071140041733218omplex @ M2 @ M2 )
= M2 ) ) ) ) ).
% projector_def
thf(fact_424_remove__def,axiom,
( remove_mat_complex
= ( ^ [X3: mat_complex,A7: set_mat_complex] : ( minus_8760755521168068590omplex @ A7 @ ( insert_mat_complex @ X3 @ bot_bo7165004461764951667omplex ) ) ) ) ).
% remove_def
thf(fact_425_permutations__of__list__impl__Nil,axiom,
( ( multis1338830684155981141pl_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% permutations_of_list_impl_Nil
thf(fact_426_permutations__of__list__impl__Nil,axiom,
( ( multis6086220127510270003omplex @ nil_complex )
= ( cons_list_complex @ nil_complex @ nil_list_complex ) ) ).
% permutations_of_list_impl_Nil
thf(fact_427_unitary__is__corthogonal,axiom,
! [U: mat_complex,N: nat] :
( ( member_mat_complex @ U @ ( carrier_mat_complex @ N @ N ) )
=> ( ( comple6660659447773130958omplex @ U )
=> ( schur_549222400177443379omplex @ U ) ) ) ).
% unitary_is_corthogonal
thf(fact_428_projector__square__eq,axiom,
! [M3: mat_complex] :
( ( linear5633924348262549461omplex @ M3 )
=> ( ( times_8009071140041733218omplex @ M3 @ M3 )
= M3 ) ) ).
% projector_square_eq
thf(fact_429_projector__hermitian,axiom,
! [M3: mat_complex] :
( ( linear5633924348262549461omplex @ M3 )
=> ( comple8306762464034002205omplex @ M3 ) ) ).
% projector_hermitian
thf(fact_430_Set_Ois__empty__def,axiom,
( is_empty_mat_complex
= ( ^ [A7: set_mat_complex] : ( A7 = bot_bo7165004461764951667omplex ) ) ) ).
% Set.is_empty_def
thf(fact_431_map__eq__map__tailrec,axiom,
map_complex_real = map_ta5686879364588479338x_real ).
% map_eq_map_tailrec
thf(fact_432_forall__vector__2,axiom,
( ( ^ [P3: finite1289000397740218697l_num1 > $o] :
! [X5: finite1289000397740218697l_num1] : ( P3 @ X5 ) )
= ( ^ [P4: finite1289000397740218697l_num1 > $o] :
! [X3: nat,Y3: nat] : ( P4 @ ( cartes7700031802712742009l_num1 @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ) ) ).
% forall_vector_2
thf(fact_433_forall__vector__2,axiom,
( ( ^ [P3: finite279839453003621671l_num1 > $o] :
! [X5: finite279839453003621671l_num1] : ( P3 @ X5 ) )
= ( ^ [P4: finite279839453003621671l_num1 > $o] :
! [X3: complex,Y3: complex] : ( P4 @ ( cartes2146103711486359127l_num1 @ ( cons_complex @ X3 @ ( cons_complex @ Y3 @ nil_complex ) ) ) ) ) ) ).
% forall_vector_2
thf(fact_434_is__singleton__def,axiom,
( is_sin1068006998250866843omplex
= ( ^ [A7: set_mat_complex] :
? [X3: mat_complex] :
( A7
= ( insert_mat_complex @ X3 @ bot_bo7165004461764951667omplex ) ) ) ) ).
% is_singleton_def
thf(fact_435_is__singletonI,axiom,
! [X: mat_complex] : ( is_sin1068006998250866843omplex @ ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) ) ).
% is_singletonI
thf(fact_436_is__singletonE,axiom,
! [A4: set_mat_complex] :
( ( is_sin1068006998250866843omplex @ A4 )
=> ~ ! [X2: mat_complex] :
( A4
!= ( insert_mat_complex @ X2 @ bot_bo7165004461764951667omplex ) ) ) ).
% is_singletonE
thf(fact_437_is__singletonI_H,axiom,
! [A4: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X2: nat,Y: nat] :
( ( member_nat2 @ X2 @ A4 )
=> ( ( member_nat2 @ Y @ A4 )
=> ( X2 = Y ) ) )
=> ( is_singleton_nat @ A4 ) ) ) ).
% is_singletonI'
thf(fact_438_is__singletonI_H,axiom,
! [A4: set_mat_complex] :
( ( A4 != bot_bo7165004461764951667omplex )
=> ( ! [X2: mat_complex,Y: mat_complex] :
( ( member_mat_complex @ X2 @ A4 )
=> ( ( member_mat_complex @ Y @ A4 )
=> ( X2 = Y ) ) )
=> ( is_sin1068006998250866843omplex @ A4 ) ) ) ).
% is_singletonI'
thf(fact_439_is__singleton__the__elem,axiom,
( is_sin1068006998250866843omplex
= ( ^ [A7: set_mat_complex] :
( A7
= ( insert_mat_complex @ ( the_elem_mat_complex @ A7 ) @ bot_bo7165004461764951667omplex ) ) ) ) ).
% is_singleton_the_elem
thf(fact_440_vector__space__over__itself_Ospan__breakdown,axiom,
! [B2: complex,S: set_complex,A3: complex] :
( ( member_complex2 @ B2 @ S )
=> ( ( member_complex2 @ A3 @ ( span_complex_complex @ times_times_complex @ S ) )
=> ? [K: complex] : ( member_complex2 @ ( minus_minus_complex @ A3 @ ( times_times_complex @ K @ B2 ) ) @ ( span_complex_complex @ times_times_complex @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ B2 @ bot_bot_set_complex ) ) ) ) ) ) ).
% vector_space_over_itself.span_breakdown
thf(fact_441_permutations__of__set__doubleton,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ( multis1655833086286526861et_nat @ ( insert_nat @ X @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
= ( insert_list_nat @ ( cons_nat @ X @ ( cons_nat @ Y2 @ nil_nat ) ) @ ( insert_list_nat @ ( cons_nat @ Y2 @ ( cons_nat @ X @ nil_nat ) ) @ bot_bot_set_list_nat ) ) ) ) ).
% permutations_of_set_doubleton
thf(fact_442_permutations__of__set__doubleton,axiom,
! [X: complex,Y2: complex] :
( ( X != Y2 )
=> ( ( multis1932168107469466731omplex @ ( insert_complex @ X @ ( insert_complex @ Y2 @ bot_bot_set_complex ) ) )
= ( insert_list_complex @ ( cons_complex @ X @ ( cons_complex @ Y2 @ nil_complex ) ) @ ( insert_list_complex @ ( cons_complex @ Y2 @ ( cons_complex @ X @ nil_complex ) ) @ bot_bo6492010485567502472omplex ) ) ) ) ).
% permutations_of_set_doubleton
thf(fact_443_permutations__of__set__doubleton,axiom,
! [X: mat_complex,Y2: mat_complex] :
( ( X != Y2 )
=> ( ( multis2234121214477010346omplex @ ( insert_mat_complex @ X @ ( insert_mat_complex @ Y2 @ bot_bo7165004461764951667omplex ) ) )
= ( insert8600743036574769149omplex @ ( cons_mat_complex @ X @ ( cons_mat_complex @ Y2 @ nil_mat_complex ) ) @ ( insert8600743036574769149omplex @ ( cons_mat_complex @ Y2 @ ( cons_mat_complex @ X @ nil_mat_complex ) ) @ bot_bo6377478972893813113omplex ) ) ) ) ).
% permutations_of_set_doubleton
thf(fact_444_permutations__of__set__singleton,axiom,
! [X: nat] :
( ( multis1655833086286526861et_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= ( insert_list_nat @ ( cons_nat @ X @ nil_nat ) @ bot_bot_set_list_nat ) ) ).
% permutations_of_set_singleton
thf(fact_445_permutations__of__set__singleton,axiom,
! [X: complex] :
( ( multis1932168107469466731omplex @ ( insert_complex @ X @ bot_bot_set_complex ) )
= ( insert_list_complex @ ( cons_complex @ X @ nil_complex ) @ bot_bo6492010485567502472omplex ) ) ).
% permutations_of_set_singleton
thf(fact_446_permutations__of__set__singleton,axiom,
! [X: mat_complex] :
( ( multis2234121214477010346omplex @ ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) )
= ( insert8600743036574769149omplex @ ( cons_mat_complex @ X @ nil_mat_complex ) @ bot_bo6377478972893813113omplex ) ) ).
% permutations_of_set_singleton
thf(fact_447_assms_I4_J,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% assms(4)
thf(fact_448_Suc_Oprems_I5_J,axiom,
ord_less_nat @ zero_zero_nat @ na ).
% Suc.prems(5)
thf(fact_449_vector__space__over__itself_Ospan__insert__0,axiom,
! [S: set_real] :
( ( span_real_real @ times_times_real @ ( insert_real @ zero_zero_real @ S ) )
= ( span_real_real @ times_times_real @ S ) ) ).
% vector_space_over_itself.span_insert_0
thf(fact_450_vector__space__over__itself_Ospan__insert__0,axiom,
! [S: set_complex] :
( ( span_complex_complex @ times_times_complex @ ( insert_complex @ zero_zero_complex @ S ) )
= ( span_complex_complex @ times_times_complex @ S ) ) ).
% vector_space_over_itself.span_insert_0
thf(fact_451_mult__sign__intros_I8_J,axiom,
! [A3: real,B2: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B2 ) ) ) ) ).
% mult_sign_intros(8)
thf(fact_452_mult__sign__intros_I7_J,axiom,
! [A3: real,B2: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(7)
thf(fact_453_mult__sign__intros_I7_J,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(7)
thf(fact_454_mult__sign__intros_I6_J,axiom,
! [A3: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(6)
thf(fact_455_mult__sign__intros_I6_J,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(6)
thf(fact_456_mult__sign__intros_I5_J,axiom,
! [A3: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B2 ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_457_mult__sign__intros_I5_J,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B2 ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_458_not__square__less__zero,axiom,
! [A3: real] :
~ ( ord_less_real @ ( times_times_real @ A3 @ A3 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_459_mult__less__0__iff,axiom,
! [A3: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_460_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B2 @ A3 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_461_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B2 @ A3 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_462_zero__less__mult__iff,axiom,
! [A3: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_463_zero__less__mult__pos,axiom,
! [A3: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_464_zero__less__mult__pos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_465_zero__less__mult__pos2,axiom,
! [B2: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_466_zero__less__mult__pos2,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_467_mult__less__cancel__left__neg,axiom,
! [C2: real,A3: real,B2: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_real @ B2 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_468_mult__less__cancel__left__pos,axiom,
! [C2: real,A3: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_real @ A3 @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_469_mult__strict__left__mono__neg,axiom,
! [B2: real,A3: real,C2: real] :
( ( ord_less_real @ B2 @ A3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_470_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: real,B2: real,C2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_471_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_472_mult__less__cancel__left__disj,axiom,
! [C2: real,A3: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A3 @ B2 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_473_mult__strict__right__mono__neg,axiom,
! [B2: real,A3: real,C2: real] :
( ( ord_less_real @ B2 @ A3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_474_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: real,B2: real,C2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_475_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_476_mult__less__cancel__right__disj,axiom,
! [A3: real,C2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A3 @ B2 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_477_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: real,B2: real,C2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_478_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_479_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_480_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_481_zero__less__one__class_Ozero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_less_one
thf(fact_482_zero__less__one__class_Ozero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_less_one
thf(fact_483_rel__simps_I68_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% rel_simps(68)
thf(fact_484_rel__simps_I68_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% rel_simps(68)
thf(fact_485_unbounded__k__infinite,axiom,
! [K3: nat,S: set_nat] :
( ! [M4: nat] :
( ( ord_less_nat @ K3 @ M4 )
=> ? [N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ( member_nat2 @ N2 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_486_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M5: nat] :
? [N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ( member_nat2 @ N3 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_487_vector__space__over__itself_Ospan__clauses_I2_J,axiom,
! [S: set_real] : ( member_real @ zero_zero_real @ ( span_real_real @ times_times_real @ S ) ) ).
% vector_space_over_itself.span_clauses(2)
thf(fact_488_vector__space__over__itself_Ospan__clauses_I2_J,axiom,
! [S: set_complex] : ( member_complex2 @ zero_zero_complex @ ( span_complex_complex @ times_times_complex @ S ) ) ).
% vector_space_over_itself.span_clauses(2)
thf(fact_489_vector__space__over__itself_Ospan__clauses_I4_J,axiom,
! [X: complex,S: set_complex,C2: complex] :
( ( member_complex2 @ X @ ( span_complex_complex @ times_times_complex @ S ) )
=> ( member_complex2 @ ( times_times_complex @ C2 @ X ) @ ( span_complex_complex @ times_times_complex @ S ) ) ) ).
% vector_space_over_itself.span_clauses(4)
thf(fact_490_vector__space__over__itself_Ospan__clauses_I1_J,axiom,
! [A3: complex,S: set_complex] :
( ( member_complex2 @ A3 @ S )
=> ( member_complex2 @ A3 @ ( span_complex_complex @ times_times_complex @ S ) ) ) ).
% vector_space_over_itself.span_clauses(1)
thf(fact_491_vector__space__over__itself_Ospan__span,axiom,
! [A4: set_complex] :
( ( span_complex_complex @ times_times_complex @ ( span_complex_complex @ times_times_complex @ A4 ) )
= ( span_complex_complex @ times_times_complex @ A4 ) ) ).
% vector_space_over_itself.span_span
thf(fact_492_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_493_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_494_rev_Osimps_I1_J,axiom,
( ( rev_nat @ nil_nat )
= nil_nat ) ).
% rev.simps(1)
thf(fact_495_rev_Osimps_I1_J,axiom,
( ( rev_complex @ nil_complex )
= nil_complex ) ).
% rev.simps(1)
thf(fact_496_Nil__is__rev__conv,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( rev_nat @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_rev_conv
thf(fact_497_Nil__is__rev__conv,axiom,
! [Xs: list_complex] :
( ( nil_complex
= ( rev_complex @ Xs ) )
= ( Xs = nil_complex ) ) ).
% Nil_is_rev_conv
thf(fact_498_rev__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rev_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rev_is_Nil_conv
thf(fact_499_rev__is__Nil__conv,axiom,
! [Xs: list_complex] :
( ( ( rev_complex @ Xs )
= nil_complex )
= ( Xs = nil_complex ) ) ).
% rev_is_Nil_conv
thf(fact_500_set__zero,axiom,
( zero_zero_set_nat
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% set_zero
thf(fact_501_set__zero,axiom,
( zero_zero_set_real
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) ).
% set_zero
thf(fact_502_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_503_rev__map,axiom,
! [F: complex > real,Xs: list_complex] :
( ( rev_real @ ( map_complex_real @ F @ Xs ) )
= ( map_complex_real @ F @ ( rev_complex @ Xs ) ) ) ).
% rev_map
thf(fact_504_rel__simps_I71_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% rel_simps(71)
thf(fact_505_rel__simps_I71_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% rel_simps(71)
thf(fact_506_finite__maxlen,axiom,
! [M3: set_list_nat] :
( ( finite8100373058378681591st_nat @ M3 )
=> ? [N4: nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ M3 )
=> ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N4 ) ) ) ).
% finite_maxlen
thf(fact_507_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys6: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys6 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys6 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_508_arithmetic__simps_I62_J,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% arithmetic_simps(62)
thf(fact_509_arithmetic__simps_I62_J,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% arithmetic_simps(62)
thf(fact_510_arithmetic__simps_I62_J,axiom,
! [A3: complex] :
( ( times_times_complex @ zero_zero_complex @ A3 )
= zero_zero_complex ) ).
% arithmetic_simps(62)
thf(fact_511_arithmetic__simps_I63_J,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% arithmetic_simps(63)
thf(fact_512_arithmetic__simps_I63_J,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% arithmetic_simps(63)
thf(fact_513_arithmetic__simps_I63_J,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ zero_zero_complex )
= zero_zero_complex ) ).
% arithmetic_simps(63)
thf(fact_514_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A3: real,B2: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ B2 @ X ) )
=> ( A3 = B2 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_515_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: complex,A3: complex,B2: complex] :
( ( X != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ B2 @ X ) )
=> ( A3 = B2 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_516_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: real,X: real,B2: real] :
( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ B2 @ X ) )
= ( ( A3 = B2 )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_517_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: complex,X: complex,B2: complex] :
( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ B2 @ X ) )
= ( ( A3 = B2 )
| ( X = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_518_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: real,X: real,Y2: real] :
( ( A3 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ A3 @ Y2 ) )
=> ( X = Y2 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_519_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: complex,X: complex,Y2: complex] :
( ( A3 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ A3 @ Y2 ) )
=> ( X = Y2 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_520_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: real,X: real,Y2: real] :
( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ A3 @ Y2 ) )
= ( ( X = Y2 )
| ( A3 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_521_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: complex,X: complex,Y2: complex] :
( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ A3 @ Y2 ) )
= ( ( X = Y2 )
| ( A3 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_522_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_523_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_524_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_525_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: complex] :
( ( times_times_complex @ zero_zero_complex @ X )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_526_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: real,X: real] :
( ( ( times_times_real @ A3 @ X )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_527_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: complex,X: complex] :
( ( ( times_times_complex @ A3 @ X )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( X = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_528_mult__not__zero,axiom,
! [A3: real,B2: real] :
( ( ( times_times_real @ A3 @ B2 )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B2 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_529_mult__not__zero,axiom,
! [A3: nat,B2: nat] :
( ( ( times_times_nat @ A3 @ B2 )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_530_mult__not__zero,axiom,
! [A3: complex,B2: complex] :
( ( ( times_times_complex @ A3 @ B2 )
!= zero_zero_complex )
=> ( ( A3 != zero_zero_complex )
& ( B2 != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_531_divisors__zero,axiom,
! [A3: real,B2: real] :
( ( ( times_times_real @ A3 @ B2 )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_532_divisors__zero,axiom,
! [A3: nat,B2: nat] :
( ( ( times_times_nat @ A3 @ B2 )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_533_divisors__zero,axiom,
! [A3: complex,B2: complex] :
( ( ( times_times_complex @ A3 @ B2 )
= zero_zero_complex )
=> ( ( A3 = zero_zero_complex )
| ( B2 = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_534_mult__eq__0__iff,axiom,
! [A3: real,B2: real] :
( ( ( times_times_real @ A3 @ B2 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_535_mult__eq__0__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( times_times_nat @ A3 @ B2 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_536_mult__eq__0__iff,axiom,
! [A3: complex,B2: complex] :
( ( ( times_times_complex @ A3 @ B2 )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( B2 = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_537_no__zero__divisors,axiom,
! [A3: real,B2: real] :
( ( A3 != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( times_times_real @ A3 @ B2 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_538_no__zero__divisors,axiom,
! [A3: nat,B2: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_539_no__zero__divisors,axiom,
! [A3: complex,B2: complex] :
( ( A3 != zero_zero_complex )
=> ( ( B2 != zero_zero_complex )
=> ( ( times_times_complex @ A3 @ B2 )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_540_mult__cancel__left,axiom,
! [C2: real,A3: real,B2: real] :
( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B2 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_541_mult__cancel__left,axiom,
! [C2: nat,A3: nat,B2: nat] :
( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B2 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_542_mult__cancel__left,axiom,
! [C2: complex,A3: complex,B2: complex] :
( ( ( times_times_complex @ C2 @ A3 )
= ( times_times_complex @ C2 @ B2 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_543_mult__left__cancel,axiom,
! [C2: real,A3: real,B2: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B2 ) )
= ( A3 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_544_mult__left__cancel,axiom,
! [C2: nat,A3: nat,B2: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B2 ) )
= ( A3 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_545_mult__left__cancel,axiom,
! [C2: complex,A3: complex,B2: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ C2 @ A3 )
= ( times_times_complex @ C2 @ B2 ) )
= ( A3 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_546_mult__cancel__right,axiom,
! [A3: real,C2: real,B2: real] :
( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B2 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_547_mult__cancel__right,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B2 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_548_mult__cancel__right,axiom,
! [A3: complex,C2: complex,B2: complex] :
( ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B2 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( A3 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_549_mult__right__cancel,axiom,
! [C2: real,A3: real,B2: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B2 @ C2 ) )
= ( A3 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_550_mult__right__cancel,axiom,
! [C2: nat,A3: nat,B2: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B2 @ C2 ) )
= ( A3 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_551_mult__right__cancel,axiom,
! [C2: complex,A3: complex,B2: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B2 @ C2 ) )
= ( A3 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_552_verit__eq__simplify_I7_J,axiom,
zero_zero_nat != one_one_nat ).
% verit_eq_simplify(7)
thf(fact_553_verit__eq__simplify_I7_J,axiom,
zero_zero_real != one_one_real ).
% verit_eq_simplify(7)
thf(fact_554_permutations__of__set__infinite,axiom,
! [A4: set_mat_complex] :
( ~ ( finite7047982916621727056omplex @ A4 )
=> ( ( multis2234121214477010346omplex @ A4 )
= bot_bo6377478972893813113omplex ) ) ).
% permutations_of_set_infinite
thf(fact_555_permutations__of__set__infinite,axiom,
! [A4: set_nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( multis1655833086286526861et_nat @ A4 )
= bot_bot_set_list_nat ) ) ).
% permutations_of_set_infinite
thf(fact_556_permutations__of__set__empty__iff,axiom,
! [A4: set_mat_complex] :
( ( ( multis2234121214477010346omplex @ A4 )
= bot_bo6377478972893813113omplex )
= ( ~ ( finite7047982916621727056omplex @ A4 ) ) ) ).
% permutations_of_set_empty_iff
thf(fact_557_permutations__of__set__empty__iff,axiom,
! [A4: set_nat] :
( ( ( multis1655833086286526861et_nat @ A4 )
= bot_bot_set_list_nat )
= ( ~ ( finite_finite_nat @ A4 ) ) ) ).
% permutations_of_set_empty_iff
thf(fact_558_vector__space__over__itself_Ospan__empty,axiom,
( ( span_real_real @ times_times_real @ bot_bot_set_real )
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) ).
% vector_space_over_itself.span_empty
thf(fact_559_vector__space__over__itself_Ospan__empty,axiom,
( ( span_complex_complex @ times_times_complex @ bot_bot_set_complex )
= ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) ).
% vector_space_over_itself.span_empty
thf(fact_560_vector__space__over__itself_Ospan__diff,axiom,
! [X: complex,S: set_complex,Y2: complex] :
( ( member_complex2 @ X @ ( span_complex_complex @ times_times_complex @ S ) )
=> ( ( member_complex2 @ Y2 @ ( span_complex_complex @ times_times_complex @ S ) )
=> ( member_complex2 @ ( minus_minus_complex @ X @ Y2 ) @ ( span_complex_complex @ times_times_complex @ S ) ) ) ) ).
% vector_space_over_itself.span_diff
thf(fact_561_vector__space__over__itself_Ospan__redundant,axiom,
! [X: complex,S: set_complex] :
( ( member_complex2 @ X @ ( span_complex_complex @ times_times_complex @ S ) )
=> ( ( span_complex_complex @ times_times_complex @ ( insert_complex @ X @ S ) )
= ( span_complex_complex @ times_times_complex @ S ) ) ) ).
% vector_space_over_itself.span_redundant
thf(fact_562_vector__space__over__itself_Oin__span__insert,axiom,
! [A3: complex,B2: complex,S: set_complex] :
( ( member_complex2 @ A3 @ ( span_complex_complex @ times_times_complex @ ( insert_complex @ B2 @ S ) ) )
=> ( ~ ( member_complex2 @ A3 @ ( span_complex_complex @ times_times_complex @ S ) )
=> ( member_complex2 @ B2 @ ( span_complex_complex @ times_times_complex @ ( insert_complex @ A3 @ S ) ) ) ) ) ).
% vector_space_over_itself.in_span_insert
thf(fact_563_vector__space__over__itself_Ospan__trans,axiom,
! [X: complex,S: set_complex,Y2: complex] :
( ( member_complex2 @ X @ ( span_complex_complex @ times_times_complex @ S ) )
=> ( ( member_complex2 @ Y2 @ ( span_complex_complex @ times_times_complex @ ( insert_complex @ X @ S ) ) )
=> ( member_complex2 @ Y2 @ ( span_complex_complex @ times_times_complex @ S ) ) ) ) ).
% vector_space_over_itself.span_trans
thf(fact_564_vector__space__over__itself_Ospan__delete__0,axiom,
! [S: set_real] :
( ( span_real_real @ times_times_real @ ( minus_minus_set_real @ S @ ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
= ( span_real_real @ times_times_real @ S ) ) ).
% vector_space_over_itself.span_delete_0
thf(fact_565_vector__space__over__itself_Ospan__delete__0,axiom,
! [S: set_complex] :
( ( span_complex_complex @ times_times_complex @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) )
= ( span_complex_complex @ times_times_complex @ S ) ) ).
% vector_space_over_itself.span_delete_0
thf(fact_566_singleton__rev__conv,axiom,
! [X: nat,Xs: list_nat] :
( ( ( cons_nat @ X @ nil_nat )
= ( rev_nat @ Xs ) )
= ( ( cons_nat @ X @ nil_nat )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_567_singleton__rev__conv,axiom,
! [X: complex,Xs: list_complex] :
( ( ( cons_complex @ X @ nil_complex )
= ( rev_complex @ Xs ) )
= ( ( cons_complex @ X @ nil_complex )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_568_rev__singleton__conv,axiom,
! [Xs: list_nat,X: nat] :
( ( ( rev_nat @ Xs )
= ( cons_nat @ X @ nil_nat ) )
= ( Xs
= ( cons_nat @ X @ nil_nat ) ) ) ).
% rev_singleton_conv
thf(fact_569_rev__singleton__conv,axiom,
! [Xs: list_complex,X: complex] :
( ( ( rev_complex @ Xs )
= ( cons_complex @ X @ nil_complex ) )
= ( Xs
= ( cons_complex @ X @ nil_complex ) ) ) ).
% rev_singleton_conv
thf(fact_570_length__concat__rev,axiom,
! [Xs: list_list_nat] :
( ( size_size_list_nat @ ( concat_nat @ ( rev_list_nat @ Xs ) ) )
= ( size_size_list_nat @ ( concat_nat @ Xs ) ) ) ).
% length_concat_rev
thf(fact_571_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_572_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_573_sorted__list__subset_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [A: nat,As: list_nat,B4: nat,Bs: list_nat] :
( ( ( A = B4 )
=> ( P @ As @ ( cons_nat @ B4 @ Bs ) ) )
=> ( ( ( A != B4 )
=> ( ( ord_less_nat @ B4 @ A )
=> ( P @ ( cons_nat @ A @ As ) @ Bs ) ) )
=> ( P @ ( cons_nat @ A @ As ) @ ( cons_nat @ B4 @ Bs ) ) ) )
=> ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [A: nat,Uv2: list_nat] : ( P @ ( cons_nat @ A @ Uv2 ) @ nil_nat )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sorted_list_subset.induct
thf(fact_574_mult__cancel__left1,axiom,
! [C2: real,B2: real] :
( ( C2
= ( times_times_real @ C2 @ B2 ) )
= ( ( C2 = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_575_mult__cancel__left1,axiom,
! [C2: complex,B2: complex] :
( ( C2
= ( times_times_complex @ C2 @ B2 ) )
= ( ( C2 = zero_zero_complex )
| ( B2 = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_576_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_577_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_578_mult__cancel__right1,axiom,
! [C2: real,B2: real] :
( ( C2
= ( times_times_real @ B2 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_579_mult__cancel__right1,axiom,
! [C2: complex,B2: complex] :
( ( C2
= ( times_times_complex @ B2 @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( B2 = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_580_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_581_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_582_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_583_permutations__of__set__empty,axiom,
( ( multis1655833086286526861et_nat @ bot_bot_set_nat )
= ( insert_list_nat @ nil_nat @ bot_bot_set_list_nat ) ) ).
% permutations_of_set_empty
thf(fact_584_permutations__of__set__empty,axiom,
( ( multis1932168107469466731omplex @ bot_bot_set_complex )
= ( insert_list_complex @ nil_complex @ bot_bo6492010485567502472omplex ) ) ).
% permutations_of_set_empty
thf(fact_585_permutations__of__set__empty,axiom,
( ( multis2234121214477010346omplex @ bot_bo7165004461764951667omplex )
= ( insert8600743036574769149omplex @ nil_mat_complex @ bot_bo6377478972893813113omplex ) ) ).
% permutations_of_set_empty
thf(fact_586_list_Osize_I3_J,axiom,
( ( size_s3451745648224563538omplex @ nil_complex )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_587_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_588_length__0__conv,axiom,
! [Xs: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= zero_zero_nat )
= ( Xs = nil_complex ) ) ).
% length_0_conv
thf(fact_589_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_590_unitary__zero,axiom,
! [A4: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ zero_zero_nat @ zero_zero_nat ) )
=> ( comple6660659447773130958omplex @ A4 ) ) ).
% unitary_zero
thf(fact_591_eq__comps__neq__0,axiom,
! [A3: nat,M: list_nat,L: list_complex] :
( ( ( cons_nat @ A3 @ M )
= ( commut93809757773076895omplex @ L ) )
=> ( A3 != zero_zero_nat ) ) ).
% eq_comps_neq_0
thf(fact_592_eq__comps__neq__0,axiom,
! [A3: nat,M: list_nat,L: list_nat] :
( ( ( cons_nat @ A3 @ M )
= ( commut2436974278740741825ps_nat @ L ) )
=> ( A3 != zero_zero_nat ) ) ).
% eq_comps_neq_0
thf(fact_593_vector__space__over__itself_Ospan__breakdown__eq,axiom,
! [X: complex,A3: complex,S: set_complex] :
( ( member_complex2 @ X @ ( span_complex_complex @ times_times_complex @ ( insert_complex @ A3 @ S ) ) )
= ( ? [K4: complex] : ( member_complex2 @ ( minus_minus_complex @ X @ ( times_times_complex @ K4 @ A3 ) ) @ ( span_complex_complex @ times_times_complex @ S ) ) ) ) ).
% vector_space_over_itself.span_breakdown_eq
thf(fact_594_vector__space__over__itself_Oeq__span__insert__eq,axiom,
! [X: complex,Y2: complex,S: set_complex] :
( ( member_complex2 @ ( minus_minus_complex @ X @ Y2 ) @ ( span_complex_complex @ times_times_complex @ S ) )
=> ( ( span_complex_complex @ times_times_complex @ ( insert_complex @ X @ S ) )
= ( span_complex_complex @ times_times_complex @ ( insert_complex @ Y2 @ S ) ) ) ) ).
% vector_space_over_itself.eq_span_insert_eq
thf(fact_595_rev__eq__Cons__iff,axiom,
! [Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( rev_nat @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( Xs
= ( append_nat @ ( rev_nat @ Ys ) @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_596_rev__eq__Cons__iff,axiom,
! [Xs: list_complex,Y2: complex,Ys: list_complex] :
( ( ( rev_complex @ Xs )
= ( cons_complex @ Y2 @ Ys ) )
= ( Xs
= ( append_complex @ ( rev_complex @ Ys ) @ ( cons_complex @ Y2 @ nil_complex ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_597_rev_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( rev_nat @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ ( rev_nat @ Xs ) @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rev.simps(2)
thf(fact_598_rev_Osimps_I2_J,axiom,
! [X: complex,Xs: list_complex] :
( ( rev_complex @ ( cons_complex @ X @ Xs ) )
= ( append_complex @ ( rev_complex @ Xs ) @ ( cons_complex @ X @ nil_complex ) ) ) ).
% rev.simps(2)
thf(fact_599_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_600_the__elem__eq,axiom,
! [X: mat_complex] :
( ( the_elem_mat_complex @ ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) )
= X ) ).
% the_elem_eq
thf(fact_601_map__tailrec__rev,axiom,
( map_ta6589641090014469329x_real
= ( ^ [F5: complex > real,As3: list_complex] : ( append_real @ ( rev_real @ ( map_complex_real @ F5 @ As3 ) ) ) ) ) ).
% map_tailrec_rev
thf(fact_602_vector__space__over__itself_Ozero__not__in__Basis,axiom,
~ ( member_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) ).
% vector_space_over_itself.zero_not_in_Basis
thf(fact_603_vector__space__over__itself_Oin__span__delete,axiom,
! [A3: complex,S: set_complex,B2: complex] :
( ( member_complex2 @ A3 @ ( span_complex_complex @ times_times_complex @ S ) )
=> ( ~ ( member_complex2 @ A3 @ ( span_complex_complex @ times_times_complex @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ B2 @ bot_bot_set_complex ) ) ) )
=> ( member_complex2 @ B2 @ ( span_complex_complex @ times_times_complex @ ( insert_complex @ A3 @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ B2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% vector_space_over_itself.in_span_delete
thf(fact_604_finite__linorder__max__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A5 )
=> ( ord_less_nat @ X4 @ B4 ) )
=> ( ( P @ A5 )
=> ( P @ ( insert_nat @ B4 @ A5 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_605_finite__linorder__min__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A5 )
=> ( ord_less_nat @ B4 @ X4 ) )
=> ( ( P @ A5 )
=> ( P @ ( insert_nat @ B4 @ A5 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_606_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_607_ex__min__if__finite,axiom,
! [S: set_mat_complex] :
( ( finite7047982916621727056omplex @ S )
=> ( ( S != bot_bo7165004461764951667omplex )
=> ? [X2: mat_complex] :
( ( member_mat_complex @ X2 @ S )
& ~ ? [Xa2: mat_complex] :
( ( member_mat_complex @ Xa2 @ S )
& ( ord_less_mat_complex @ Xa2 @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_608_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ S )
& ~ ? [Xa2: nat] :
( ( member_nat2 @ Xa2 @ S )
& ( ord_less_nat @ Xa2 @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_609_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ X6 )
=> ? [Xa2: nat] :
( ( member_nat2 @ Xa2 @ X6 )
& ( ord_less_nat @ X2 @ Xa2 ) ) )
=> ~ ( finite_finite_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_610_poly__cancel__eq__conv,axiom,
! [X: real,A3: real,Y2: real,B2: real] :
( ( X = zero_zero_real )
=> ( ( A3 != zero_zero_real )
=> ( ( Y2 = zero_zero_real )
= ( ( minus_minus_real @ ( times_times_real @ A3 @ Y2 ) @ ( times_times_real @ B2 @ X ) )
= zero_zero_real ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_611_poly__cancel__eq__conv,axiom,
! [X: complex,A3: complex,Y2: complex,B2: complex] :
( ( X = zero_zero_complex )
=> ( ( A3 != zero_zero_complex )
=> ( ( Y2 = zero_zero_complex )
= ( ( minus_minus_complex @ ( times_times_complex @ A3 @ Y2 ) @ ( times_times_complex @ B2 @ X ) )
= zero_zero_complex ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_612_not__psubset__empty,axiom,
! [A4: set_mat_complex] :
~ ( ord_le5598786136212072115omplex @ A4 @ bot_bo7165004461764951667omplex ) ).
% not_psubset_empty
thf(fact_613_finite__psubset__induct,axiom,
! [A4: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ A4 )
=> ( ! [A5: set_mat_complex] :
( ( finite7047982916621727056omplex @ A5 )
=> ( ! [B5: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ B5 @ A5 )
=> ( P @ B5 ) )
=> ( P @ A5 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_614_finite__psubset__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [B5: set_nat] :
( ( ord_less_set_nat @ B5 @ A5 )
=> ( P @ B5 ) )
=> ( P @ A5 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_615_times__nat_Osimps_I1_J,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% times_nat.simps(1)
thf(fact_616_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_617_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_618_mult__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K3 @ M )
= ( times_times_nat @ K3 @ N ) )
= ( ( M = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_619_mult__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ( times_times_nat @ M @ K3 )
= ( times_times_nat @ N @ K3 ) )
= ( ( M = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_620_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_621_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_622_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_623_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_624_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T3: set_nat] :
( ( ord_less_set_nat @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ ( minus_minus_set_nat @ S @ T3 ) )
& ( P @ ( insert_nat @ X4 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_625_finite__induct__select,axiom,
! [S: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ S )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [T3: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X4: mat_complex] :
( ( member_mat_complex @ X4 @ ( minus_8760755521168068590omplex @ S @ T3 ) )
& ( P @ ( insert_mat_complex @ X4 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_626_mult__less__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ) ).
% mult_less_mono1
thf(fact_627_mult__less__mono2,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_628_mult__less__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel1
thf(fact_629_mult__less__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_630_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_631_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_632_mult__if__delta,axiom,
! [P: $o,Q2: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q2 )
= Q2 ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q2 )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_633_mult__if__delta,axiom,
! [P: $o,Q2: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q2 )
= Q2 ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q2 )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_634_mult__if__delta,axiom,
! [P: $o,Q2: complex] :
( ( P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q2 )
= Q2 ) )
& ( ~ P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q2 )
= zero_zero_complex ) ) ) ).
% mult_if_delta
thf(fact_635_nat__distrib_I4_J,axiom,
! [K3: nat,M: nat,N: nat] :
( ( times_times_nat @ K3 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) ) ) ).
% nat_distrib(4)
thf(fact_636_nat__distrib_I3_J,axiom,
! [M: nat,N: nat,K3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K3 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) ) ) ).
% nat_distrib(3)
thf(fact_637_nat__mult__eq__cancel__disj,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K3 @ M )
= ( times_times_nat @ K3 @ N ) )
= ( ( K3 = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_638_nat__mult__less__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_639_nat__mult__eq__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ( times_times_nat @ K3 @ M )
= ( times_times_nat @ K3 @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_640_mult__less__iff1,axiom,
! [Z3: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y2 @ Z3 ) )
= ( ord_less_real @ X @ Y2 ) ) ) ).
% mult_less_iff1
thf(fact_641_permutations__of__list__impl__aux__correct,axiom,
! [Xs: list_nat] :
( ( multis6640212516482799089ux_nat @ nil_nat @ Xs )
= ( map_li7225945977422193158st_nat @ rev_nat @ ( multis1338830684155981141pl_nat @ Xs ) ) ) ).
% permutations_of_list_impl_aux_correct
thf(fact_642_permutations__of__list__impl__aux__correct,axiom,
! [Xs: list_complex] :
( ( multis1292119725739966671omplex @ nil_complex @ Xs )
= ( map_li2870275437539113154omplex @ rev_complex @ ( multis6086220127510270003omplex @ Xs ) ) ) ).
% permutations_of_list_impl_aux_correct
thf(fact_643_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_644_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_complex @ N @ nil_complex )
= ( cons_list_complex @ nil_complex @ nil_list_complex ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_complex @ N @ nil_complex )
= nil_list_complex ) ) ) ).
% n_lists_Nil
thf(fact_645_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_646_n__lists_Osimps_I1_J,axiom,
! [Xs: list_complex] :
( ( n_lists_complex @ zero_zero_nat @ Xs )
= ( cons_list_complex @ nil_complex @ nil_list_complex ) ) ).
% n_lists.simps(1)
thf(fact_647_listset_Osimps_I1_J,axiom,
( ( listset_nat @ nil_set_nat )
= ( insert_list_nat @ nil_nat @ bot_bot_set_list_nat ) ) ).
% listset.simps(1)
thf(fact_648_listset_Osimps_I1_J,axiom,
( ( listset_complex @ nil_set_complex )
= ( insert_list_complex @ nil_complex @ bot_bo6492010485567502472omplex ) ) ).
% listset.simps(1)
thf(fact_649_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_650_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M5: nat] :
! [X3: nat] :
( ( member_nat2 @ X3 @ N6 )
=> ( ord_less_nat @ X3 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_651_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat2 @ X4 @ S )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_652_arg__min__if__finite_I2_J,axiom,
! [S: set_mat_complex,F: mat_complex > nat] :
( ( finite7047982916621727056omplex @ S )
=> ( ( S != bot_bo7165004461764951667omplex )
=> ~ ? [X4: mat_complex] :
( ( member_mat_complex @ X4 @ S )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic8691243167872488990ex_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_653_diag__elems__ne,axiom,
! [B: mat_complex,N: nat] :
( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( projec2809893096078145286omplex @ B )
!= bot_bot_set_complex ) ) ) ).
% diag_elems_ne
thf(fact_654_diag__elems__ne,axiom,
! [B: mat_mat_complex,N: nat] :
( ( member7752848204589936667omplex @ B @ ( carrie8442657464762054641omplex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( projec1765981369499306831omplex @ B )
!= bot_bo7165004461764951667omplex ) ) ) ).
% diag_elems_ne
thf(fact_655_bot_Onot__eq__extremum,axiom,
! [A3: set_mat_complex] :
( ( A3 != bot_bo7165004461764951667omplex )
= ( ord_le5598786136212072115omplex @ bot_bo7165004461764951667omplex @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_656_bot_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_657_bot_Oextremum__strict,axiom,
! [A3: set_mat_complex] :
~ ( ord_le5598786136212072115omplex @ A3 @ bot_bo7165004461764951667omplex ) ).
% bot.extremum_strict
thf(fact_658_bot_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_659_bot__set__def,axiom,
( bot_bo7165004461764951667omplex
= ( collect_mat_complex @ bot_bo2514468519737825834plex_o ) ) ).
% bot_set_def
thf(fact_660_diag__elems__finite,axiom,
! [B: mat_mat_complex] : ( finite7047982916621727056omplex @ ( projec1765981369499306831omplex @ B ) ) ).
% diag_elems_finite
thf(fact_661_diag__elems__finite,axiom,
! [B: mat_nat] : ( finite_finite_nat @ ( projec8639844951311350312ms_nat @ B ) ) ).
% diag_elems_finite
thf(fact_662_class__field_Ozero__not__one,axiom,
zero_zero_real != one_one_real ).
% class_field.zero_not_one
thf(fact_663_mult__delta__right,axiom,
! [B2: $o,X: real,Y2: real] :
( ( B2
=> ( ( times_times_real @ X @ ( if_real @ B2 @ Y2 @ zero_zero_real ) )
= ( times_times_real @ X @ Y2 ) ) )
& ( ~ B2
=> ( ( times_times_real @ X @ ( if_real @ B2 @ Y2 @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_664_mult__delta__right,axiom,
! [B2: $o,X: nat,Y2: nat] :
( ( B2
=> ( ( times_times_nat @ X @ ( if_nat @ B2 @ Y2 @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y2 ) ) )
& ( ~ B2
=> ( ( times_times_nat @ X @ ( if_nat @ B2 @ Y2 @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_665_mult__delta__right,axiom,
! [B2: $o,X: complex,Y2: complex] :
( ( B2
=> ( ( times_times_complex @ X @ ( if_complex @ B2 @ Y2 @ zero_zero_complex ) )
= ( times_times_complex @ X @ Y2 ) ) )
& ( ~ B2
=> ( ( times_times_complex @ X @ ( if_complex @ B2 @ Y2 @ zero_zero_complex ) )
= zero_zero_complex ) ) ) ).
% mult_delta_right
thf(fact_666_class__cring_Ofactors__equal,axiom,
! [A3: complex,B2: complex,C2: complex,D2: complex] :
( ( A3 = B2 )
=> ( ( C2 = D2 )
=> ( ( times_times_complex @ A3 @ C2 )
= ( times_times_complex @ B2 @ D2 ) ) ) ) ).
% class_cring.factors_equal
thf(fact_667_mult__delta__left,axiom,
! [B2: $o,X: real,Y2: real] :
( ( B2
=> ( ( times_times_real @ ( if_real @ B2 @ X @ zero_zero_real ) @ Y2 )
= ( times_times_real @ X @ Y2 ) ) )
& ( ~ B2
=> ( ( times_times_real @ ( if_real @ B2 @ X @ zero_zero_real ) @ Y2 )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_668_mult__delta__left,axiom,
! [B2: $o,X: nat,Y2: nat] :
( ( B2
=> ( ( times_times_nat @ ( if_nat @ B2 @ X @ zero_zero_nat ) @ Y2 )
= ( times_times_nat @ X @ Y2 ) ) )
& ( ~ B2
=> ( ( times_times_nat @ ( if_nat @ B2 @ X @ zero_zero_nat ) @ Y2 )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_669_mult__delta__left,axiom,
! [B2: $o,X: complex,Y2: complex] :
( ( B2
=> ( ( times_times_complex @ ( if_complex @ B2 @ X @ zero_zero_complex ) @ Y2 )
= ( times_times_complex @ X @ Y2 ) ) )
& ( ~ B2
=> ( ( times_times_complex @ ( if_complex @ B2 @ X @ zero_zero_complex ) @ Y2 )
= zero_zero_complex ) ) ) ).
% mult_delta_left
thf(fact_670_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat2 @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_671_bot__empty__eq,axiom,
( bot_bo2514468519737825834plex_o
= ( ^ [X3: mat_complex] : ( member_mat_complex @ X3 @ bot_bo7165004461764951667omplex ) ) ) ).
% bot_empty_eq
thf(fact_672_Collect__empty__eq__bot,axiom,
! [P: mat_complex > $o] :
( ( ( collect_mat_complex @ P )
= bot_bo7165004461764951667omplex )
= ( P = bot_bo2514468519737825834plex_o ) ) ).
% Collect_empty_eq_bot
thf(fact_673_mult__hom_Ohom__zero,axiom,
! [C2: real] :
( ( times_times_real @ C2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_674_mult__hom_Ohom__zero,axiom,
! [C2: nat] :
( ( times_times_nat @ C2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_675_mult__hom_Ohom__zero,axiom,
! [C2: complex] :
( ( times_times_complex @ C2 @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_hom.hom_zero
thf(fact_676_eq__comps__singleton__elems,axiom,
! [L: list_mat_complex,A3: nat] :
( ( ( commut5736191610077499254omplex @ L )
= ( cons_nat @ A3 @ nil_nat ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5969786470865220249omplex @ L ) )
=> ( ( nth_mat_complex @ L @ I2 )
= ( nth_mat_complex @ L @ zero_zero_nat ) ) ) ) ).
% eq_comps_singleton_elems
thf(fact_677_eq__comps__singleton__elems,axiom,
! [L: list_real,A3: nat] :
( ( ( commut8680161604938074397s_real @ L )
= ( cons_nat @ A3 @ nil_nat ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_real @ L ) )
=> ( ( nth_real @ L @ I2 )
= ( nth_real @ L @ zero_zero_nat ) ) ) ) ).
% eq_comps_singleton_elems
thf(fact_678_eq__comps__singleton__elems,axiom,
! [L: list_complex,A3: nat] :
( ( ( commut93809757773076895omplex @ L )
= ( cons_nat @ A3 @ nil_nat ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ L ) )
=> ( ( nth_complex @ L @ I2 )
= ( nth_complex @ L @ zero_zero_nat ) ) ) ) ).
% eq_comps_singleton_elems
thf(fact_679_eq__comps__singleton__elems,axiom,
! [L: list_nat,A3: nat] :
( ( ( commut2436974278740741825ps_nat @ L )
= ( cons_nat @ A3 @ nil_nat ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
=> ( ( nth_nat @ L @ I2 )
= ( nth_nat @ L @ zero_zero_nat ) ) ) ) ).
% eq_comps_singleton_elems
thf(fact_680_nth__Cons__0,axiom,
! [X: mat_complex,Xs: list_mat_complex] :
( ( nth_mat_complex @ ( cons_mat_complex @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_681_nth__Cons__0,axiom,
! [X: real,Xs: list_real] :
( ( nth_real @ ( cons_real @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_682_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_683_nth__Cons__0,axiom,
! [X: complex,Xs: list_complex] :
( ( nth_complex @ ( cons_complex @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_684_nth__equalityI,axiom,
! [Xs: list_mat_complex,Ys: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ( nth_mat_complex @ Xs @ I3 )
= ( nth_mat_complex @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_685_nth__equalityI,axiom,
! [Xs: list_real,Ys: list_real] :
( ( ( size_size_list_real @ Xs )
= ( size_size_list_real @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
=> ( ( nth_real @ Xs @ I3 )
= ( nth_real @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_686_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_687_Skolem__list__nth,axiom,
! [K3: nat,P: nat > mat_complex > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ? [X7: mat_complex] : ( P @ I4 @ X7 ) ) )
= ( ? [Xs4: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs4 )
= K3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ( P @ I4 @ ( nth_mat_complex @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_688_Skolem__list__nth,axiom,
! [K3: nat,P: nat > real > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ? [X7: real] : ( P @ I4 @ X7 ) ) )
= ( ? [Xs4: list_real] :
( ( ( size_size_list_real @ Xs4 )
= K3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ( P @ I4 @ ( nth_real @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_689_Skolem__list__nth,axiom,
! [K3: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ? [X7: nat] : ( P @ I4 @ X7 ) ) )
= ( ? [Xs4: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= K3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ( P @ I4 @ ( nth_nat @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_690_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_mat_complex,Z4: list_mat_complex] : ( Y5 = Z4 ) )
= ( ^ [Xs4: list_mat_complex,Ys3: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Xs4 )
= ( size_s5969786470865220249omplex @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5969786470865220249omplex @ Xs4 ) )
=> ( ( nth_mat_complex @ Xs4 @ I4 )
= ( nth_mat_complex @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_691_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_real,Z4: list_real] : ( Y5 = Z4 ) )
= ( ^ [Xs4: list_real,Ys3: list_real] :
( ( ( size_size_list_real @ Xs4 )
= ( size_size_list_real @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs4 ) )
=> ( ( nth_real @ Xs4 @ I4 )
= ( nth_real @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_692_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_nat,Z4: list_nat] : ( Y5 = Z4 ) )
= ( ^ [Xs4: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs4 ) )
=> ( ( nth_nat @ Xs4 @ I4 )
= ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_693_nth__map__conv,axiom,
! [Xs: list_mat_complex,Ys: list_complex,F: mat_complex > real,G: complex > real] :
( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ( F @ ( nth_mat_complex @ Xs @ I3 ) )
= ( G @ ( nth_complex @ Ys @ I3 ) ) ) )
=> ( ( map_mat_complex_real @ F @ Xs )
= ( map_complex_real @ G @ Ys ) ) ) ) ).
% nth_map_conv
thf(fact_694_nth__map__conv,axiom,
! [Xs: list_real,Ys: list_complex,F: real > real,G: complex > real] :
( ( ( size_size_list_real @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
=> ( ( F @ ( nth_real @ Xs @ I3 ) )
= ( G @ ( nth_complex @ Ys @ I3 ) ) ) )
=> ( ( map_real_real @ F @ Xs )
= ( map_complex_real @ G @ Ys ) ) ) ) ).
% nth_map_conv
thf(fact_695_nth__map__conv,axiom,
! [Xs: list_complex,Ys: list_mat_complex,F: complex > real,G: mat_complex > real] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( ( F @ ( nth_complex @ Xs @ I3 ) )
= ( G @ ( nth_mat_complex @ Ys @ I3 ) ) ) )
=> ( ( map_complex_real @ F @ Xs )
= ( map_mat_complex_real @ G @ Ys ) ) ) ) ).
% nth_map_conv
thf(fact_696_nth__map__conv,axiom,
! [Xs: list_complex,Ys: list_real,F: complex > real,G: real > real] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_real @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( ( F @ ( nth_complex @ Xs @ I3 ) )
= ( G @ ( nth_real @ Ys @ I3 ) ) ) )
=> ( ( map_complex_real @ F @ Xs )
= ( map_real_real @ G @ Ys ) ) ) ) ).
% nth_map_conv
thf(fact_697_nth__map__conv,axiom,
! [Xs: list_complex,Ys: list_complex,F: complex > real,G: complex > real] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( ( F @ ( nth_complex @ Xs @ I3 ) )
= ( G @ ( nth_complex @ Ys @ I3 ) ) ) )
=> ( ( map_complex_real @ F @ Xs )
= ( map_complex_real @ G @ Ys ) ) ) ) ).
% nth_map_conv
thf(fact_698_nth__map__conv,axiom,
! [Xs: list_complex,Ys: list_nat,F: complex > real,G: nat > real] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( ( F @ ( nth_complex @ Xs @ I3 ) )
= ( G @ ( nth_nat @ Ys @ I3 ) ) ) )
=> ( ( map_complex_real @ F @ Xs )
= ( map_nat_real @ G @ Ys ) ) ) ) ).
% nth_map_conv
thf(fact_699_nth__map__conv,axiom,
! [Xs: list_nat,Ys: list_complex,F: nat > real,G: complex > real] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( F @ ( nth_nat @ Xs @ I3 ) )
= ( G @ ( nth_complex @ Ys @ I3 ) ) ) )
=> ( ( map_nat_real @ F @ Xs )
= ( map_complex_real @ G @ Ys ) ) ) ) ).
% nth_map_conv
thf(fact_700_map__equality__iff,axiom,
! [F: mat_complex > real,Xs: list_mat_complex,G: complex > real,Ys: list_complex] :
( ( ( map_mat_complex_real @ F @ Xs )
= ( map_complex_real @ G @ Ys ) )
= ( ( ( size_s5969786470865220249omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( F @ ( nth_mat_complex @ Xs @ I4 ) )
= ( G @ ( nth_complex @ Ys @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_701_map__equality__iff,axiom,
! [F: real > real,Xs: list_real,G: complex > real,Ys: list_complex] :
( ( ( map_real_real @ F @ Xs )
= ( map_complex_real @ G @ Ys ) )
= ( ( ( size_size_list_real @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( F @ ( nth_real @ Xs @ I4 ) )
= ( G @ ( nth_complex @ Ys @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_702_map__equality__iff,axiom,
! [F: complex > real,Xs: list_complex,G: mat_complex > real,Ys: list_mat_complex] :
( ( ( map_complex_real @ F @ Xs )
= ( map_mat_complex_real @ G @ Ys ) )
= ( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s5969786470865220249omplex @ Ys ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5969786470865220249omplex @ Ys ) )
=> ( ( F @ ( nth_complex @ Xs @ I4 ) )
= ( G @ ( nth_mat_complex @ Ys @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_703_map__equality__iff,axiom,
! [F: complex > real,Xs: list_complex,G: real > real,Ys: list_real] :
( ( ( map_complex_real @ F @ Xs )
= ( map_real_real @ G @ Ys ) )
= ( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_real @ Ys ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_real @ Ys ) )
=> ( ( F @ ( nth_complex @ Xs @ I4 ) )
= ( G @ ( nth_real @ Ys @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_704_map__equality__iff,axiom,
! [F: complex > real,Xs: list_complex,G: complex > real,Ys: list_complex] :
( ( ( map_complex_real @ F @ Xs )
= ( map_complex_real @ G @ Ys ) )
= ( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( F @ ( nth_complex @ Xs @ I4 ) )
= ( G @ ( nth_complex @ Ys @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_705_map__equality__iff,axiom,
! [F: complex > real,Xs: list_complex,G: nat > real,Ys: list_nat] :
( ( ( map_complex_real @ F @ Xs )
= ( map_nat_real @ G @ Ys ) )
= ( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_complex @ Xs @ I4 ) )
= ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_706_map__equality__iff,axiom,
! [F: nat > real,Xs: list_nat,G: complex > real,Ys: list_complex] :
( ( ( map_nat_real @ F @ Xs )
= ( map_complex_real @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I4 ) )
= ( G @ ( nth_complex @ Ys @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_707_nth__map,axiom,
! [N: nat,Xs: list_mat_complex,F: mat_complex > nat] :
( ( ord_less_nat @ N @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ( nth_nat @ ( map_mat_complex_nat @ F @ Xs ) @ N )
= ( F @ ( nth_mat_complex @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_708_nth__map,axiom,
! [N: nat,Xs: list_real,F: real > nat] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( ( nth_nat @ ( map_real_nat @ F @ Xs ) @ N )
= ( F @ ( nth_real @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_709_nth__map,axiom,
! [N: nat,Xs: list_mat_complex,F: mat_complex > mat_complex] :
( ( ord_less_nat @ N @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ( nth_mat_complex @ ( map_ma6165852935686130436omplex @ F @ Xs ) @ N )
= ( F @ ( nth_mat_complex @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_710_nth__map,axiom,
! [N: nat,Xs: list_real,F: real > mat_complex] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( ( nth_mat_complex @ ( map_real_mat_complex @ F @ Xs ) @ N )
= ( F @ ( nth_real @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_711_nth__map,axiom,
! [N: nat,Xs: list_mat_complex,F: mat_complex > real] :
( ( ord_less_nat @ N @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ( nth_real @ ( map_mat_complex_real @ F @ Xs ) @ N )
= ( F @ ( nth_mat_complex @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_712_nth__map,axiom,
! [N: nat,Xs: list_real,F: real > real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( ( nth_real @ ( map_real_real @ F @ Xs ) @ N )
= ( F @ ( nth_real @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_713_nth__map,axiom,
! [N: nat,Xs: list_complex,F: complex > real] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( ( nth_real @ ( map_complex_real @ F @ Xs ) @ N )
= ( F @ ( nth_complex @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_714_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_715_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > mat_complex] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_mat_complex @ ( map_nat_mat_complex @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_716_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > real] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_real @ ( map_nat_real @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_717_nth__append__length,axiom,
! [Xs: list_mat_complex,X: mat_complex,Ys: list_mat_complex] :
( ( nth_mat_complex @ ( append_mat_complex @ Xs @ ( cons_mat_complex @ X @ Ys ) ) @ ( size_s5969786470865220249omplex @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_718_nth__append__length,axiom,
! [Xs: list_real,X: real,Ys: list_real] :
( ( nth_real @ ( append_real @ Xs @ ( cons_real @ X @ Ys ) ) @ ( size_size_list_real @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_719_nth__append__length,axiom,
! [Xs: list_complex,X: complex,Ys: list_complex] :
( ( nth_complex @ ( append_complex @ Xs @ ( cons_complex @ X @ Ys ) ) @ ( size_s3451745648224563538omplex @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_720_nth__append__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_721_nth__Cons_H,axiom,
! [N: nat,X: mat_complex,Xs: list_mat_complex] :
( ( ( N = zero_zero_nat )
=> ( ( nth_mat_complex @ ( cons_mat_complex @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_mat_complex @ ( cons_mat_complex @ X @ Xs ) @ N )
= ( nth_mat_complex @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_722_nth__Cons_H,axiom,
! [N: nat,X: real,Xs: list_real] :
( ( ( N = zero_zero_nat )
=> ( ( nth_real @ ( cons_real @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_real @ ( cons_real @ X @ Xs ) @ N )
= ( nth_real @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_723_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_724_nth__Cons_H,axiom,
! [N: nat,X: complex,Xs: list_complex] :
( ( ( N = zero_zero_nat )
=> ( ( nth_complex @ ( cons_complex @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_complex @ ( cons_complex @ X @ Xs ) @ N )
= ( nth_complex @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_725_nth__append,axiom,
! [N: nat,Xs: list_mat_complex,Ys: list_mat_complex] :
( ( ( ord_less_nat @ N @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ( nth_mat_complex @ ( append_mat_complex @ Xs @ Ys ) @ N )
= ( nth_mat_complex @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ( nth_mat_complex @ ( append_mat_complex @ Xs @ Ys ) @ N )
= ( nth_mat_complex @ Ys @ ( minus_minus_nat @ N @ ( size_s5969786470865220249omplex @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_726_nth__append,axiom,
! [N: nat,Xs: list_real,Ys: list_real] :
( ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( ( nth_real @ ( append_real @ Xs @ Ys ) @ N )
= ( nth_real @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( ( nth_real @ ( append_real @ Xs @ Ys ) @ N )
= ( nth_real @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_real @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_727_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_728_nth__Cons__pos,axiom,
! [N: nat,X: mat_complex,Xs: list_mat_complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_mat_complex @ ( cons_mat_complex @ X @ Xs ) @ N )
= ( nth_mat_complex @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_729_nth__Cons__pos,axiom,
! [N: nat,X: real,Xs: list_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_real @ ( cons_real @ X @ Xs ) @ N )
= ( nth_real @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_730_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_731_nth__Cons__pos,axiom,
! [N: nat,X: complex,Xs: list_complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_complex @ ( cons_complex @ X @ Xs ) @ N )
= ( nth_complex @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_732_nth__non__equal__first__eq,axiom,
! [X: mat_complex,Y2: mat_complex,Xs: list_mat_complex,N: nat] :
( ( X != Y2 )
=> ( ( ( nth_mat_complex @ ( cons_mat_complex @ X @ Xs ) @ N )
= Y2 )
= ( ( ( nth_mat_complex @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_733_nth__non__equal__first__eq,axiom,
! [X: real,Y2: real,Xs: list_real,N: nat] :
( ( X != Y2 )
=> ( ( ( nth_real @ ( cons_real @ X @ Xs ) @ N )
= Y2 )
= ( ( ( nth_real @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_734_nth__non__equal__first__eq,axiom,
! [X: nat,Y2: nat,Xs: list_nat,N: nat] :
( ( X != Y2 )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= Y2 )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_735_nth__non__equal__first__eq,axiom,
! [X: complex,Y2: complex,Xs: list_complex,N: nat] :
( ( X != Y2 )
=> ( ( ( nth_complex @ ( cons_complex @ X @ Xs ) @ N )
= Y2 )
= ( ( ( nth_complex @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_736_per__diag__diag__mat,axiom,
! [A4: mat_nat,N: nat,I: nat,F: nat > nat] :
( ( member_mat_nat @ A4 @ ( carrier_mat_nat @ N @ N ) )
=> ( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ ( F @ I ) @ N )
=> ( ( nth_nat @ ( diag_mat_nat @ ( commut5604902300900073841ag_nat @ A4 @ F ) ) @ I )
= ( nth_nat @ ( diag_mat_nat @ A4 ) @ ( F @ I ) ) ) ) ) ) ).
% per_diag_diag_mat
thf(fact_737_per__diag__diag__mat,axiom,
! [A4: mat_mat_complex,N: nat,I: nat,F: nat > nat] :
( ( member7752848204589936667omplex @ A4 @ ( carrie8442657464762054641omplex @ N @ N ) )
=> ( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ ( F @ I ) @ N )
=> ( ( nth_mat_complex @ ( diag_mat_mat_complex @ ( commut3385207333667201222omplex @ A4 @ F ) ) @ I )
= ( nth_mat_complex @ ( diag_mat_mat_complex @ A4 ) @ ( F @ I ) ) ) ) ) ) ).
% per_diag_diag_mat
thf(fact_738_per__diag__diag__mat,axiom,
! [A4: mat_real,N: nat,I: nat,F: nat > nat] :
( ( member_mat_real @ A4 @ ( carrier_mat_real @ N @ N ) )
=> ( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ ( F @ I ) @ N )
=> ( ( nth_real @ ( diag_mat_real @ ( commut7814478499001091021g_real @ A4 @ F ) ) @ I )
= ( nth_real @ ( diag_mat_real @ A4 ) @ ( F @ I ) ) ) ) ) ) ).
% per_diag_diag_mat
thf(fact_739_per__diag__diag__mat,axiom,
! [A4: mat_complex,N: nat,I: nat,F: nat > nat] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ ( F @ I ) @ N )
=> ( ( nth_complex @ ( diag_mat_complex @ ( commut4119912100034661455omplex @ A4 @ F ) ) @ I )
= ( nth_complex @ ( diag_mat_complex @ A4 ) @ ( F @ I ) ) ) ) ) ) ).
% per_diag_diag_mat
thf(fact_740_extract__subdiags__comp__commute,axiom,
! [A4: mat_complex,N: nat,I: nat,B: mat_complex] :
( ( diagonal_mat_complex @ A4 )
=> ( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( commut93809757773076895omplex @ ( diag_mat_complex @ A4 ) ) ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ ( nth_nat @ ( commut93809757773076895omplex @ ( diag_mat_complex @ A4 ) ) @ I ) @ ( nth_nat @ ( commut93809757773076895omplex @ ( diag_mat_complex @ A4 ) ) @ I ) ) )
=> ( ( times_8009071140041733218omplex @ ( nth_mat_complex @ ( commut6900707758132580272omplex @ A4 @ ( commut93809757773076895omplex @ ( diag_mat_complex @ A4 ) ) ) @ I ) @ B )
= ( times_8009071140041733218omplex @ B @ ( nth_mat_complex @ ( commut6900707758132580272omplex @ A4 @ ( commut93809757773076895omplex @ ( diag_mat_complex @ A4 ) ) ) @ I ) ) ) ) ) ) ) ) ).
% extract_subdiags_comp_commute
thf(fact_741_undef__vec__def,axiom,
( undef_2495355514574404529omplex
= ( nth_mat_complex @ nil_mat_complex ) ) ).
% undef_vec_def
thf(fact_742_undef__vec__def,axiom,
( undef_vec_real
= ( nth_real @ nil_real ) ) ).
% undef_vec_def
thf(fact_743_undef__vec__def,axiom,
( undef_vec_nat
= ( nth_nat @ nil_nat ) ) ).
% undef_vec_def
thf(fact_744_undef__vec__def,axiom,
( undef_vec_complex
= ( nth_complex @ nil_complex ) ) ).
% undef_vec_def
thf(fact_745_extract__subdiags__length,axiom,
! [B: mat_complex,L: list_nat] :
( ( size_s5969786470865220249omplex @ ( commut6900707758132580272omplex @ B @ L ) )
= ( size_size_list_nat @ L ) ) ).
% extract_subdiags_length
thf(fact_746_extract__subdiags_Osimps_I1_J,axiom,
! [B: mat_complex] :
( ( commut6900707758132580272omplex @ B @ nil_nat )
= nil_mat_complex ) ).
% extract_subdiags.simps(1)
thf(fact_747_extract__subdiags__neq__Nil,axiom,
! [B: mat_complex,A3: nat,L: list_nat] :
( ( commut6900707758132580272omplex @ B @ ( cons_nat @ A3 @ L ) )
!= nil_mat_complex ) ).
% extract_subdiags_neq_Nil
thf(fact_748_extract__subdiags__carrier,axiom,
! [I: nat,L: list_nat,B: mat_complex] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
=> ( member_mat_complex @ ( nth_mat_complex @ ( commut6900707758132580272omplex @ B @ L ) @ I ) @ ( carrier_mat_complex @ ( nth_nat @ L @ I ) @ ( nth_nat @ L @ I ) ) ) ) ).
% extract_subdiags_carrier
thf(fact_749_mat__delete__carrier,axiom,
! [A4: mat_complex,M: nat,N: nat,I: nat,J: nat] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ M @ N ) )
=> ( member_mat_complex @ ( mat_delete_complex @ A4 @ I @ J ) @ ( carrier_mat_complex @ ( minus_minus_nat @ M @ one_one_nat ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% mat_delete_carrier
thf(fact_750_append__one__prefix,axiom,
! [Xs: list_mat_complex,Ys: list_mat_complex] :
( ( prefix_mat_complex @ Xs @ Ys )
=> ( ( ord_less_nat @ ( size_s5969786470865220249omplex @ Xs ) @ ( size_s5969786470865220249omplex @ Ys ) )
=> ( prefix_mat_complex @ ( append_mat_complex @ Xs @ ( cons_mat_complex @ ( nth_mat_complex @ Ys @ ( size_s5969786470865220249omplex @ Xs ) ) @ nil_mat_complex ) ) @ Ys ) ) ) ).
% append_one_prefix
thf(fact_751_append__one__prefix,axiom,
! [Xs: list_real,Ys: list_real] :
( ( prefix_real @ Xs @ Ys )
=> ( ( ord_less_nat @ ( size_size_list_real @ Xs ) @ ( size_size_list_real @ Ys ) )
=> ( prefix_real @ ( append_real @ Xs @ ( cons_real @ ( nth_real @ Ys @ ( size_size_list_real @ Xs ) ) @ nil_real ) ) @ Ys ) ) ) ).
% append_one_prefix
thf(fact_752_append__one__prefix,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( prefix_complex @ Xs @ Ys )
=> ( ( ord_less_nat @ ( size_s3451745648224563538omplex @ Xs ) @ ( size_s3451745648224563538omplex @ Ys ) )
=> ( prefix_complex @ ( append_complex @ Xs @ ( cons_complex @ ( nth_complex @ Ys @ ( size_s3451745648224563538omplex @ Xs ) ) @ nil_complex ) ) @ Ys ) ) ) ).
% append_one_prefix
thf(fact_753_append__one__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( ( ord_less_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( prefix_nat @ ( append_nat @ Xs @ ( cons_nat @ ( nth_nat @ Ys @ ( size_size_list_nat @ Xs ) ) @ nil_nat ) ) @ Ys ) ) ) ).
% append_one_prefix
thf(fact_754_diagonal__extract__eq,axiom,
! [B: mat_nat,N: nat] :
( ( member_mat_nat @ B @ ( carrier_mat_nat @ N @ N ) )
=> ( ( diagonal_mat_nat @ B )
=> ( B
= ( diag_block_mat_nat @ ( commut2742254738641004242gs_nat @ B @ ( commut2436974278740741825ps_nat @ ( diag_mat_nat @ B ) ) ) ) ) ) ) ).
% diagonal_extract_eq
thf(fact_755_diagonal__extract__eq,axiom,
! [B: mat_complex,N: nat] :
( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( diagonal_mat_complex @ B )
=> ( B
= ( diag_b9145358668110806138omplex @ ( commut6900707758132580272omplex @ B @ ( commut93809757773076895omplex @ ( diag_mat_complex @ B ) ) ) ) ) ) ) ).
% diagonal_extract_eq
thf(fact_756_extract__subdiags__eq__comp,axiom,
! [A4: mat_complex,N: nat,I: nat] :
( ( diagonal_mat_complex @ A4 )
=> ( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( commut93809757773076895omplex @ ( diag_mat_complex @ A4 ) ) ) )
=> ? [K: complex] :
( ( nth_mat_complex @ ( commut6900707758132580272omplex @ A4 @ ( commut93809757773076895omplex @ ( diag_mat_complex @ A4 ) ) ) @ I )
= ( smult_mat_complex @ K @ ( one_mat_complex @ ( nth_nat @ ( commut93809757773076895omplex @ ( diag_mat_complex @ A4 ) ) @ I ) ) ) ) ) ) ) ) ).
% extract_subdiags_eq_comp
thf(fact_757_prefix__Nil,axiom,
! [Xs: list_nat] :
( ( prefix_nat @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% prefix_Nil
thf(fact_758_prefix__Nil,axiom,
! [Xs: list_complex] :
( ( prefix_complex @ Xs @ nil_complex )
= ( Xs = nil_complex ) ) ).
% prefix_Nil
thf(fact_759_Nil__prefix,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% Nil_prefix
thf(fact_760_Nil__prefix,axiom,
! [Xs: list_complex] : ( prefix_complex @ nil_complex @ Xs ) ).
% Nil_prefix
thf(fact_761_prefix__bot_Oextremum__uniqueI,axiom,
! [A3: list_nat] :
( ( prefix_nat @ A3 @ nil_nat )
=> ( A3 = nil_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_762_prefix__bot_Oextremum__uniqueI,axiom,
! [A3: list_complex] :
( ( prefix_complex @ A3 @ nil_complex )
=> ( A3 = nil_complex ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_763_prefix__bot_Oextremum__unique,axiom,
! [A3: list_nat] :
( ( prefix_nat @ A3 @ nil_nat )
= ( A3 = nil_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_764_prefix__bot_Oextremum__unique,axiom,
! [A3: list_complex] :
( ( prefix_complex @ A3 @ nil_complex )
= ( A3 = nil_complex ) ) ).
% prefix_bot.extremum_unique
thf(fact_765_prefix__bot_Obot__least,axiom,
! [A3: list_nat] : ( prefix_nat @ nil_nat @ A3 ) ).
% prefix_bot.bot_least
thf(fact_766_prefix__bot_Obot__least,axiom,
! [A3: list_complex] : ( prefix_complex @ nil_complex @ A3 ) ).
% prefix_bot.bot_least
thf(fact_767_prefix__code_I1_J,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% prefix_code(1)
thf(fact_768_prefix__code_I1_J,axiom,
! [Xs: list_complex] : ( prefix_complex @ nil_complex @ Xs ) ).
% prefix_code(1)
thf(fact_769_smult__smult__mat,axiom,
! [A4: mat_complex,Nr: nat,N: nat,K3: complex,L: complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( smult_mat_complex @ K3 @ ( smult_mat_complex @ L @ A4 ) )
= ( smult_mat_complex @ ( times_times_complex @ K3 @ L ) @ A4 ) ) ) ).
% smult_smult_mat
thf(fact_770_smult__smult__times,axiom,
! [A3: nat,K3: nat,A4: mat_nat] :
( ( smult_mat_nat @ A3 @ ( smult_mat_nat @ K3 @ A4 ) )
= ( smult_mat_nat @ ( times_times_nat @ A3 @ K3 ) @ A4 ) ) ).
% smult_smult_times
thf(fact_771_smult__smult__times,axiom,
! [A3: complex,K3: complex,A4: mat_complex] :
( ( smult_mat_complex @ A3 @ ( smult_mat_complex @ K3 @ A4 ) )
= ( smult_mat_complex @ ( times_times_complex @ A3 @ K3 ) @ A4 ) ) ).
% smult_smult_times
thf(fact_772_diagonal__mat__smult,axiom,
! [A4: mat_complex,X: complex] :
( ( diagonal_mat_complex @ A4 )
=> ( diagonal_mat_complex @ ( smult_mat_complex @ X @ A4 ) ) ) ).
% diagonal_mat_smult
thf(fact_773_smult__carrier__mat,axiom,
! [A4: mat_complex,Nr: nat,Nc: nat,K3: complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( smult_mat_complex @ K3 @ A4 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% smult_carrier_mat
thf(fact_774_unitary__one,axiom,
! [N: nat] : ( comple6660659447773130958omplex @ ( one_mat_complex @ N ) ) ).
% unitary_one
thf(fact_775_Linear__Algebra__Complements_Osmult__one,axiom,
! [A4: mat_complex] :
( ( smult_mat_complex @ one_one_complex @ A4 )
= A4 ) ).
% Linear_Algebra_Complements.smult_one
thf(fact_776_Linear__Algebra__Complements_Osmult__one,axiom,
! [A4: mat_nat] :
( ( smult_mat_nat @ one_one_nat @ A4 )
= A4 ) ).
% Linear_Algebra_Complements.smult_one
thf(fact_777_Linear__Algebra__Complements_Osmult__one,axiom,
! [A4: mat_real] :
( ( smult_mat_real @ one_one_real @ A4 )
= A4 ) ).
% Linear_Algebra_Complements.smult_one
thf(fact_778_hermitian__one,axiom,
! [N: nat] : ( comple8306762464034002205omplex @ ( one_mat_complex @ N ) ) ).
% hermitian_one
thf(fact_779_map__mono__prefix,axiom,
! [Xs: list_complex,Ys: list_complex,F: complex > real] :
( ( prefix_complex @ Xs @ Ys )
=> ( prefix_real @ ( map_complex_real @ F @ Xs ) @ ( map_complex_real @ F @ Ys ) ) ) ).
% map_mono_prefix
thf(fact_780_prefix__map__rightE,axiom,
! [Xs: list_real,F: complex > real,Ys: list_complex] :
( ( prefix_real @ Xs @ ( map_complex_real @ F @ Ys ) )
=> ? [Xs3: list_complex] :
( ( prefix_complex @ Xs3 @ Ys )
& ( Xs
= ( map_complex_real @ F @ Xs3 ) ) ) ) ).
% prefix_map_rightE
thf(fact_781_Cons__prefix__Cons,axiom,
! [X: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( prefix_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= ( ( X = Y2 )
& ( prefix_nat @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_782_Cons__prefix__Cons,axiom,
! [X: complex,Xs: list_complex,Y2: complex,Ys: list_complex] :
( ( prefix_complex @ ( cons_complex @ X @ Xs ) @ ( cons_complex @ Y2 @ Ys ) )
= ( ( X = Y2 )
& ( prefix_complex @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_783_one__carrier__mat,axiom,
! [N: nat] : ( member_mat_complex @ ( one_mat_complex @ N ) @ ( carrier_mat_complex @ N @ N ) ) ).
% one_carrier_mat
thf(fact_784_right__mult__one__mat,axiom,
! [A4: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A4 @ ( one_mat_complex @ Nc ) )
= A4 ) ) ).
% right_mult_one_mat
thf(fact_785_left__mult__one__mat,axiom,
! [A4: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( one_mat_complex @ Nr ) @ A4 )
= A4 ) ) ).
% left_mult_one_mat
thf(fact_786_mat__mult__left__right__inverse,axiom,
! [A4: mat_complex,N: nat,B: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ( times_8009071140041733218omplex @ A4 @ B )
= ( one_mat_complex @ N ) )
=> ( ( times_8009071140041733218omplex @ B @ A4 )
= ( one_mat_complex @ N ) ) ) ) ) ).
% mat_mult_left_right_inverse
thf(fact_787_prefix__code_I2_J,axiom,
! [X: nat,Xs: list_nat] :
~ ( prefix_nat @ ( cons_nat @ X @ Xs ) @ nil_nat ) ).
% prefix_code(2)
thf(fact_788_prefix__code_I2_J,axiom,
! [X: complex,Xs: list_complex] :
~ ( prefix_complex @ ( cons_complex @ X @ Xs ) @ nil_complex ) ).
% prefix_code(2)
thf(fact_789_prefix__Cons,axiom,
! [Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) )
= ( ( Xs = nil_nat )
| ? [Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Zs2 ) )
& ( prefix_nat @ Zs2 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_790_prefix__Cons,axiom,
! [Xs: list_complex,Y2: complex,Ys: list_complex] :
( ( prefix_complex @ Xs @ ( cons_complex @ Y2 @ Ys ) )
= ( ( Xs = nil_complex )
| ? [Zs2: list_complex] :
( ( Xs
= ( cons_complex @ Y2 @ Zs2 ) )
& ( prefix_complex @ Zs2 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_791_not__prefix__cases,axiom,
! [Ps: list_nat,Ls: list_nat] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ( ( Ps != nil_nat )
=> ( Ls != nil_nat ) )
=> ( ! [A: nat,As: list_nat] :
( ( Ps
= ( cons_nat @ A @ As ) )
=> ! [X2: nat,Xs2: list_nat] :
( ( Ls
= ( cons_nat @ X2 @ Xs2 ) )
=> ( ( X2 = A )
=> ( prefix_nat @ As @ Xs2 ) ) ) )
=> ~ ! [A: nat] :
( ? [As: list_nat] :
( Ps
= ( cons_nat @ A @ As ) )
=> ! [X2: nat] :
( ? [Xs2: list_nat] :
( Ls
= ( cons_nat @ X2 @ Xs2 ) )
=> ( X2 = A ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_792_not__prefix__cases,axiom,
! [Ps: list_complex,Ls: list_complex] :
( ~ ( prefix_complex @ Ps @ Ls )
=> ( ( ( Ps != nil_complex )
=> ( Ls != nil_complex ) )
=> ( ! [A: complex,As: list_complex] :
( ( Ps
= ( cons_complex @ A @ As ) )
=> ! [X2: complex,Xs2: list_complex] :
( ( Ls
= ( cons_complex @ X2 @ Xs2 ) )
=> ( ( X2 = A )
=> ( prefix_complex @ As @ Xs2 ) ) ) )
=> ~ ! [A: complex] :
( ? [As: list_complex] :
( Ps
= ( cons_complex @ A @ As ) )
=> ! [X2: complex] :
( ? [Xs2: list_complex] :
( Ls
= ( cons_complex @ X2 @ Xs2 ) )
=> ( X2 = A ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_793_not__prefix__induct,axiom,
! [Ps: list_nat,Ls: list_nat,P: list_nat > list_nat > $o] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ! [X2: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X2 @ Xs2 ) @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( X2 != Y )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( X2 = Y )
=> ( ~ ( prefix_nat @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_794_not__prefix__induct,axiom,
! [Ps: list_complex,Ls: list_complex,P: list_complex > list_complex > $o] :
( ~ ( prefix_complex @ Ps @ Ls )
=> ( ! [X2: complex,Xs2: list_complex] : ( P @ ( cons_complex @ X2 @ Xs2 ) @ nil_complex )
=> ( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex] :
( ( X2 != Y )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) ) )
=> ( ! [X2: complex,Xs2: list_complex,Y: complex,Ys2: list_complex] :
( ( X2 = Y )
=> ( ~ ( prefix_complex @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_complex @ X2 @ Xs2 ) @ ( cons_complex @ Y @ Ys2 ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_795_same__prefix__nil,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys ) @ Xs )
= ( Ys = nil_nat ) ) ).
% same_prefix_nil
thf(fact_796_same__prefix__nil,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( prefix_complex @ ( append_complex @ Xs @ Ys ) @ Xs )
= ( Ys = nil_complex ) ) ).
% same_prefix_nil
thf(fact_797_mat__assoc__test_I3_J,axiom,
! [A4: mat_complex,N: nat,B: mat_complex,C: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A4 @ ( one_mat_complex @ N ) ) @ ( one_mat_complex @ N ) ) @ B ) @ ( one_mat_complex @ N ) )
= ( times_8009071140041733218omplex @ A4 @ B ) ) ) ) ) ) ).
% mat_assoc_test(3)
thf(fact_798_mult__smult__assoc__mat,axiom,
! [A4: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat,K3: complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( smult_mat_complex @ K3 @ A4 ) @ B )
= ( smult_mat_complex @ K3 @ ( times_8009071140041733218omplex @ A4 @ B ) ) ) ) ) ).
% mult_smult_assoc_mat
thf(fact_799_mult__smult__distrib,axiom,
! [A4: mat_complex,Nr: nat,N: nat,B: mat_complex,Nc: nat,K3: complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A4 @ ( smult_mat_complex @ K3 @ B ) )
= ( smult_mat_complex @ K3 @ ( times_8009071140041733218omplex @ A4 @ B ) ) ) ) ) ).
% mult_smult_distrib
thf(fact_800_smult__distrib__left__minus__mat,axiom,
! [A4: mat_complex,N: nat,B: mat_complex,C2: complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( smult_mat_complex @ C2 @ ( minus_2412168080157227406omplex @ B @ A4 ) )
= ( minus_2412168080157227406omplex @ ( smult_mat_complex @ C2 @ B ) @ ( smult_mat_complex @ C2 @ A4 ) ) ) ) ) ).
% smult_distrib_left_minus_mat
thf(fact_801_prefix__snoc,axiom,
! [Xs: list_nat,Ys: list_nat,Y2: nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
| ( prefix_nat @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_802_prefix__snoc,axiom,
! [Xs: list_complex,Ys: list_complex,Y2: complex] :
( ( prefix_complex @ Xs @ ( append_complex @ Ys @ ( cons_complex @ Y2 @ nil_complex ) ) )
= ( ( Xs
= ( append_complex @ Ys @ ( cons_complex @ Y2 @ nil_complex ) ) )
| ( prefix_complex @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_803_diag__block__mat__length__1,axiom,
! [Al: list_mat_complex] :
( ( ( size_s5969786470865220249omplex @ Al )
= one_one_nat )
=> ( ( diag_b9145358668110806138omplex @ Al )
= ( nth_mat_complex @ Al @ zero_zero_nat ) ) ) ).
% diag_block_mat_length_1
thf(fact_804_diag__compat__extract__subdiag,axiom,
! [B: mat_complex,N: nat,L: list_nat] :
( ( member_mat_complex @ B @ ( carrier_mat_complex @ N @ N ) )
=> ( ( commut5261563022830629508omplex @ B @ L )
=> ( B
= ( diag_b9145358668110806138omplex @ ( commut6900707758132580272omplex @ B @ L ) ) ) ) ) ).
% diag_compat_extract_subdiag
thf(fact_805_vector__space__over__itself_Ospan__Basis,axiom,
( ( span_real_real @ times_times_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) )
= top_top_set_real ) ).
% vector_space_over_itself.span_Basis
thf(fact_806_vector__space__over__itself_Ospan__Basis,axiom,
( ( span_complex_complex @ times_times_complex @ ( insert_complex @ one_one_complex @ bot_bot_set_complex ) )
= top_top_set_complex ) ).
% vector_space_over_itself.span_Basis
thf(fact_807_nth__map__out__of__bound,axiom,
! [Xs: list_complex,I: nat,F: complex > real] :
( ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ I )
=> ( ( nth_real @ ( map_complex_real @ F @ Xs ) @ I )
= ( nth_real @ nil_real @ ( minus_minus_nat @ I @ ( size_s3451745648224563538omplex @ Xs ) ) ) ) ) ).
% nth_map_out_of_bound
thf(fact_808_nth__map__out__of__bound,axiom,
! [Xs: list_nat,I: nat,F: nat > mat_complex] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( nth_mat_complex @ ( map_nat_mat_complex @ F @ Xs ) @ I )
= ( nth_mat_complex @ nil_mat_complex @ ( minus_minus_nat @ I @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_map_out_of_bound
thf(fact_809_nth__map__out__of__bound,axiom,
! [Xs: list_nat,I: nat,F: nat > real] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( nth_real @ ( map_nat_real @ F @ Xs ) @ I )
= ( nth_real @ nil_real @ ( minus_minus_nat @ I @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_map_out_of_bound
thf(fact_810_nth__map__out__of__bound,axiom,
! [Xs: list_nat,I: nat,F: nat > nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ I )
= ( nth_nat @ nil_nat @ ( minus_minus_nat @ I @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_map_out_of_bound
thf(fact_811_nth__map__out__of__bound,axiom,
! [Xs: list_nat,I: nat,F: nat > complex] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( nth_complex @ ( map_nat_complex @ F @ Xs ) @ I )
= ( nth_complex @ nil_complex @ ( minus_minus_nat @ I @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_map_out_of_bound
thf(fact_812_rev__nth,axiom,
! [N: nat,Xs: list_mat_complex] :
( ( ord_less_nat @ N @ ( size_s5969786470865220249omplex @ Xs ) )
=> ( ( nth_mat_complex @ ( rev_mat_complex @ Xs ) @ N )
= ( nth_mat_complex @ Xs @ ( minus_minus_nat @ ( size_s5969786470865220249omplex @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_813_rev__nth,axiom,
! [N: nat,Xs: list_real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( ( nth_real @ ( rev_real @ Xs ) @ N )
= ( nth_real @ Xs @ ( minus_minus_nat @ ( size_size_list_real @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_814_rev__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rev_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_815_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_complex] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s3451745648224563538omplex @ Xs ) )
= ( ? [X3: complex,Ys3: list_complex] :
( ( Xs
= ( cons_complex @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_s3451745648224563538omplex @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_816_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_817_prod__decode__aux_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [K: nat,M4: nat] :
( ( ~ ( ord_less_eq_nat @ M4 @ K )
=> ( P @ ( suc @ K ) @ ( minus_minus_nat @ M4 @ ( suc @ K ) ) ) )
=> ( P @ K @ M4 ) )
=> ( P @ A0 @ A1 ) ) ).
% prod_decode_aux.induct
thf(fact_818_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_819_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_820_finite__Plus__UNIV__iff,axiom,
( ( finite6995816559193130876omplex @ top_to442280769018103243omplex )
= ( ( finite7047982916621727056omplex @ top_to1861530291043981143omplex )
& ( finite7047982916621727056omplex @ top_to1861530291043981143omplex ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_821_finite__Plus__UNIV__iff,axiom,
( ( finite3494777059389363281ex_nat @ top_to2112016286549842904ex_nat )
= ( ( finite7047982916621727056omplex @ top_to1861530291043981143omplex )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_822_finite__Plus__UNIV__iff,axiom,
( ( finite6535610298543002082t_unit @ top_to1024881433771703089t_unit )
= ( ( finite7047982916621727056omplex @ top_to1861530291043981143omplex )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_823_finite__Plus__UNIV__iff,axiom,
( ( finite1813174609741014903omplex @ top_to5982683327856507902omplex )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite7047982916621727056omplex @ top_to1861530291043981143omplex ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_824_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_825_finite__Plus__UNIV__iff,axiom,
( ( finite4327512606132785245t_unit @ top_to5465250082899874788t_unit )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_826_finite__Plus__UNIV__iff,axiom,
( ( finite9054228731462424214omplex @ top_to302141804076443109omplex )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite7047982916621727056omplex @ top_to1861530291043981143omplex ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_827_finite__Plus__UNIV__iff,axiom,
( ( finite4401952911629260215it_nat @ top_to2894617605782473790it_nat )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_828_finite__Plus__UNIV__iff,axiom,
( ( finite3146551501593861116t_unit @ top_to2771918933716375115t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_829_finite__Prod__UNIV,axiom,
( ( finite7047982916621727056omplex @ top_to1861530291043981143omplex )
=> ( ( finite7047982916621727056omplex @ top_to1861530291043981143omplex )
=> ( finite7366070217807234576omplex @ top_to6470415304537177623omplex ) ) ) ).
% finite_Prod_UNIV
thf(fact_830_finite__Prod__UNIV,axiom,
( ( finite7047982916621727056omplex @ top_to1861530291043981143omplex )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite5159396851285225149ex_nat @ top_to3326070779871765132ex_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_831_finite__Prod__UNIV,axiom,
( ( finite7047982916621727056omplex @ top_to1861530291043981143omplex )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite4703860625191999094t_unit @ top_to843579388925295997t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_832_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite7047982916621727056omplex @ top_to1861530291043981143omplex )
=> ( finite3477794401636876771omplex @ top_to7196737821178430130omplex ) ) ) ).
% finite_Prod_UNIV
thf(fact_833_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_834_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite5113082511001691337t_unit @ top_to8544742955230171288t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_835_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite7047982916621727056omplex @ top_to1861530291043981143omplex )
=> ( finite7222479058111421226omplex @ top_to120839759230036017omplex ) ) ) ).
% finite_Prod_UNIV
thf(fact_836_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_837_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_838_finite__prod,axiom,
( ( finite7366070217807234576omplex @ top_to6470415304537177623omplex )
= ( ( finite7047982916621727056omplex @ top_to1861530291043981143omplex )
& ( finite7047982916621727056omplex @ top_to1861530291043981143omplex ) ) ) ).
% finite_prod
thf(fact_839_finite__prod,axiom,
( ( finite5159396851285225149ex_nat @ top_to3326070779871765132ex_nat )
= ( ( finite7047982916621727056omplex @ top_to1861530291043981143omplex )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_840_finite__prod,axiom,
( ( finite4703860625191999094t_unit @ top_to843579388925295997t_unit )
= ( ( finite7047982916621727056omplex @ top_to1861530291043981143omplex )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_841_finite__prod,axiom,
( ( finite3477794401636876771omplex @ top_to7196737821178430130omplex )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite7047982916621727056omplex @ top_to1861530291043981143omplex ) ) ) ).
% finite_prod
thf(fact_842_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_843_finite__prod,axiom,
( ( finite5113082511001691337t_unit @ top_to8544742955230171288t_unit )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_844_finite__prod,axiom,
( ( finite7222479058111421226omplex @ top_to120839759230036017omplex )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite7047982916621727056omplex @ top_to1861530291043981143omplex ) ) ) ).
% finite_prod
thf(fact_845_finite__prod,axiom,
( ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_846_finite__prod,axiom,
( ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_847_Finite__Set_Ofinite__set,axiom,
( ( finite1349200545324696496omplex @ top_to5556459383890298295omplex )
= ( finite7047982916621727056omplex @ top_to1861530291043981143omplex ) ) ).
% Finite_Set.finite_set
thf(fact_848_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_849_Finite__Set_Ofinite__set,axiom,
( ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% Finite_Set.finite_set
thf(fact_850_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_851_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_852_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N4: nat] :
( X
!= ( suc @ N4 ) ) ) ).
% list_decode.cases
thf(fact_853_finite__has__minimal2,axiom,
! [A4: set_mat_complex,A3: mat_complex] :
( ( finite7047982916621727056omplex @ A4 )
=> ( ( member_mat_complex @ A3 @ A4 )
=> ? [X2: mat_complex] :
( ( member_mat_complex @ X2 @ A4 )
& ( ord_le1403324449407493959omplex @ X2 @ A3 )
& ! [Xa2: mat_complex] :
( ( member_mat_complex @ Xa2 @ A4 )
=> ( ( ord_le1403324449407493959omplex @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_854_finite__has__minimal2,axiom,
! [A4: set_nat,A3: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat2 @ A3 @ A4 )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A4 )
& ( ord_less_eq_nat @ X2 @ A3 )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ A4 )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_855_finite__has__minimal2,axiom,
! [A4: set_real,A3: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A3 @ A4 )
=> ? [X2: real] :
( ( member_real @ X2 @ A4 )
& ( ord_less_eq_real @ X2 @ A3 )
& ! [Xa2: real] :
( ( member_real @ Xa2 @ A4 )
=> ( ( ord_less_eq_real @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_856_finite__has__maximal2,axiom,
! [A4: set_mat_complex,A3: mat_complex] :
( ( finite7047982916621727056omplex @ A4 )
=> ( ( member_mat_complex @ A3 @ A4 )
=> ? [X2: mat_complex] :
( ( member_mat_complex @ X2 @ A4 )
& ( ord_le1403324449407493959omplex @ A3 @ X2 )
& ! [Xa2: mat_complex] :
( ( member_mat_complex @ Xa2 @ A4 )
=> ( ( ord_le1403324449407493959omplex @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_857_finite__has__maximal2,axiom,
! [A4: set_nat,A3: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat2 @ A3 @ A4 )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A4 )
& ( ord_less_eq_nat @ A3 @ X2 )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ A4 )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_858_finite__has__maximal2,axiom,
! [A4: set_real,A3: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A3 @ A4 )
=> ? [X2: real] :
( ( member_real @ X2 @ A4 )
& ( ord_less_eq_real @ A3 @ X2 )
& ! [Xa2: real] :
( ( member_real @ Xa2 @ A4 )
=> ( ( ord_less_eq_real @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_859_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_860_ex__new__if__finite,axiom,
! [A4: set_mat_complex] :
( ~ ( finite7047982916621727056omplex @ top_to1861530291043981143omplex )
=> ( ( finite7047982916621727056omplex @ A4 )
=> ? [A: mat_complex] :
~ ( member_mat_complex @ A @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_861_ex__new__if__finite,axiom,
! [A4: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A4 )
=> ? [A: nat] :
~ ( member_nat2 @ A @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_862_ex__new__if__finite,axiom,
! [A4: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ A4 )
=> ? [A: product_unit] :
~ ( member_Product_unit @ A @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_863_finite__class_Ofinite__UNIV,axiom,
finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ).
% finite_class.finite_UNIV
thf(fact_864_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M5: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( member_nat2 @ N3 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_865_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K: nat] :
? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( F @ K @ I2 ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ! [K2: nat] :
? [K5: nat] :
( ( ord_less_eq_nat @ K2 @ K5 )
& ( F @ K5 @ I3 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_866_Suc__mult__le__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K3 ) @ M ) @ ( times_times_nat @ ( suc @ K3 ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_867_Suc__mult__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K3 ) @ M )
= ( times_times_nat @ ( suc @ K3 ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_868_mult__le__mono2,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J ) ) ) ).
% mult_le_mono2
thf(fact_869_mult__le__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ).
% mult_le_mono1
thf(fact_870_mult__le__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_871_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_872_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_873_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_874_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M5: nat] :
! [X3: nat] :
( ( member_nat2 @ X3 @ N6 )
=> ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_875_class__cring_Om__ac_I3_J,axiom,
! [X: complex,Y2: complex,Z3: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( member_complex2 @ Y2 @ top_top_set_complex )
=> ( ( member_complex2 @ Z3 @ top_top_set_complex )
=> ( ( times_times_complex @ X @ ( times_times_complex @ Y2 @ Z3 ) )
= ( times_times_complex @ Y2 @ ( times_times_complex @ X @ Z3 ) ) ) ) ) ) ).
% class_cring.m_ac(3)
thf(fact_876_class__cring_Om__ac_I2_J,axiom,
! [X: complex,Y2: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( member_complex2 @ Y2 @ top_top_set_complex )
=> ( ( times_times_complex @ X @ Y2 )
= ( times_times_complex @ Y2 @ X ) ) ) ) ).
% class_cring.m_ac(2)
thf(fact_877_class__cring_Om__ac_I1_J,axiom,
! [X: complex,Y2: complex,Z3: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( member_complex2 @ Y2 @ top_top_set_complex )
=> ( ( member_complex2 @ Z3 @ top_top_set_complex )
=> ( ( times_times_complex @ ( times_times_complex @ X @ Y2 ) @ Z3 )
= ( times_times_complex @ X @ ( times_times_complex @ Y2 @ Z3 ) ) ) ) ) ) ).
% class_cring.m_ac(1)
thf(fact_878_class__ring_Oring__simprules_I11_J,axiom,
! [X: complex,Y2: complex,Z3: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( member_complex2 @ Y2 @ top_top_set_complex )
=> ( ( member_complex2 @ Z3 @ top_top_set_complex )
=> ( ( times_times_complex @ ( times_times_complex @ X @ Y2 ) @ Z3 )
= ( times_times_complex @ X @ ( times_times_complex @ Y2 @ Z3 ) ) ) ) ) ) ).
% class_ring.ring_simprules(11)
thf(fact_879_class__ring_Oring__simprules_I5_J,axiom,
! [X: complex,Y2: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( member_complex2 @ Y2 @ top_top_set_complex )
=> ( member_complex2 @ ( times_times_complex @ X @ Y2 ) @ top_top_set_complex ) ) ) ).
% class_ring.ring_simprules(5)
thf(fact_880_class__cring_Ocring__simprules_I5_J,axiom,
! [X: complex,Y2: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( member_complex2 @ Y2 @ top_top_set_complex )
=> ( member_complex2 @ ( times_times_complex @ X @ Y2 ) @ top_top_set_complex ) ) ) ).
% class_cring.cring_simprules(5)
thf(fact_881_class__semiring_Osemiring__simprules_I8_J,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( member_nat2 @ X @ top_top_set_nat )
=> ( ( member_nat2 @ Y2 @ top_top_set_nat )
=> ( ( member_nat2 @ Z3 @ top_top_set_nat )
=> ( ( times_times_nat @ ( times_times_nat @ X @ Y2 ) @ Z3 )
= ( times_times_nat @ X @ ( times_times_nat @ Y2 @ Z3 ) ) ) ) ) ) ).
% class_semiring.semiring_simprules(8)
thf(fact_882_class__semiring_Osemiring__simprules_I8_J,axiom,
! [X: complex,Y2: complex,Z3: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( member_complex2 @ Y2 @ top_top_set_complex )
=> ( ( member_complex2 @ Z3 @ top_top_set_complex )
=> ( ( times_times_complex @ ( times_times_complex @ X @ Y2 ) @ Z3 )
= ( times_times_complex @ X @ ( times_times_complex @ Y2 @ Z3 ) ) ) ) ) ) ).
% class_semiring.semiring_simprules(8)
thf(fact_883_class__semiring_Osemiring__simprules_I3_J,axiom,
! [X: nat,Y2: nat] :
( ( member_nat2 @ X @ top_top_set_nat )
=> ( ( member_nat2 @ Y2 @ top_top_set_nat )
=> ( member_nat2 @ ( times_times_nat @ X @ Y2 ) @ top_top_set_nat ) ) ) ).
% class_semiring.semiring_simprules(3)
thf(fact_884_class__semiring_Osemiring__simprules_I3_J,axiom,
! [X: complex,Y2: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( member_complex2 @ Y2 @ top_top_set_complex )
=> ( member_complex2 @ ( times_times_complex @ X @ Y2 ) @ top_top_set_complex ) ) ) ).
% class_semiring.semiring_simprules(3)
thf(fact_885_class__semiring_Oone__closed,axiom,
member_real @ one_one_real @ top_top_set_real ).
% class_semiring.one_closed
thf(fact_886_class__semiring_Oone__closed,axiom,
member_nat2 @ one_one_nat @ top_top_set_nat ).
% class_semiring.one_closed
thf(fact_887_class__cring_Ocring__simprules_I6_J,axiom,
member_real @ one_one_real @ top_top_set_real ).
% class_cring.cring_simprules(6)
thf(fact_888_class__ring_Oring__simprules_I6_J,axiom,
member_real @ one_one_real @ top_top_set_real ).
% class_ring.ring_simprules(6)
thf(fact_889_UNIV__not__empty,axiom,
top_to1861530291043981143omplex != bot_bo7165004461764951667omplex ).
% UNIV_not_empty
thf(fact_890_UNIV__not__empty,axiom,
top_top_set_nat != bot_bot_set_nat ).
% UNIV_not_empty
thf(fact_891_UNIV__not__empty,axiom,
top_to1996260823553986621t_unit != bot_bo3957492148770167129t_unit ).
% UNIV_not_empty
thf(fact_892_le__bot,axiom,
! [A3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A3 @ bot_bo7165004461764951667omplex )
=> ( A3 = bot_bo7165004461764951667omplex ) ) ).
% le_bot
thf(fact_893_le__bot,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
=> ( A3 = bot_bot_nat ) ) ).
% le_bot
thf(fact_894_bot__least,axiom,
! [A3: set_mat_complex] : ( ord_le3632134057777142183omplex @ bot_bo7165004461764951667omplex @ A3 ) ).
% bot_least
thf(fact_895_bot__least,axiom,
! [A3: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A3 ) ).
% bot_least
thf(fact_896_bot__unique,axiom,
! [A3: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A3 @ bot_bo7165004461764951667omplex )
= ( A3 = bot_bo7165004461764951667omplex ) ) ).
% bot_unique
thf(fact_897_bot__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
= ( A3 = bot_bot_nat ) ) ).
% bot_unique
thf(fact_898_class__semiring_Ocarrier__not__empty,axiom,
top_top_set_nat != bot_bot_set_nat ).
% class_semiring.carrier_not_empty
thf(fact_899_zero__compare__simps_I4_J,axiom,
! [A3: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_compare_simps(4)
thf(fact_900_zero__compare__simps_I8_J,axiom,
! [A3: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% zero_compare_simps(8)
thf(fact_901_mult__sign__intros_I4_J,axiom,
! [A3: complex,B2: complex] :
( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B2 ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_902_mult__sign__intros_I4_J,axiom,
! [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B2 ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_903_mult__sign__intros_I3_J,axiom,
! [A3: complex,B2: complex] :
( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B2 ) @ zero_zero_complex ) ) ) ).
% mult_sign_intros(3)
thf(fact_904_mult__sign__intros_I3_J,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(3)
thf(fact_905_mult__sign__intros_I3_J,axiom,
! [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(3)
thf(fact_906_mult__sign__intros_I2_J,axiom,
! [A3: complex,B2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ B2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B2 ) @ zero_zero_complex ) ) ) ).
% mult_sign_intros(2)
thf(fact_907_mult__sign__intros_I2_J,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(2)
thf(fact_908_mult__sign__intros_I2_J,axiom,
! [A3: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(2)
thf(fact_909_mult__sign__intros_I1_J,axiom,
! [A3: complex,B2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B2 )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B2 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_910_mult__sign__intros_I1_J,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B2 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_911_mult__sign__intros_I1_J,axiom,
! [A3: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B2 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_912_mult__mono,axiom,
! [A3: complex,B2: complex,C2: complex,D2: complex] :
( ( ord_less_eq_complex @ A3 @ B2 )
=> ( ( ord_less_eq_complex @ C2 @ D2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_913_mult__mono,axiom,
! [A3: nat,B2: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_914_mult__mono,axiom,
! [A3: real,B2: real,C2: real,D2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_915_mult__mono_H,axiom,
! [A3: complex,B2: complex,C2: complex,D2: complex] :
( ( ord_less_eq_complex @ A3 @ B2 )
=> ( ( ord_less_eq_complex @ C2 @ D2 )
=> ( ( 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 @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_916_mult__mono_H,axiom,
! [A3: nat,B2: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ( 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 @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_917_mult__mono_H,axiom,
! [A3: real,B2: real,C2: real,D2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D2 )
=> ( ( 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 @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_918_zero__le__square,axiom,
! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_919_split__mult__pos__le,axiom,
! [A3: complex,B2: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
& ( ord_less_eq_complex @ zero_zero_complex @ B2 ) )
| ( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
& ( ord_less_eq_complex @ B2 @ zero_zero_complex ) ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_920_split__mult__pos__le,axiom,
! [A3: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_921_mult__left__mono__neg,axiom,
! [B2: complex,A3: complex,C2: complex] :
( ( ord_less_eq_complex @ B2 @ A3 )
=> ( ( ord_less_eq_complex @ C2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_922_mult__left__mono__neg,axiom,
! [B2: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_923_mult__left__mono,axiom,
! [A3: complex,B2: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_924_mult__left__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_925_mult__left__mono,axiom,
! [A3: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_926_mult__right__mono__neg,axiom,
! [B2: complex,A3: complex,C2: complex] :
( ( ord_less_eq_complex @ B2 @ A3 )
=> ( ( ord_less_eq_complex @ C2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B2 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_927_mult__right__mono__neg,axiom,
! [B2: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_928_mult__right__mono,axiom,
! [A3: complex,B2: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C2 ) @ ( times_times_complex @ B2 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_929_mult__right__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_930_mult__right__mono,axiom,
! [A3: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_931_split__mult__neg__le,axiom,
! [A3: complex,B2: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
& ( ord_less_eq_complex @ B2 @ zero_zero_complex ) )
| ( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
& ( ord_less_eq_complex @ zero_zero_complex @ B2 ) ) )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B2 ) @ zero_zero_complex ) ) ).
% split_mult_neg_le
thf(fact_932_split__mult__neg__le,axiom,
! [A3: nat,B2: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
& ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_933_split__mult__neg__le,axiom,
! [A3: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B2 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_934_mult__nonneg__nonpos2,axiom,
! [A3: complex,B2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ B2 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ B2 @ A3 ) @ zero_zero_complex ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_935_mult__nonneg__nonpos2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_936_mult__nonneg__nonpos2,axiom,
! [A3: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B2 @ A3 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_937_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: complex,B2: complex,C2: complex] :
( ( ord_less_eq_complex @ A3 @ B2 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C2 )
=> ( ord_less_eq_complex @ ( times_times_complex @ C2 @ A3 ) @ ( times_times_complex @ C2 @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_938_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_939_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_940_semiring__norm_I112_J,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% semiring_norm(112)
thf(fact_941_semiring__norm_I112_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% semiring_norm(112)
thf(fact_942_semiring__norm_I111_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% semiring_norm(111)
thf(fact_943_semiring__norm_I111_J,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% semiring_norm(111)
thf(fact_944_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_945_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_946_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_947_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_948_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_949_finite__has__minimal,axiom,
! [A4: set_mat_complex] :
( ( finite7047982916621727056omplex @ A4 )
=> ( ( A4 != bot_bo7165004461764951667omplex )
=> ? [X2: mat_complex] :
( ( member_mat_complex @ X2 @ A4 )
& ! [Xa2: mat_complex] :
( ( member_mat_complex @ Xa2 @ A4 )
=> ( ( ord_le1403324449407493959omplex @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_950_finite__has__minimal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A4 )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ A4 )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_951_finite__has__minimal,axiom,
! [A4: set_real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ? [X2: real] :
( ( member_real @ X2 @ A4 )
& ! [Xa2: real] :
( ( member_real @ Xa2 @ A4 )
=> ( ( ord_less_eq_real @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_952_finite__has__maximal,axiom,
! [A4: set_mat_complex] :
( ( finite7047982916621727056omplex @ A4 )
=> ( ( A4 != bot_bo7165004461764951667omplex )
=> ? [X2: mat_complex] :
( ( member_mat_complex @ X2 @ A4 )
& ! [Xa2: mat_complex] :
( ( member_mat_complex @ Xa2 @ A4 )
=> ( ( ord_le1403324449407493959omplex @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_953_finite__has__maximal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A4 )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ A4 )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_954_finite__has__maximal,axiom,
! [A4: set_real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ? [X2: real] :
( ( member_real @ X2 @ A4 )
& ! [Xa2: real] :
( ( member_real @ Xa2 @ A4 )
=> ( ( ord_less_eq_real @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_955_Suc__length__conv,axiom,
! [N: nat,Xs: list_complex] :
( ( ( suc @ N )
= ( size_s3451745648224563538omplex @ Xs ) )
= ( ? [Y3: complex,Ys3: list_complex] :
( ( Xs
= ( cons_complex @ Y3 @ Ys3 ) )
& ( ( size_s3451745648224563538omplex @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_956_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_957_length__Suc__conv,axiom,
! [Xs: list_complex,N: nat] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( suc @ N ) )
= ( ? [Y3: complex,Ys3: list_complex] :
( ( Xs
= ( cons_complex @ Y3 @ Ys3 ) )
& ( ( size_s3451745648224563538omplex @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_958_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_959_numeral__nat_I7_J,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% numeral_nat(7)
thf(fact_960_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_961_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_962_Suc__mult__less__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K3 ) @ M ) @ ( times_times_nat @ ( suc @ K3 ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_963_nth__Cons__Suc,axiom,
! [X: mat_complex,Xs: list_mat_complex,N: nat] :
( ( nth_mat_complex @ ( cons_mat_complex @ X @ Xs ) @ ( suc @ N ) )
= ( nth_mat_complex @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_964_nth__Cons__Suc,axiom,
! [X: real,Xs: list_real,N: nat] :
( ( nth_real @ ( cons_real @ X @ Xs ) @ ( suc @ N ) )
= ( nth_real @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_965_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_966_nth__Cons__Suc,axiom,
! [X: complex,Xs: list_complex,N: nat] :
( ( nth_complex @ ( cons_complex @ X @ Xs ) @ ( suc @ N ) )
= ( nth_complex @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_967_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_968_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_969_impossible__Cons,axiom,
! [Xs: list_complex,Ys: list_complex,X: complex] :
( ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ ( size_s3451745648224563538omplex @ Ys ) )
=> ( Xs
!= ( cons_complex @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_970_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_971_class__ring_Oring__simprules_I25_J,axiom,
! [X: real] :
( ( member_real @ X @ top_top_set_real )
=> ( ( times_times_real @ X @ zero_zero_real )
= zero_zero_real ) ) ).
% class_ring.ring_simprules(25)
thf(fact_972_class__ring_Oring__simprules_I25_J,axiom,
! [X: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( times_times_complex @ X @ zero_zero_complex )
= zero_zero_complex ) ) ).
% class_ring.ring_simprules(25)
thf(fact_973_class__ring_Oring__simprules_I24_J,axiom,
! [X: real] :
( ( member_real @ X @ top_top_set_real )
=> ( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ) ).
% class_ring.ring_simprules(24)
thf(fact_974_class__ring_Oring__simprules_I24_J,axiom,
! [X: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( times_times_complex @ zero_zero_complex @ X )
= zero_zero_complex ) ) ).
% class_ring.ring_simprules(24)
thf(fact_975_class__cring_Ocring__simprules_I27_J,axiom,
! [X: real] :
( ( member_real @ X @ top_top_set_real )
=> ( ( times_times_real @ X @ zero_zero_real )
= zero_zero_real ) ) ).
% class_cring.cring_simprules(27)
thf(fact_976_class__cring_Ocring__simprules_I27_J,axiom,
! [X: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( times_times_complex @ X @ zero_zero_complex )
= zero_zero_complex ) ) ).
% class_cring.cring_simprules(27)
thf(fact_977_class__cring_Ocring__simprules_I26_J,axiom,
! [X: real] :
( ( member_real @ X @ top_top_set_real )
=> ( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ) ).
% class_cring.cring_simprules(26)
thf(fact_978_class__cring_Ocring__simprules_I26_J,axiom,
! [X: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( times_times_complex @ zero_zero_complex @ X )
= zero_zero_complex ) ) ).
% class_cring.cring_simprules(26)
thf(fact_979_class__semiring_Osemiring__simprules_I15_J,axiom,
! [X: real] :
( ( member_real @ X @ top_top_set_real )
=> ( ( times_times_real @ X @ zero_zero_real )
= zero_zero_real ) ) ).
% class_semiring.semiring_simprules(15)
thf(fact_980_class__semiring_Osemiring__simprules_I15_J,axiom,
! [X: nat] :
( ( member_nat2 @ X @ top_top_set_nat )
=> ( ( times_times_nat @ X @ zero_zero_nat )
= zero_zero_nat ) ) ).
% class_semiring.semiring_simprules(15)
thf(fact_981_class__semiring_Osemiring__simprules_I15_J,axiom,
! [X: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( times_times_complex @ X @ zero_zero_complex )
= zero_zero_complex ) ) ).
% class_semiring.semiring_simprules(15)
thf(fact_982_class__semiring_Osemiring__simprules_I14_J,axiom,
! [X: real] :
( ( member_real @ X @ top_top_set_real )
=> ( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ) ).
% class_semiring.semiring_simprules(14)
thf(fact_983_class__semiring_Osemiring__simprules_I14_J,axiom,
! [X: nat] :
( ( member_nat2 @ X @ top_top_set_nat )
=> ( ( times_times_nat @ zero_zero_nat @ X )
= zero_zero_nat ) ) ).
% class_semiring.semiring_simprules(14)
thf(fact_984_class__semiring_Osemiring__simprules_I14_J,axiom,
! [X: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( times_times_complex @ zero_zero_complex @ X )
= zero_zero_complex ) ) ).
% class_semiring.semiring_simprules(14)
thf(fact_985_class__field_Oconc,axiom,
! [A3: real,B2: real,C2: real] :
( ( A3 != zero_zero_real )
=> ( ( member_real @ A3 @ top_top_set_real )
=> ( ( member_real @ B2 @ top_top_set_real )
=> ( ( member_real @ C2 @ top_top_set_real )
=> ( ( ( times_times_real @ B2 @ A3 )
= ( times_times_real @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ) ) ) ).
% class_field.conc
thf(fact_986_class__field_Oconc,axiom,
! [A3: complex,B2: complex,C2: complex] :
( ( A3 != zero_zero_complex )
=> ( ( member_complex2 @ A3 @ top_top_set_complex )
=> ( ( member_complex2 @ B2 @ top_top_set_complex )
=> ( ( member_complex2 @ C2 @ top_top_set_complex )
=> ( ( ( times_times_complex @ B2 @ A3 )
= ( times_times_complex @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ) ) ) ).
% class_field.conc
thf(fact_987_class__field_Ointegral,axiom,
! [A3: real,B2: real] :
( ( ( times_times_real @ A3 @ B2 )
= zero_zero_real )
=> ( ( member_real @ A3 @ top_top_set_real )
=> ( ( member_real @ B2 @ top_top_set_real )
=> ( ( A3 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ) ) ).
% class_field.integral
thf(fact_988_class__field_Ointegral,axiom,
! [A3: complex,B2: complex] :
( ( ( times_times_complex @ A3 @ B2 )
= zero_zero_complex )
=> ( ( member_complex2 @ A3 @ top_top_set_complex )
=> ( ( member_complex2 @ B2 @ top_top_set_complex )
=> ( ( A3 = zero_zero_complex )
| ( B2 = zero_zero_complex ) ) ) ) ) ).
% class_field.integral
thf(fact_989_class__field_Om__lcancel,axiom,
! [A3: real,B2: real,C2: real] :
( ( A3 != zero_zero_real )
=> ( ( member_real @ A3 @ top_top_set_real )
=> ( ( member_real @ B2 @ top_top_set_real )
=> ( ( member_real @ C2 @ top_top_set_real )
=> ( ( ( times_times_real @ A3 @ B2 )
= ( times_times_real @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ) ) ) ).
% class_field.m_lcancel
thf(fact_990_class__field_Om__lcancel,axiom,
! [A3: complex,B2: complex,C2: complex] :
( ( A3 != zero_zero_complex )
=> ( ( member_complex2 @ A3 @ top_top_set_complex )
=> ( ( member_complex2 @ B2 @ top_top_set_complex )
=> ( ( member_complex2 @ C2 @ top_top_set_complex )
=> ( ( ( times_times_complex @ A3 @ B2 )
= ( times_times_complex @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ) ) ) ).
% class_field.m_lcancel
thf(fact_991_class__field_Ointegral__iff,axiom,
! [A3: real,B2: real] :
( ( member_real @ A3 @ top_top_set_real )
=> ( ( member_real @ B2 @ top_top_set_real )
=> ( ( ( times_times_real @ A3 @ B2 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ) ) ).
% class_field.integral_iff
thf(fact_992_class__field_Ointegral__iff,axiom,
! [A3: complex,B2: complex] :
( ( member_complex2 @ A3 @ top_top_set_complex )
=> ( ( member_complex2 @ B2 @ top_top_set_complex )
=> ( ( ( times_times_complex @ A3 @ B2 )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( B2 = zero_zero_complex ) ) ) ) ) ).
% class_field.integral_iff
thf(fact_993_class__semiring_Oone__unique,axiom,
! [U2: real] :
( ( member_real @ U2 @ top_top_set_real )
=> ( ! [X2: real] :
( ( member_real @ X2 @ top_top_set_real )
=> ( ( times_times_real @ U2 @ X2 )
= X2 ) )
=> ( U2 = one_one_real ) ) ) ).
% class_semiring.one_unique
thf(fact_994_class__semiring_Oone__unique,axiom,
! [U2: nat] :
( ( member_nat2 @ U2 @ top_top_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ top_top_set_nat )
=> ( ( times_times_nat @ U2 @ X2 )
= X2 ) )
=> ( U2 = one_one_nat ) ) ) ).
% class_semiring.one_unique
thf(fact_995_class__semiring_Oone__unique,axiom,
! [U2: complex] :
( ( member_complex2 @ U2 @ top_top_set_complex )
=> ( ! [X2: complex] :
( ( member_complex2 @ X2 @ top_top_set_complex )
=> ( ( times_times_complex @ U2 @ X2 )
= X2 ) )
=> ( U2 = one_one_complex ) ) ) ).
% class_semiring.one_unique
thf(fact_996_class__semiring_Oinv__unique,axiom,
! [Y2: real,X: real,Y6: real] :
( ( ( times_times_real @ Y2 @ X )
= one_one_real )
=> ( ( ( times_times_real @ X @ Y6 )
= one_one_real )
=> ( ( member_real @ X @ top_top_set_real )
=> ( ( member_real @ Y2 @ top_top_set_real )
=> ( ( member_real @ Y6 @ top_top_set_real )
=> ( Y2 = Y6 ) ) ) ) ) ) ).
% class_semiring.inv_unique
thf(fact_997_class__semiring_Oinv__unique,axiom,
! [Y2: nat,X: nat,Y6: nat] :
( ( ( times_times_nat @ Y2 @ X )
= one_one_nat )
=> ( ( ( times_times_nat @ X @ Y6 )
= one_one_nat )
=> ( ( member_nat2 @ X @ top_top_set_nat )
=> ( ( member_nat2 @ Y2 @ top_top_set_nat )
=> ( ( member_nat2 @ Y6 @ top_top_set_nat )
=> ( Y2 = Y6 ) ) ) ) ) ) ).
% class_semiring.inv_unique
thf(fact_998_class__semiring_Oinv__unique,axiom,
! [Y2: complex,X: complex,Y6: complex] :
( ( ( times_times_complex @ Y2 @ X )
= one_one_complex )
=> ( ( ( times_times_complex @ X @ Y6 )
= one_one_complex )
=> ( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( member_complex2 @ Y2 @ top_top_set_complex )
=> ( ( member_complex2 @ Y6 @ top_top_set_complex )
=> ( Y2 = Y6 ) ) ) ) ) ) ).
% class_semiring.inv_unique
thf(fact_999_class__semiring_Or__one,axiom,
! [X: real] :
( ( member_real @ X @ top_top_set_real )
=> ( ( times_times_real @ X @ one_one_real )
= X ) ) ).
% class_semiring.r_one
thf(fact_1000_class__semiring_Or__one,axiom,
! [X: nat] :
( ( member_nat2 @ X @ top_top_set_nat )
=> ( ( times_times_nat @ X @ one_one_nat )
= X ) ) ).
% class_semiring.r_one
thf(fact_1001_class__semiring_Or__one,axiom,
! [X: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( times_times_complex @ X @ one_one_complex )
= X ) ) ).
% class_semiring.r_one
thf(fact_1002_class__semiring_Ol__one,axiom,
! [X: real] :
( ( member_real @ X @ top_top_set_real )
=> ( ( times_times_real @ one_one_real @ X )
= X ) ) ).
% class_semiring.l_one
thf(fact_1003_class__semiring_Ol__one,axiom,
! [X: nat] :
( ( member_nat2 @ X @ top_top_set_nat )
=> ( ( times_times_nat @ one_one_nat @ X )
= X ) ) ).
% class_semiring.l_one
thf(fact_1004_class__semiring_Ol__one,axiom,
! [X: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( times_times_complex @ one_one_complex @ X )
= X ) ) ).
% class_semiring.l_one
thf(fact_1005_class__cring_Ocring__simprules_I12_J,axiom,
! [X: real] :
( ( member_real @ X @ top_top_set_real )
=> ( ( times_times_real @ one_one_real @ X )
= X ) ) ).
% class_cring.cring_simprules(12)
thf(fact_1006_class__cring_Ocring__simprules_I12_J,axiom,
! [X: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( times_times_complex @ one_one_complex @ X )
= X ) ) ).
% class_cring.cring_simprules(12)
thf(fact_1007_class__ring_Oring__simprules_I12_J,axiom,
! [X: real] :
( ( member_real @ X @ top_top_set_real )
=> ( ( times_times_real @ one_one_real @ X )
= X ) ) ).
% class_ring.ring_simprules(12)
thf(fact_1008_class__ring_Oring__simprules_I12_J,axiom,
! [X: complex] :
( ( member_complex2 @ X @ top_top_set_complex )
=> ( ( times_times_complex @ one_one_complex @ X )
= X ) ) ).
% class_ring.ring_simprules(12)
thf(fact_1009_prefix__length__le,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% prefix_length_le
thf(fact_1010_prefix__length__prefix,axiom,
! [Ps: list_nat,Xs: list_nat,Qs: list_nat] :
( ( prefix_nat @ Ps @ Xs )
=> ( ( prefix_nat @ Qs @ Xs )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ps ) @ ( size_size_list_nat @ Qs ) )
=> ( prefix_nat @ Ps @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_1011_Diff__UNIV,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_1012_Diff__UNIV,axiom,
! [A4: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A4 @ top_to1996260823553986621t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Diff_UNIV
thf(fact_1013_Diff__UNIV,axiom,
! [A4: set_mat_complex] :
( ( minus_8760755521168068590omplex @ A4 @ top_to1861530291043981143omplex )
= bot_bo7165004461764951667omplex ) ).
% Diff_UNIV
thf(fact_1014_vector__space__over__itself_Ospan__UNIV,axiom,
( ( span_complex_complex @ times_times_complex @ top_top_set_complex )
= top_top_set_complex ) ).
% vector_space_over_itself.span_UNIV
thf(fact_1015_length__suffixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( suffixes_nat @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_suffixes
thf(fact_1016_gen__length__code_I2_J,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1017_gen__length__code_I2_J,axiom,
! [N: nat,X: complex,Xs: list_complex] :
( ( gen_length_complex @ N @ ( cons_complex @ X @ Xs ) )
= ( gen_length_complex @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1018_mult__le__cancel__left,axiom,
! [C2: real,A3: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1019_mult__le__cancel__right,axiom,
! [A3: real,C2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1020_mult__left__less__imp__less,axiom,
! [C2: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A3 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_1021_mult__left__less__imp__less,axiom,
! [C2: real,A3: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_1022_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: nat,B2: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1023_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: real,B2: real,C2: real,D2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( ( ord_less_real @ C2 @ D2 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1024_mult__less__cancel__left,axiom,
! [C2: real,A3: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1025_mult__right__less__imp__less,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A3 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_1026_mult__right__less__imp__less,axiom,
! [A3: real,C2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_1027_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: nat,B2: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ( 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 @ B2 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_1028_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: real,B2: real,C2: real,D2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( ( ord_less_real @ C2 @ D2 )
=> ( ( 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 @ B2 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_1029_mult__less__cancel__right,axiom,
! [A3: real,C2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A3 @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1030_mult__le__cancel__left__neg,axiom,
! [C2: real,A3: real,B2: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_eq_real @ B2 @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1031_mult__le__cancel__left__pos,axiom,
! [C2: real,A3: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_eq_real @ A3 @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1032_mult__left__le__imp__le,axiom,
! [C2: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1033_mult__left__le__imp__le,axiom,
! [C2: real,A3: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1034_mult__right__le__imp__le,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1035_mult__right__le__imp__le,axiom,
! [A3: real,C2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A3 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1036_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: nat,B2: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ( 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 @ B2 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1037_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: real,B2: real,C2: real,D2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
=> ( ( ord_less_real @ C2 @ D2 )
=> ( ( 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 @ B2 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_1038_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: nat,B2: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ( 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 @ B2 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1039_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: real,B2: real,C2: real,D2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D2 )
=> ( ( 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 @ B2 @ D2 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_1040_mult__le__cancel__iff2,axiom,
! [Z3: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ ( times_times_real @ Z3 @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1041_mult__le__cancel__iff1,axiom,
! [Z3: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y2 @ Z3 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1042_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_1043_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_1044_mult__le__one,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B2 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1045_mult__le__one,axiom,
! [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ B2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B2 ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_1046_mult__right__le__one__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y2 ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1047_mult__left__le__one__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y2 @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1048_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y4: nat] :
( ( member_nat2 @ Y4 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X2 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1049_finite__ranking__induct,axiom,
! [S: set_mat_complex,P: set_mat_complex > $o,F: mat_complex > nat] :
( ( finite7047982916621727056omplex @ S )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [X2: mat_complex,S3: set_mat_complex] :
( ( finite7047982916621727056omplex @ S3 )
=> ( ! [Y4: mat_complex] :
( ( member_mat_complex @ Y4 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_mat_complex @ X2 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1050_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y4: nat] :
( ( member_nat2 @ Y4 @ S3 )
=> ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X2 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1051_finite__ranking__induct,axiom,
! [S: set_mat_complex,P: set_mat_complex > $o,F: mat_complex > real] :
( ( finite7047982916621727056omplex @ S )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [X2: mat_complex,S3: set_mat_complex] :
( ( finite7047982916621727056omplex @ S3 )
=> ( ! [Y4: mat_complex] :
( ( member_mat_complex @ Y4 @ S3 )
=> ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_mat_complex @ X2 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1052_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1053_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1054_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1055_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1056_nat__mult__le__cancel__disj,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1057_nat__mult__le__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1058_mult__le__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1059_Longest__common__prefix__ex,axiom,
! [L3: set_list_nat] :
( ( L3 != bot_bot_set_list_nat )
=> ? [Ps2: list_nat] :
( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ L3 )
=> ( prefix_nat @ Ps2 @ X4 ) )
& ! [Qs2: list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ L3 )
=> ( prefix_nat @ Qs2 @ X2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Qs2 ) @ ( size_size_list_nat @ Ps2 ) ) ) ) ) ).
% Longest_common_prefix_ex
thf(fact_1060_Longest__common__prefix__unique,axiom,
! [L3: set_list_nat] :
( ( L3 != bot_bot_set_list_nat )
=> ? [X2: list_nat] :
( ! [Xa2: list_nat] :
( ( member_list_nat @ Xa2 @ L3 )
=> ( prefix_nat @ X2 @ Xa2 ) )
& ! [Qs2: list_nat] :
( ! [Xa3: list_nat] :
( ( member_list_nat @ Xa3 @ L3 )
=> ( prefix_nat @ Qs2 @ Xa3 ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Qs2 ) @ ( size_size_list_nat @ X2 ) ) )
& ! [Y4: list_nat] :
( ( ! [Xa3: list_nat] :
( ( member_list_nat @ Xa3 @ L3 )
=> ( prefix_nat @ Y4 @ Xa3 ) )
& ! [Qs3: list_nat] :
( ! [Xa2: list_nat] :
( ( member_list_nat @ Xa2 @ L3 )
=> ( prefix_nat @ Qs3 @ Xa2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Qs3 ) @ ( size_size_list_nat @ Y4 ) ) ) )
=> ( Y4 = X2 ) ) ) ) ).
% Longest_common_prefix_unique
thf(fact_1061_eq__comps__elem__le__length,axiom,
! [A3: nat,M: list_nat,L: list_complex] :
( ( ( cons_nat @ A3 @ M )
= ( commut93809757773076895omplex @ L ) )
=> ( ord_less_eq_nat @ A3 @ ( size_s3451745648224563538omplex @ L ) ) ) ).
% eq_comps_elem_le_length
thf(fact_1062_eq__comps__elem__le__length,axiom,
! [A3: nat,M: list_nat,L: list_nat] :
( ( ( cons_nat @ A3 @ M )
= ( commut2436974278740741825ps_nat @ L ) )
=> ( ord_less_eq_nat @ A3 @ ( size_size_list_nat @ L ) ) ) ).
% eq_comps_elem_le_length
thf(fact_1063_eq__comps__length,axiom,
! [L: list_complex] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( commut93809757773076895omplex @ L ) ) @ ( size_s3451745648224563538omplex @ L ) ) ).
% eq_comps_length
thf(fact_1064_eq__comps__length,axiom,
! [L: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( commut2436974278740741825ps_nat @ L ) ) @ ( size_size_list_nat @ L ) ) ).
% eq_comps_length
thf(fact_1065_arg__min__least,axiom,
! [S: set_nat,Y2: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat2 @ Y2 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_1066_arg__min__least,axiom,
! [S: set_mat_complex,Y2: mat_complex,F: mat_complex > nat] :
( ( finite7047982916621727056omplex @ S )
=> ( ( S != bot_bo7165004461764951667omplex )
=> ( ( member_mat_complex @ Y2 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic8691243167872488990ex_nat @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_1067_arg__min__least,axiom,
! [S: set_nat,Y2: nat,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat2 @ Y2 @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic488527866317076247t_real @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_1068_arg__min__least,axiom,
! [S: set_mat_complex,Y2: mat_complex,F: mat_complex > real] :
( ( finite7047982916621727056omplex @ S )
=> ( ( S != bot_bo7165004461764951667omplex )
=> ( ( member_mat_complex @ Y2 @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic2495253371305538042x_real @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_1069_mult__le__cancel__left1,axiom,
! [C2: real,B2: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1070_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_1071_mult__le__cancel__right1,axiom,
! [C2: real,B2: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1072_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_1073_mult__less__cancel__left1,axiom,
! [C2: real,B2: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1074_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_1075_mult__less__cancel__right1,axiom,
! [C2: real,B2: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1076_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_1077_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1078_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1079_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1080_length__Suc__conv__rev,axiom,
! [Xs: list_complex,N: nat] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( suc @ N ) )
= ( ? [Y3: complex,Ys3: list_complex] :
( ( Xs
= ( append_complex @ Ys3 @ ( cons_complex @ Y3 @ nil_complex ) ) )
& ( ( size_s3451745648224563538omplex @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1081_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ Y3 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1082_class__semiring_Oone__zeroD,axiom,
( ( one_one_real = zero_zero_real )
=> ( top_top_set_real
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) ) ).
% class_semiring.one_zeroD
thf(fact_1083_class__semiring_Oone__zeroD,axiom,
( ( one_one_nat = zero_zero_nat )
=> ( top_top_set_nat
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% class_semiring.one_zeroD
thf(fact_1084_class__semiring_Oone__zeroI,axiom,
( ( top_top_set_real
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) )
=> ( one_one_real = zero_zero_real ) ) ).
% class_semiring.one_zeroI
thf(fact_1085_class__semiring_Oone__zeroI,axiom,
( ( top_top_set_nat
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
=> ( one_one_nat = zero_zero_nat ) ) ).
% class_semiring.one_zeroI
thf(fact_1086_class__ring_Onontrivial__ring,axiom,
( ( top_top_set_real
!= ( insert_real @ zero_zero_real @ bot_bot_set_real ) )
=> ( zero_zero_real != one_one_real ) ) ).
% class_ring.nontrivial_ring
thf(fact_1087_class__semiring_Ocarrier__one__zero,axiom,
( ( top_top_set_real
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) )
= ( one_one_real = zero_zero_real ) ) ).
% class_semiring.carrier_one_zero
thf(fact_1088_class__semiring_Ocarrier__one__zero,axiom,
( ( top_top_set_nat
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
= ( one_one_nat = zero_zero_nat ) ) ).
% class_semiring.carrier_one_zero
thf(fact_1089_class__semiring_Ocarrier__one__not__zero,axiom,
( ( top_top_set_real
!= ( insert_real @ zero_zero_real @ bot_bot_set_real ) )
= ( one_one_real != zero_zero_real ) ) ).
% class_semiring.carrier_one_not_zero
thf(fact_1090_class__semiring_Ocarrier__one__not__zero,axiom,
( ( top_top_set_nat
!= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
= ( one_one_nat != zero_zero_nat ) ) ).
% class_semiring.carrier_one_not_zero
thf(fact_1091_permutation__insert__expand,axiom,
( permut4060954620988167523t_real
= ( ^ [I4: real,J2: nat,P5: real > nat,I5: real] : ( if_nat @ ( ord_less_real @ I5 @ I4 ) @ ( if_nat @ ( ord_less_nat @ ( P5 @ I5 ) @ J2 ) @ ( P5 @ I5 ) @ ( suc @ ( P5 @ I5 ) ) ) @ ( if_nat @ ( I5 = I4 ) @ J2 @ ( if_nat @ ( ord_less_nat @ ( P5 @ ( minus_minus_real @ I5 @ one_one_real ) ) @ J2 ) @ ( P5 @ ( minus_minus_real @ I5 @ one_one_real ) ) @ ( suc @ ( P5 @ ( minus_minus_real @ I5 @ one_one_real ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_1092_permutation__insert__expand,axiom,
( permut3695043542826343943rt_nat
= ( ^ [I4: nat,J2: nat,P5: nat > nat,I5: nat] : ( if_nat @ ( ord_less_nat @ I5 @ I4 ) @ ( if_nat @ ( ord_less_nat @ ( P5 @ I5 ) @ J2 ) @ ( P5 @ I5 ) @ ( suc @ ( P5 @ I5 ) ) ) @ ( if_nat @ ( I5 = I4 ) @ J2 @ ( if_nat @ ( ord_less_nat @ ( P5 @ ( minus_minus_nat @ I5 @ one_one_nat ) ) @ J2 ) @ ( P5 @ ( minus_minus_nat @ I5 @ one_one_nat ) ) @ ( suc @ ( P5 @ ( minus_minus_nat @ I5 @ one_one_nat ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_1093_length__append__singleton,axiom,
! [Xs: list_complex,X: complex] :
( ( size_s3451745648224563538omplex @ ( append_complex @ Xs @ ( cons_complex @ X @ nil_complex ) ) )
= ( suc @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1094_length__append__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1095_field__le__mult__one__interval,axiom,
! [X: real,Y2: real] :
( ! [Z2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ Z2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ Y2 ) ) )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% field_le_mult_one_interval
thf(fact_1096_rev__finite__subset,axiom,
! [B: set_mat_complex,A4: set_mat_complex] :
( ( finite7047982916621727056omplex @ B )
=> ( ( ord_le3632134057777142183omplex @ A4 @ B )
=> ( finite7047982916621727056omplex @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_1097_rev__finite__subset,axiom,
! [B: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A4 @ B )
=> ( finite_finite_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_1098_infinite__super,axiom,
! [S: set_mat_complex,T: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ S @ T )
=> ( ~ ( finite7047982916621727056omplex @ S )
=> ~ ( finite7047982916621727056omplex @ T ) ) ) ).
% infinite_super
thf(fact_1099_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_1100_finite__subset,axiom,
! [A4: set_mat_complex,B: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A4 @ B )
=> ( ( finite7047982916621727056omplex @ B )
=> ( finite7047982916621727056omplex @ A4 ) ) ) ).
% finite_subset
thf(fact_1101_finite__subset,axiom,
! [A4: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_1102_subset__empty,axiom,
! [A4: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A4 @ bot_bo7165004461764951667omplex )
= ( A4 = bot_bo7165004461764951667omplex ) ) ).
% subset_empty
thf(fact_1103_empty__subsetI,axiom,
! [A4: set_mat_complex] : ( ord_le3632134057777142183omplex @ bot_bo7165004461764951667omplex @ A4 ) ).
% empty_subsetI
thf(fact_1104_subset__singletonD,axiom,
! [A4: set_mat_complex,X: mat_complex] :
( ( ord_le3632134057777142183omplex @ A4 @ ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) )
=> ( ( A4 = bot_bo7165004461764951667omplex )
| ( A4
= ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) ) ) ) ).
% subset_singletonD
thf(fact_1105_subset__singleton__iff,axiom,
! [X6: set_mat_complex,A3: mat_complex] :
( ( ord_le3632134057777142183omplex @ X6 @ ( insert_mat_complex @ A3 @ bot_bo7165004461764951667omplex ) )
= ( ( X6 = bot_bo7165004461764951667omplex )
| ( X6
= ( insert_mat_complex @ A3 @ bot_bo7165004461764951667omplex ) ) ) ) ).
% subset_singleton_iff
thf(fact_1106_singleton__insert__inj__eq,axiom,
! [B2: mat_complex,A3: mat_complex,A4: set_mat_complex] :
( ( ( insert_mat_complex @ B2 @ bot_bo7165004461764951667omplex )
= ( insert_mat_complex @ A3 @ A4 ) )
= ( ( A3 = B2 )
& ( ord_le3632134057777142183omplex @ A4 @ ( insert_mat_complex @ B2 @ bot_bo7165004461764951667omplex ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1107_singleton__insert__inj__eq_H,axiom,
! [A3: mat_complex,A4: set_mat_complex,B2: mat_complex] :
( ( ( insert_mat_complex @ A3 @ A4 )
= ( insert_mat_complex @ B2 @ bot_bo7165004461764951667omplex ) )
= ( ( A3 = B2 )
& ( ord_le3632134057777142183omplex @ A4 @ ( insert_mat_complex @ B2 @ bot_bo7165004461764951667omplex ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1108_last__subset,axiom,
! [A4: set_mat_complex,A3: mat_complex,B2: mat_complex] :
( ( ord_le3632134057777142183omplex @ A4 @ ( insert_mat_complex @ A3 @ ( insert_mat_complex @ B2 @ bot_bo7165004461764951667omplex ) ) )
=> ( ( A3 != B2 )
=> ( ( A4
!= ( insert_mat_complex @ A3 @ ( insert_mat_complex @ B2 @ bot_bo7165004461764951667omplex ) ) )
=> ( ( A4 != bot_bo7165004461764951667omplex )
=> ( ( A4
!= ( insert_mat_complex @ A3 @ bot_bo7165004461764951667omplex ) )
=> ( A4
= ( insert_mat_complex @ B2 @ bot_bo7165004461764951667omplex ) ) ) ) ) ) ) ).
% last_subset
thf(fact_1109_Diff__eq__empty__iff,axiom,
! [A4: set_mat_complex,B: set_mat_complex] :
( ( ( minus_8760755521168068590omplex @ A4 @ B )
= bot_bo7165004461764951667omplex )
= ( ord_le3632134057777142183omplex @ A4 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1110_vector__space__over__itself_Ospan__superset,axiom,
! [S: set_complex] : ( ord_le211207098394363844omplex @ S @ ( span_complex_complex @ times_times_complex @ S ) ) ).
% vector_space_over_itself.span_superset
thf(fact_1111_vector__space__over__itself_Ospan__mono,axiom,
! [A4: set_complex,B: set_complex] :
( ( ord_le211207098394363844omplex @ A4 @ B )
=> ( ord_le211207098394363844omplex @ ( span_complex_complex @ times_times_complex @ A4 ) @ ( span_complex_complex @ times_times_complex @ B ) ) ) ).
% vector_space_over_itself.span_mono
thf(fact_1112_vector__space__over__itself_Ospan__eq,axiom,
! [S: set_complex,T: set_complex] :
( ( ( span_complex_complex @ times_times_complex @ S )
= ( span_complex_complex @ times_times_complex @ T ) )
= ( ( ord_le211207098394363844omplex @ S @ ( span_complex_complex @ times_times_complex @ T ) )
& ( ord_le211207098394363844omplex @ T @ ( span_complex_complex @ times_times_complex @ S ) ) ) ) ).
% vector_space_over_itself.span_eq
thf(fact_1113_finite__subset__induct,axiom,
! [F4: set_nat,A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat2 @ A @ A4 )
=> ( ~ ( member_nat2 @ A @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1114_finite__subset__induct,axiom,
! [F4: set_mat_complex,A4: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ F4 )
=> ( ( ord_le3632134057777142183omplex @ F4 @ A4 )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [A: mat_complex,F3: set_mat_complex] :
( ( finite7047982916621727056omplex @ F3 )
=> ( ( member_mat_complex @ A @ A4 )
=> ( ~ ( member_mat_complex @ A @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_mat_complex @ A @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1115_finite__subset__induct_H,axiom,
! [F4: set_nat,A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat2 @ A @ A4 )
=> ( ( ord_less_eq_set_nat @ F3 @ A4 )
=> ( ~ ( member_nat2 @ A @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1116_finite__subset__induct_H,axiom,
! [F4: set_mat_complex,A4: set_mat_complex,P: set_mat_complex > $o] :
( ( finite7047982916621727056omplex @ F4 )
=> ( ( ord_le3632134057777142183omplex @ F4 @ A4 )
=> ( ( P @ bot_bo7165004461764951667omplex )
=> ( ! [A: mat_complex,F3: set_mat_complex] :
( ( finite7047982916621727056omplex @ F3 )
=> ( ( member_mat_complex @ A @ A4 )
=> ( ( ord_le3632134057777142183omplex @ F3 @ A4 )
=> ( ~ ( member_mat_complex @ A @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_mat_complex @ A @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1117_subset__insert__iff,axiom,
! [A4: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat2 @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat2 @ X @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1118_subset__insert__iff,axiom,
! [A4: set_mat_complex,X: mat_complex,B: set_mat_complex] :
( ( ord_le3632134057777142183omplex @ A4 @ ( insert_mat_complex @ X @ B ) )
= ( ( ( member_mat_complex @ X @ A4 )
=> ( ord_le3632134057777142183omplex @ ( minus_8760755521168068590omplex @ A4 @ ( insert_mat_complex @ X @ bot_bo7165004461764951667omplex ) ) @ B ) )
& ( ~ ( member_mat_complex @ X @ A4 )
=> ( ord_le3632134057777142183omplex @ A4 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1119_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X2: nat > real] :
( ( P @ X2 )
=> ( P @ ( F @ X2 ) ) )
=> ( ! [X2: nat > real] :
( ( P @ X2 )
=> ! [I3: nat] :
( ( Q @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I3 ) )
& ( ord_less_eq_real @ ( X2 @ I3 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X4: nat > real,I2: nat] : ( ord_less_eq_nat @ ( L2 @ X4 @ I2 ) @ one_one_nat )
& ! [X4: nat > real,I2: nat] :
( ( ( P @ X4 )
& ( Q @ I2 )
& ( ( X4 @ I2 )
= zero_zero_real ) )
=> ( ( L2 @ X4 @ I2 )
= zero_zero_nat ) )
& ! [X4: nat > real,I2: nat] :
( ( ( P @ X4 )
& ( Q @ I2 )
& ( ( X4 @ I2 )
= one_one_real ) )
=> ( ( L2 @ X4 @ I2 )
= one_one_nat ) )
& ! [X4: nat > real,I2: nat] :
( ( ( P @ X4 )
& ( Q @ I2 )
& ( ( L2 @ X4 @ I2 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X4 @ I2 ) @ ( F @ X4 @ I2 ) ) )
& ! [X4: nat > real,I2: nat] :
( ( ( P @ X4 )
& ( Q @ I2 )
& ( ( L2 @ X4 @ I2 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X4 @ I2 ) @ ( X4 @ I2 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1120_Suc_I3_J,axiom,
ord_less_eq_nat @ ( finite32325147380013007omplex @ afa ) @ ( suc @ ia ) ).
% Suc(3)
thf(fact_1121__092_060open_062card_AAfp_A_092_060le_062_Ai_092_060close_062,axiom,
ord_less_eq_nat @ ( finite32325147380013007omplex @ afp ) @ ia ).
% \<open>card Afp \<le> i\<close>
thf(fact_1122_i__def,axiom,
( i
= ( finite32325147380013007omplex @ af ) ) ).
% i_def
thf(fact_1123__092_060open_062card_AAf_A_092_060le_062_Ai_____092_060close_062,axiom,
ord_less_eq_nat @ ( finite32325147380013007omplex @ af ) @ i ).
% \<open>card Af \<le> i__\<close>
thf(fact_1124_Suc_Ohyps,axiom,
! [Af: set_mat_complex,N: nat] :
( ( finite7047982916621727056omplex @ Af )
=> ( ( ord_less_eq_nat @ ( finite32325147380013007omplex @ Af ) @ ia )
=> ( ( Af != bot_bo7165004461764951667omplex )
=> ( ! [A5: mat_complex] :
( ( member_mat_complex @ A5 @ Af )
=> ( member_mat_complex @ A5 @ ( carrier_mat_complex @ N @ N ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ! [A5: mat_complex] :
( ( member_mat_complex @ A5 @ Af )
=> ( comple8306762464034002205omplex @ A5 ) )
=> ( ! [A5: mat_complex] :
( ( member_mat_complex @ A5 @ Af )
=> ! [B6: mat_complex] :
( ( member_mat_complex @ B6 @ Af )
=> ( ( times_8009071140041733218omplex @ A5 @ B6 )
= ( times_8009071140041733218omplex @ B6 @ A5 ) ) ) )
=> ? [U3: mat_complex] :
! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ Af )
=> ? [B6: mat_complex] : ( spectr5409772854192057952omplex @ X4 @ B6 @ U3 ) ) ) ) ) ) ) ) ) ).
% Suc.hyps
thf(fact_1125_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_1126_real__diag__decomp__hermitian,axiom,
! [A4: mat_complex,B: mat_complex,U: mat_complex] :
( ( spectr5409772854192057952omplex @ A4 @ B @ U )
=> ( comple8306762464034002205omplex @ A4 ) ) ).
% real_diag_decomp_hermitian
thf(fact_1127_hermitian__real__diag__decomp,axiom,
! [A4: mat_complex,N: nat] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( comple8306762464034002205omplex @ A4 )
=> ~ ! [B6: mat_complex,U3: mat_complex] :
~ ( spectr5409772854192057952omplex @ A4 @ B6 @ U3 ) ) ) ) ).
% hermitian_real_diag_decomp
thf(fact_1128_rd,axiom,
( ( spectr5409772854192057952omplex @ ap @ bs @ us )
& ( sorted_wrt_real @ ord_less_eq_real @ ( map_complex_real @ re @ ( diag_mat_complex @ bs ) ) ) ) ).
% rd
thf(fact_1129__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Bs_AUs_O_Areal__diag__decomp_AAp_ABs_AUs_A_092_060and_062_Asorted_A_Imap_ARe_A_Idiag__mat_ABs_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Bs3: mat_complex,Us3: mat_complex] :
~ ( ( spectr5409772854192057952omplex @ ap @ Bs3 @ Us3 )
& ( sorted_wrt_real @ ord_less_eq_real @ ( map_complex_real @ re @ ( diag_mat_complex @ Bs3 ) ) ) ) ).
% \<open>\<And>thesis. (\<And>Bs Us. real_diag_decomp Ap Bs Us \<and> sorted (map Re (diag_mat Bs)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1130_real__diag__decomp__eq__comps__props_I4_J,axiom,
! [Ap2: mat_complex,N: nat,Bs4: mat_complex,Us4: mat_complex] :
( ( member_mat_complex @ Ap2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( spectr5409772854192057952omplex @ Ap2 @ Bs4 @ Us4 )
& ( sorted_wrt_real @ ord_less_eq_real @ ( map_complex_real @ re @ ( diag_mat_complex @ Bs4 ) ) ) )
=> ( member_mat_complex @ Us4 @ ( carrier_mat_complex @ N @ N ) ) ) ) ) ).
% real_diag_decomp_eq_comps_props(4)
thf(fact_1131_real__diag__decomp__eq__comps__props_I1_J,axiom,
! [Ap2: mat_complex,N: nat,Bs4: mat_complex,Us4: mat_complex] :
( ( member_mat_complex @ Ap2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( spectr5409772854192057952omplex @ Ap2 @ Bs4 @ Us4 )
& ( sorted_wrt_real @ ord_less_eq_real @ ( map_complex_real @ re @ ( diag_mat_complex @ Bs4 ) ) ) )
=> ( member_mat_complex @ Bs4 @ ( carrier_mat_complex @ N @ N ) ) ) ) ) ).
% real_diag_decomp_eq_comps_props(1)
thf(fact_1132_per__diag__diag__mat__Re,axiom,
! [A4: mat_complex,N: nat,I: nat,F: nat > nat] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ ( F @ I ) @ N )
=> ( ( nth_real @ ( map_complex_real @ re @ ( diag_mat_complex @ ( commut4119912100034661455omplex @ A4 @ F ) ) ) @ I )
= ( nth_real @ ( map_complex_real @ re @ ( diag_mat_complex @ A4 ) ) @ ( F @ I ) ) ) ) ) ) ).
% per_diag_diag_mat_Re
thf(fact_1133_hermitian__real__diag__sorted,axiom,
! [A4: mat_complex,N: nat] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( comple8306762464034002205omplex @ A4 )
=> ~ ! [Bs3: mat_complex,Us3: mat_complex] :
~ ( ( spectr5409772854192057952omplex @ A4 @ Bs3 @ Us3 )
& ( sorted_wrt_real @ ord_less_eq_real @ ( map_complex_real @ re @ ( diag_mat_complex @ Bs3 ) ) ) ) ) ) ) ).
% hermitian_real_diag_sorted
thf(fact_1134_real__diag__decomp__eq__comps__props_I3_J,axiom,
! [Ap2: mat_complex,N: nat,Bs4: mat_complex,Us4: mat_complex] :
( ( member_mat_complex @ Ap2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( spectr5409772854192057952omplex @ Ap2 @ Bs4 @ Us4 )
& ( sorted_wrt_real @ ord_less_eq_real @ ( map_complex_real @ re @ ( diag_mat_complex @ Bs4 ) ) ) )
=> ( comple6660659447773130958omplex @ Us4 ) ) ) ) ).
% real_diag_decomp_eq_comps_props(3)
thf(fact_1135_real__diag__decomp__eq__comps__props_I7_J,axiom,
! [Ap2: mat_complex,N: nat,Bs4: mat_complex,Us4: mat_complex] :
( ( member_mat_complex @ Ap2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( spectr5409772854192057952omplex @ Ap2 @ Bs4 @ Us4 )
& ( sorted_wrt_real @ ord_less_eq_real @ ( map_complex_real @ re @ ( diag_mat_complex @ Bs4 ) ) ) )
=> ( ( diag_mat_complex @ Bs4 )
!= nil_complex ) ) ) ) ).
% real_diag_decomp_eq_comps_props(7)
thf(fact_1136_real__diag__decomp__eq__comps__props_I2_J,axiom,
! [Ap2: mat_complex,N: nat,Bs4: mat_complex,Us4: mat_complex] :
( ( member_mat_complex @ Ap2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( spectr5409772854192057952omplex @ Ap2 @ Bs4 @ Us4 )
& ( sorted_wrt_real @ ord_less_eq_real @ ( map_complex_real @ re @ ( diag_mat_complex @ Bs4 ) ) ) )
=> ( diagonal_mat_complex @ Bs4 ) ) ) ) ).
% real_diag_decomp_eq_comps_props(2)
thf(fact_1137_real__diag__decomp__eq__comps__props_I6_J,axiom,
! [Ap2: mat_complex,N: nat,Bs4: mat_complex,Us4: mat_complex] :
( ( member_mat_complex @ Ap2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( spectr5409772854192057952omplex @ Ap2 @ Bs4 @ Us4 )
& ( sorted_wrt_real @ ord_less_eq_real @ ( map_complex_real @ re @ ( diag_mat_complex @ Bs4 ) ) ) )
=> ( ( commut93809757773076895omplex @ ( diag_mat_complex @ Bs4 ) )
!= nil_nat ) ) ) ) ).
% real_diag_decomp_eq_comps_props(6)
thf(fact_1138_real__diag__decomp__eq__comps__props_I5_J,axiom,
! [Ap2: mat_complex,N: nat,Bs4: mat_complex,Us4: mat_complex] :
( ( member_mat_complex @ Ap2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( spectr5409772854192057952omplex @ Ap2 @ Bs4 @ Us4 )
& ( sorted_wrt_real @ ord_less_eq_real @ ( map_complex_real @ re @ ( diag_mat_complex @ Bs4 ) ) ) )
=> ( commut4502369927624756007omplex @ Bs4 @ ( commut93809757773076895omplex @ ( diag_mat_complex @ Bs4 ) ) ) ) ) ) ).
% real_diag_decomp_eq_comps_props(5)
thf(fact_1139_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_1140_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
= ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% nat_mult_max_left
thf(fact_1141_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
= ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% nat_mult_max_right
thf(fact_1142_max__list_Oelims,axiom,
! [X: list_nat,Y2: nat] :
( ( ( max_list @ X )
= Y2 )
=> ( ( ( X = nil_nat )
=> ( Y2 != zero_zero_nat ) )
=> ~ ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ( Y2
!= ( ord_max_nat @ X2 @ ( max_list @ Xs2 ) ) ) ) ) ) ).
% max_list.elims
thf(fact_1143_max__list_Osimps_I1_J,axiom,
( ( max_list @ nil_nat )
= zero_zero_nat ) ).
% max_list.simps(1)
thf(fact_1144_max__list_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( max_list @ ( cons_nat @ X @ Xs ) )
= ( ord_max_nat @ X @ ( max_list @ Xs ) ) ) ).
% max_list.simps(2)
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y2: complex] :
( ( if_complex @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y2: complex] :
( ( if_complex @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
eqcl != nil_nat ).
%------------------------------------------------------------------------------