TPTP Problem File: SLH0502^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Commuting_Hermitian/0001_Spectral_Theory_Complements/prob_00352_012426__19275438_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1246 ( 541 unt; 160 typ; 0 def)
% Number of atoms : 3074 (1464 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 11212 ( 311 ~; 60 |; 241 &;9308 @)
% ( 0 <=>;1292 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 6 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 578 ( 578 >; 0 *; 0 +; 0 <<)
% Number of symbols : 142 ( 139 usr; 15 con; 0-6 aty)
% Number of variables : 3132 ( 73 ^;3006 !; 53 ?;3132 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:35:03.680
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc4471711990508489141at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc7248412053542808358at_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
mat_Pr4418187712550559894l_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Matrix__Omat_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
mat_Pr441993697091756986al_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
mat_Pr6475371594752929594t_real: $tType ).
thf(ty_n_t__Matrix__Omat_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
mat_Pr3994417008679617630at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
produc2422161461964618553l_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
produc3741383161447143261al_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
produc7716430852924023517t_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Real__Oreal_J,type,
mat_real: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Matrix__Omat_Itf__a_J,type,
mat_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (139)
thf(sy_c_Column__Operations_Oadd__col__sub__row_001t__Real__Oreal,type,
column3494657893274022100w_real: real > nat > nat > mat_real > mat_real ).
thf(sy_c_Column__Operations_Omat__addcol_001t__Nat__Onat,type,
column5442440509538803650ol_nat: nat > nat > nat > mat_nat > mat_nat ).
thf(sy_c_Column__Operations_Omat__addcol_001t__Real__Oreal,type,
column5677306341442300318l_real: real > nat > nat > mat_real > mat_real ).
thf(sy_c_Column__Operations_Omat__multcol_001t__Nat__Onat,type,
column384608550491945071ol_nat: nat > nat > mat_nat > mat_nat ).
thf(sy_c_Column__Operations_Omat__multcol_001t__Real__Oreal,type,
column7747928533466807243l_real: nat > real > mat_real > mat_real ).
thf(sy_c_Column__Operations_Omat__swapcols_001t__Nat__Onat,type,
column8975334967120514601ls_nat: nat > nat > mat_nat > mat_nat ).
thf(sy_c_Column__Operations_Omat__swapcols_001t__Real__Oreal,type,
column2501654400089035909s_real: nat > nat > mat_real > mat_real ).
thf(sy_c_Column__Operations_Omat__swapcols_001tf__a,type,
column2528828918332591333cols_a: nat > nat > mat_a > mat_a ).
thf(sy_c_Column__Operations_Oswap__cols__rows_001t__Nat__Onat,type,
column141131285749525182ws_nat: nat > nat > mat_nat > mat_nat ).
thf(sy_c_Column__Operations_Oswap__cols__rows_001t__Real__Oreal,type,
column2532385344177419930s_real: nat > nat > mat_real > mat_real ).
thf(sy_c_Column__Operations_Oswap__cols__rows_001tf__a,type,
column5129559316938501008rows_a: nat > nat > mat_a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__addrow__gen_001t__Nat__Onat,type,
gauss_8885043348566651034en_nat: ( nat > nat > nat ) > ( nat > nat > nat ) > nat > nat > nat > mat_nat > mat_nat ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__addrow__gen_001t__Real__Oreal,type,
gauss_4246877906280926838n_real: ( real > real > real ) > ( real > real > real ) > real > nat > nat > mat_real > mat_real ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__addrow__gen_001tf__a,type,
gauss_3441994962245461172_gen_a: ( a > a > a ) > ( a > a > a ) > a > nat > nat > mat_a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__multrow__gen_001t__Nat__Onat,type,
gauss_2409696420326117733en_nat: ( nat > nat > nat ) > nat > nat > mat_nat > mat_nat ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__multrow__gen_001t__Real__Oreal,type,
gauss_1037889766561479105n_real: ( real > real > real ) > nat > real > mat_real > mat_real ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__multrow__gen_001tf__a,type,
gauss_5154200947219177641_gen_a: ( a > a > a ) > nat > a > mat_a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__swaprows_001t__Nat__Onat,type,
gauss_2892196111178452267ws_nat: nat > nat > mat_nat > mat_nat ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__swaprows_001t__Real__Oreal,type,
gauss_821192380332421767s_real: nat > nat > mat_real > mat_real ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__swaprows_001tf__a,type,
gauss_2482569599970757219rows_a: nat > nat > mat_a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Omultrow__mat_001t__Nat__Onat,type,
gauss_3195076542185637913at_nat: nat > nat > nat > mat_nat ).
thf(sy_c_Gauss__Jordan__Elimination_Omultrow__mat_001t__Real__Oreal,type,
gauss_7241202418770761333t_real: nat > nat > real > mat_real ).
thf(sy_c_Gauss__Jordan__Elimination_Opivot__fun_001t__Nat__Onat,type,
gauss_8416567519840421984un_nat: mat_nat > ( nat > nat ) > nat > $o ).
thf(sy_c_Gauss__Jordan__Elimination_Opivot__fun_001t__Real__Oreal,type,
gauss_5041415250090615612n_real: mat_real > ( nat > nat ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
minus_minus_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Nat__Onat_J,type,
minus_minus_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Real__Oreal_J,type,
minus_minus_mat_real: mat_real > mat_real > mat_real ).
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__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
minus_9067931446424981591at_nat: produc8199716216217303280at_nat > produc8199716216217303280at_nat > produc8199716216217303280at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
minus_4365393887724441320at_nat: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
minus_5557628854490389828t_real: produc7716430852924023517t_real > produc7716430852924023517t_real > produc7716430852924023517t_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
minus_1582581163013509572al_nat: produc3741383161447143261al_nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
minus_885040589139849760l_real: produc2422161461964618553l_real > produc2422161461964618553l_real > produc2422161461964618553l_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Nat__Onat_J,type,
plus_plus_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Real__Oreal_J,type,
plus_plus_mat_real: mat_real > mat_real > mat_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
plus_p8900843186509212308t_real: produc7716430852924023517t_real > produc7716430852924023517t_real > produc7716430852924023517t_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
plus_p4925795495032332052al_nat: produc3741383161447143261al_nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
plus_p1196244663705802608l_real: produc2422161461964618553l_real > produc2422161461964618553l_real > produc2422161461964618553l_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Nat__Onat_J,type,
times_times_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Real__Oreal_J,type,
times_times_mat_real: mat_real > mat_real > mat_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
zero_z3979849011205770936at_nat: product_prod_nat_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
zero_z738777567634093332t_real: produc7716430852924023517t_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
zero_z5987101913011988884al_nat: produc3741383161447143261al_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
zero_z1365759597461889520l_real: produc2422161461964618553l_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Jordan__Normal__Form__Existence_Odiff__ev_001t__Real__Oreal,type,
jordan8934236962569034858v_real: mat_real > nat > nat > $o ).
thf(sy_c_Jordan__Normal__Form__Existence_Oidentify__block_001t__Nat__Onat,type,
jordan8923406848002823307ck_nat: mat_nat > nat > nat ).
thf(sy_c_Jordan__Normal__Form__Existence_Oidentify__block_001t__Real__Oreal,type,
jordan6672758942465739239k_real: mat_real > nat > nat ).
thf(sy_c_Jordan__Normal__Form__Existence_Oswap__cols__rows__block_001tf__a,type,
jordan7507754584721484182lock_a: nat > nat > mat_a > mat_a ).
thf(sy_c_Jordan__Normal__Form__Existence_Ouppert_001t__Real__Oreal,type,
jordan3508124462612338182t_real: mat_real > nat > nat > $o ).
thf(sy_c_Matrix_Odiag__mat_001tf__a,type,
diag_mat_a: mat_a > list_a ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Nat__Onat,type,
diagonal_mat_nat: mat_nat > $o ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
diagon239625111944231554at_nat: mat_Pr3994417008679617630at_nat > $o ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
diagon7349657413217491934t_real: mat_Pr6475371594752929594t_real > $o ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
diagon3374609721740611678al_nat: mat_Pr441993697091756986al_nat > $o ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
diagon116145390467335098l_real: mat_Pr4418187712550559894l_real > $o ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Real__Oreal,type,
diagonal_mat_real: mat_real > $o ).
thf(sy_c_Matrix_Odiagonal__mat_001tf__a,type,
diagonal_mat_a: mat_a > $o ).
thf(sy_c_Matrix_Odim__col_001t__Nat__Onat,type,
dim_col_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__col_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
dim_co6492538100599295771at_nat: mat_Pr3994417008679617630at_nat > nat ).
thf(sy_c_Matrix_Odim__col_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
dim_co2449572004255790071t_real: mat_Pr6475371594752929594t_real > nat ).
thf(sy_c_Matrix_Odim__col_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
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thf(sy_c_Matrix_Odim__col_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
dim_co5882570131174956883l_real: mat_Pr4418187712550559894l_real > nat ).
thf(sy_c_Matrix_Odim__col_001t__Real__Oreal,type,
dim_col_real: mat_real > nat ).
thf(sy_c_Matrix_Odim__col_001tf__a,type,
dim_col_a: mat_a > nat ).
thf(sy_c_Matrix_Odim__row_001t__Nat__Onat,type,
dim_row_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__row_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
dim_ro1249899285275649537at_nat: mat_Pr3994417008679617630at_nat > nat ).
thf(sy_c_Matrix_Odim__row_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
dim_ro7135456796746250205t_real: mat_Pr6475371594752929594t_real > nat ).
thf(sy_c_Matrix_Odim__row_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
dim_ro3160409105269369949al_nat: mat_Pr441993697091756986al_nat > nat ).
thf(sy_c_Matrix_Odim__row_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
dim_ro6251147279260880953l_real: mat_Pr4418187712550559894l_real > nat ).
thf(sy_c_Matrix_Odim__row_001t__Real__Oreal,type,
dim_row_real: mat_real > nat ).
thf(sy_c_Matrix_Odim__row_001tf__a,type,
dim_row_a: mat_a > nat ).
thf(sy_c_Matrix_Oelements__mat_001t__Nat__Onat,type,
elements_mat_nat: mat_nat > set_nat ).
thf(sy_c_Matrix_Oelements__mat_001t__Real__Oreal,type,
elements_mat_real: mat_real > set_real ).
thf(sy_c_Matrix_Oelements__mat_001tf__a,type,
elements_mat_a: mat_a > set_a ).
thf(sy_c_Matrix_Oindex__mat_001t__Nat__Onat,type,
index_mat_nat: mat_nat > product_prod_nat_nat > nat ).
thf(sy_c_Matrix_Oindex__mat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Matrix_Oindex__mat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
index_4442700198795501153t_real: mat_Pr6475371594752929594t_real > product_prod_nat_nat > produc7716430852924023517t_real ).
thf(sy_c_Matrix_Oindex__mat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
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thf(sy_c_Matrix_Oindex__mat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(sy_c_Matrix_Oindex__mat_001t__Real__Oreal,type,
index_mat_real: mat_real > product_prod_nat_nat > real ).
thf(sy_c_Matrix_Oindex__mat_001tf__a,type,
index_mat_a: mat_a > product_prod_nat_nat > a ).
thf(sy_c_Matrix_Omap__mat_001t__Nat__Onat_001t__Nat__Onat,type,
map_mat_nat_nat: ( nat > nat ) > mat_nat > mat_nat ).
thf(sy_c_Matrix_Omap__mat_001t__Nat__Onat_001t__Real__Oreal,type,
map_mat_nat_real: ( nat > real ) > mat_nat > mat_real ).
thf(sy_c_Matrix_Omap__mat_001t__Nat__Onat_001tf__a,type,
map_mat_nat_a: ( nat > a ) > mat_nat > mat_a ).
thf(sy_c_Matrix_Omap__mat_001t__Real__Oreal_001t__Nat__Onat,type,
map_mat_real_nat: ( real > nat ) > mat_real > mat_nat ).
thf(sy_c_Matrix_Omap__mat_001t__Real__Oreal_001t__Real__Oreal,type,
map_mat_real_real: ( real > real ) > mat_real > mat_real ).
thf(sy_c_Matrix_Omap__mat_001t__Real__Oreal_001tf__a,type,
map_mat_real_a: ( real > a ) > mat_real > mat_a ).
thf(sy_c_Matrix_Omap__mat_001tf__a_001t__Nat__Onat,type,
map_mat_a_nat: ( a > nat ) > mat_a > mat_nat ).
thf(sy_c_Matrix_Omap__mat_001tf__a_001t__Real__Oreal,type,
map_mat_a_real: ( a > real ) > mat_a > mat_real ).
thf(sy_c_Matrix_Omap__mat_001tf__a_001tf__a,type,
map_mat_a_a: ( a > a ) > mat_a > mat_a ).
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_Otranspose__mat_001t__Nat__Onat,type,
transpose_mat_nat: mat_nat > mat_nat ).
thf(sy_c_Matrix_Otranspose__mat_001t__Real__Oreal,type,
transpose_mat_real: mat_real > mat_real ).
thf(sy_c_Matrix_Otranspose__mat_001tf__a,type,
transpose_mat_a: mat_a > mat_a ).
thf(sy_c_Matrix_Oupdate__mat_001t__Nat__Onat,type,
update_mat_nat: mat_nat > product_prod_nat_nat > nat > mat_nat ).
thf(sy_c_Matrix_Oupdate__mat_001t__Real__Oreal,type,
update_mat_real: mat_real > product_prod_nat_nat > real > mat_real ).
thf(sy_c_Matrix_Oupdate__mat_001tf__a,type,
update_mat_a: mat_a > product_prod_nat_nat > a > mat_a ).
thf(sy_c_Matrix_Oupper__triangular_001t__Nat__Onat,type,
upper_triangular_nat: mat_nat > $o ).
thf(sy_c_Matrix_Oupper__triangular_001t__Real__Oreal,type,
upper_8570057991637822137r_real: mat_real > $o ).
thf(sy_c_Matrix_Oupper__triangular_001tf__a,type,
upper_triangular_a: mat_a > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Omat_It__Nat__Onat_J,type,
ord_less_eq_mat_nat: mat_nat > mat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Omat_It__Real__Oreal_J,type,
ord_less_eq_mat_real: mat_real > mat_real > $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_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc72220940542539688at_nat: ( nat > nat ) > nat > produc8199716216217303280at_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Real__Oreal,type,
produc7837566107596912789t_real: nat > real > produc7716430852924023517t_real ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
produc4511245868158468465l_real: real > real > produc2422161461964618553l_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: mat_a ).
thf(sy_v_B,type,
b: mat_a ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_j____,type,
j: nat ).
% Relevant facts (1080)
thf(fact_0_assms_I1_J,axiom,
( ( diag_mat_a @ a2 )
= ( diag_mat_a @ b ) ) ).
% assms(1)
thf(fact_1_r,axiom,
( ( dim_row_a @ a2 )
= ( dim_row_a @ b ) ) ).
% r
thf(fact_2_assms_I4_J,axiom,
( ( dim_col_a @ a2 )
= ( dim_col_a @ b ) ) ).
% assms(4)
thf(fact_3_False,axiom,
i != j ).
% False
thf(fact_4_assms_I3_J,axiom,
diagonal_mat_a @ b ).
% assms(3)
thf(fact_5_assms_I2_J,axiom,
diagonal_mat_a @ a2 ).
% assms(2)
thf(fact_6__092_060open_062i_A_060_Adim__row_AB_092_060close_062,axiom,
ord_less_nat @ i @ ( dim_row_a @ b ) ).
% \<open>i < dim_row B\<close>
thf(fact_7__092_060open_062j_A_060_Adim__col_AB_092_060close_062,axiom,
ord_less_nat @ j @ ( dim_col_a @ b ) ).
% \<open>j < dim_col B\<close>
thf(fact_8_bezw_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [X2: nat,Y: nat] :
( X
!= ( product_Pair_nat_nat @ X2 @ Y ) ) ).
% bezw.cases
thf(fact_9_prod_Osimps_I1_J,axiom,
! [X1: real,X22: real,Y1: real,Y2: real] :
( ( ( produc4511245868158468465l_real @ X1 @ X22 )
= ( produc4511245868158468465l_real @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.simps(1)
thf(fact_10_prod_Osimps_I1_J,axiom,
! [X1: real,X22: nat,Y1: real,Y2: nat] :
( ( ( produc3181502643871035669al_nat @ X1 @ X22 )
= ( produc3181502643871035669al_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.simps(1)
thf(fact_11_prod_Osimps_I1_J,axiom,
! [X1: nat,X22: real,Y1: nat,Y2: real] :
( ( ( produc7837566107596912789t_real @ X1 @ X22 )
= ( produc7837566107596912789t_real @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.simps(1)
thf(fact_12_prod_Osimps_I1_J,axiom,
! [X1: nat > nat > nat,X22: produc7248412053542808358at_nat,Y1: nat > nat > nat,Y2: produc7248412053542808358at_nat] :
( ( ( produc3209952032786966637at_nat @ X1 @ X22 )
= ( produc3209952032786966637at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.simps(1)
thf(fact_13_prod_Osimps_I1_J,axiom,
! [X1: nat,X22: product_prod_nat_nat,Y1: nat,Y2: product_prod_nat_nat] :
( ( ( produc487386426758144856at_nat @ X1 @ X22 )
= ( produc487386426758144856at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.simps(1)
thf(fact_14_prod_Osimps_I1_J,axiom,
! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X22 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.simps(1)
thf(fact_15_prod_Osimps_I1_J,axiom,
! [X1: nat > nat,X22: nat,Y1: nat > nat,Y2: nat] :
( ( ( produc72220940542539688at_nat @ X1 @ X22 )
= ( produc72220940542539688at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.simps(1)
thf(fact_16_old_Oprod_Osimps_I1_J,axiom,
! [A: real,B: real,A2: real,B2: real] :
( ( ( produc4511245868158468465l_real @ A @ B )
= ( produc4511245868158468465l_real @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.simps(1)
thf(fact_17_old_Oprod_Osimps_I1_J,axiom,
! [A: real,B: nat,A2: real,B2: nat] :
( ( ( produc3181502643871035669al_nat @ A @ B )
= ( produc3181502643871035669al_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.simps(1)
thf(fact_18_old_Oprod_Osimps_I1_J,axiom,
! [A: nat,B: real,A2: nat,B2: real] :
( ( ( produc7837566107596912789t_real @ A @ B )
= ( produc7837566107596912789t_real @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.simps(1)
thf(fact_19_old_Oprod_Osimps_I1_J,axiom,
! [A: nat > nat > nat,B: produc7248412053542808358at_nat,A2: nat > nat > nat,B2: produc7248412053542808358at_nat] :
( ( ( produc3209952032786966637at_nat @ A @ B )
= ( produc3209952032786966637at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.simps(1)
thf(fact_20_old_Oprod_Osimps_I1_J,axiom,
! [A: nat,B: product_prod_nat_nat,A2: nat,B2: product_prod_nat_nat] :
( ( ( produc487386426758144856at_nat @ A @ B )
= ( produc487386426758144856at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.simps(1)
thf(fact_21_old_Oprod_Osimps_I1_J,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.simps(1)
thf(fact_22_old_Oprod_Osimps_I1_J,axiom,
! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
( ( ( produc72220940542539688at_nat @ A @ B )
= ( produc72220940542539688at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.simps(1)
thf(fact_23_prod_Oinduct,axiom,
! [P: produc2422161461964618553l_real > $o,Prod: produc2422161461964618553l_real] :
( ! [A3: real,B3: real] : ( P @ ( produc4511245868158468465l_real @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_24_prod_Oinduct,axiom,
! [P: produc3741383161447143261al_nat > $o,Prod: produc3741383161447143261al_nat] :
( ! [A3: real,B3: nat] : ( P @ ( produc3181502643871035669al_nat @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_25_prod_Oinduct,axiom,
! [P: produc7716430852924023517t_real > $o,Prod: produc7716430852924023517t_real] :
( ! [A3: nat,B3: real] : ( P @ ( produc7837566107596912789t_real @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_26_prod_Oinduct,axiom,
! [P: produc4471711990508489141at_nat > $o,Prod: produc4471711990508489141at_nat] :
( ! [A3: nat > nat > nat,B3: produc7248412053542808358at_nat] : ( P @ ( produc3209952032786966637at_nat @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_27_prod_Oinduct,axiom,
! [P: produc7248412053542808358at_nat > $o,Prod: produc7248412053542808358at_nat] :
( ! [A3: nat,B3: product_prod_nat_nat] : ( P @ ( produc487386426758144856at_nat @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_28_prod_Oinduct,axiom,
! [P: product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_29_prod_Oinduct,axiom,
! [P: produc8199716216217303280at_nat > $o,Prod: produc8199716216217303280at_nat] :
( ! [A3: nat > nat,B3: nat] : ( P @ ( produc72220940542539688at_nat @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_30_prod_Oexhaust,axiom,
! [Y3: produc2422161461964618553l_real] :
~ ! [X12: real,X23: real] :
( Y3
!= ( produc4511245868158468465l_real @ X12 @ X23 ) ) ).
% prod.exhaust
thf(fact_31_prod_Oexhaust,axiom,
! [Y3: produc3741383161447143261al_nat] :
~ ! [X12: real,X23: nat] :
( Y3
!= ( produc3181502643871035669al_nat @ X12 @ X23 ) ) ).
% prod.exhaust
thf(fact_32_prod_Oexhaust,axiom,
! [Y3: produc7716430852924023517t_real] :
~ ! [X12: nat,X23: real] :
( Y3
!= ( produc7837566107596912789t_real @ X12 @ X23 ) ) ).
% prod.exhaust
thf(fact_33_prod_Oexhaust,axiom,
! [Y3: produc4471711990508489141at_nat] :
~ ! [X12: nat > nat > nat,X23: produc7248412053542808358at_nat] :
( Y3
!= ( produc3209952032786966637at_nat @ X12 @ X23 ) ) ).
% prod.exhaust
thf(fact_34_prod_Oexhaust,axiom,
! [Y3: produc7248412053542808358at_nat] :
~ ! [X12: nat,X23: product_prod_nat_nat] :
( Y3
!= ( produc487386426758144856at_nat @ X12 @ X23 ) ) ).
% prod.exhaust
thf(fact_35_prod_Oexhaust,axiom,
! [Y3: product_prod_nat_nat] :
~ ! [X12: nat,X23: nat] :
( Y3
!= ( product_Pair_nat_nat @ X12 @ X23 ) ) ).
% prod.exhaust
thf(fact_36_prod_Oexhaust,axiom,
! [Y3: produc8199716216217303280at_nat] :
~ ! [X12: nat > nat,X23: nat] :
( Y3
!= ( produc72220940542539688at_nat @ X12 @ X23 ) ) ).
% prod.exhaust
thf(fact_37_prod__induct4,axiom,
! [P: produc4471711990508489141at_nat > $o,X: produc4471711990508489141at_nat] :
( ! [A3: nat > nat > nat,B3: nat,C: nat,D: nat] : ( P @ ( produc3209952032786966637at_nat @ A3 @ ( produc487386426758144856at_nat @ B3 @ ( product_Pair_nat_nat @ C @ D ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_38_prod__induct3,axiom,
! [P: produc4471711990508489141at_nat > $o,X: produc4471711990508489141at_nat] :
( ! [A3: nat > nat > nat,B3: nat,C: product_prod_nat_nat] : ( P @ ( produc3209952032786966637at_nat @ A3 @ ( produc487386426758144856at_nat @ B3 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_39_prod__induct3,axiom,
! [P: produc7248412053542808358at_nat > $o,X: produc7248412053542808358at_nat] :
( ! [A3: nat,B3: nat,C: nat] : ( P @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_40_prod__cases4,axiom,
! [Y3: produc4471711990508489141at_nat] :
~ ! [A3: nat > nat > nat,B3: nat,C: nat,D: nat] :
( Y3
!= ( produc3209952032786966637at_nat @ A3 @ ( produc487386426758144856at_nat @ B3 @ ( product_Pair_nat_nat @ C @ D ) ) ) ) ).
% prod_cases4
thf(fact_41_prod__cases3,axiom,
! [Y3: produc4471711990508489141at_nat] :
~ ! [A3: nat > nat > nat,B3: nat,C: product_prod_nat_nat] :
( Y3
!= ( produc3209952032786966637at_nat @ A3 @ ( produc487386426758144856at_nat @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_42_prod__cases3,axiom,
! [Y3: produc7248412053542808358at_nat] :
~ ! [A3: nat,B3: nat,C: nat] :
( Y3
!= ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_43_Pair__inject,axiom,
! [A: real,B: real,A2: real,B2: real] :
( ( ( produc4511245868158468465l_real @ A @ B )
= ( produc4511245868158468465l_real @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_44_Pair__inject,axiom,
! [A: real,B: nat,A2: real,B2: nat] :
( ( ( produc3181502643871035669al_nat @ A @ B )
= ( produc3181502643871035669al_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_45_Pair__inject,axiom,
! [A: nat,B: real,A2: nat,B2: real] :
( ( ( produc7837566107596912789t_real @ A @ B )
= ( produc7837566107596912789t_real @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_46_Pair__inject,axiom,
! [A: nat > nat > nat,B: produc7248412053542808358at_nat,A2: nat > nat > nat,B2: produc7248412053542808358at_nat] :
( ( ( produc3209952032786966637at_nat @ A @ B )
= ( produc3209952032786966637at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_47_Pair__inject,axiom,
! [A: nat,B: product_prod_nat_nat,A2: nat,B2: product_prod_nat_nat] :
( ( ( produc487386426758144856at_nat @ A @ B )
= ( produc487386426758144856at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_48_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_49_Pair__inject,axiom,
! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
( ( ( produc72220940542539688at_nat @ A @ B )
= ( produc72220940542539688at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_50_prod__cases,axiom,
! [P: produc2422161461964618553l_real > $o,P2: produc2422161461964618553l_real] :
( ! [A3: real,B3: real] : ( P @ ( produc4511245868158468465l_real @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_51_prod__cases,axiom,
! [P: produc3741383161447143261al_nat > $o,P2: produc3741383161447143261al_nat] :
( ! [A3: real,B3: nat] : ( P @ ( produc3181502643871035669al_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_52_prod__cases,axiom,
! [P: produc7716430852924023517t_real > $o,P2: produc7716430852924023517t_real] :
( ! [A3: nat,B3: real] : ( P @ ( produc7837566107596912789t_real @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_53_prod__cases,axiom,
! [P: produc4471711990508489141at_nat > $o,P2: produc4471711990508489141at_nat] :
( ! [A3: nat > nat > nat,B3: produc7248412053542808358at_nat] : ( P @ ( produc3209952032786966637at_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_54_prod__cases,axiom,
! [P: produc7248412053542808358at_nat > $o,P2: produc7248412053542808358at_nat] :
( ! [A3: nat,B3: product_prod_nat_nat] : ( P @ ( produc487386426758144856at_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_55_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_56_prod__cases,axiom,
! [P: produc8199716216217303280at_nat > $o,P2: produc8199716216217303280at_nat] :
( ! [A3: nat > nat,B3: nat] : ( P @ ( produc72220940542539688at_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_57_surj__pair,axiom,
! [P2: produc2422161461964618553l_real] :
? [X2: real,Y: real] :
( P2
= ( produc4511245868158468465l_real @ X2 @ Y ) ) ).
% surj_pair
thf(fact_58_surj__pair,axiom,
! [P2: produc3741383161447143261al_nat] :
? [X2: real,Y: nat] :
( P2
= ( produc3181502643871035669al_nat @ X2 @ Y ) ) ).
% surj_pair
thf(fact_59_surj__pair,axiom,
! [P2: produc7716430852924023517t_real] :
? [X2: nat,Y: real] :
( P2
= ( produc7837566107596912789t_real @ X2 @ Y ) ) ).
% surj_pair
thf(fact_60_surj__pair,axiom,
! [P2: produc4471711990508489141at_nat] :
? [X2: nat > nat > nat,Y: produc7248412053542808358at_nat] :
( P2
= ( produc3209952032786966637at_nat @ X2 @ Y ) ) ).
% surj_pair
thf(fact_61_surj__pair,axiom,
! [P2: produc7248412053542808358at_nat] :
? [X2: nat,Y: product_prod_nat_nat] :
( P2
= ( produc487386426758144856at_nat @ X2 @ Y ) ) ).
% surj_pair
thf(fact_62_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X2: nat,Y: nat] :
( P2
= ( product_Pair_nat_nat @ X2 @ Y ) ) ).
% surj_pair
thf(fact_63_surj__pair,axiom,
! [P2: produc8199716216217303280at_nat] :
? [X2: nat > nat,Y: nat] :
( P2
= ( produc72220940542539688at_nat @ X2 @ Y ) ) ).
% surj_pair
thf(fact_64_mat__eq__iff,axiom,
( ( ^ [Y4: mat_real,Z: mat_real] : ( Y4 = Z ) )
= ( ^ [X3: mat_real,Y5: mat_real] :
( ( ( dim_row_real @ X3 )
= ( dim_row_real @ Y5 ) )
& ( ( dim_col_real @ X3 )
= ( dim_col_real @ Y5 ) )
& ! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_real @ Y5 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_real @ Y5 ) )
=> ( ( index_mat_real @ X3 @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_real @ Y5 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ).
% mat_eq_iff
thf(fact_65_mat__eq__iff,axiom,
( ( ^ [Y4: mat_nat,Z: mat_nat] : ( Y4 = Z ) )
= ( ^ [X3: mat_nat,Y5: mat_nat] :
( ( ( dim_row_nat @ X3 )
= ( dim_row_nat @ Y5 ) )
& ( ( dim_col_nat @ X3 )
= ( dim_col_nat @ Y5 ) )
& ! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ Y5 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ Y5 ) )
=> ( ( index_mat_nat @ X3 @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_nat @ Y5 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ).
% mat_eq_iff
thf(fact_66_mat__eq__iff,axiom,
( ( ^ [Y4: mat_a,Z: mat_a] : ( Y4 = Z ) )
= ( ^ [X3: mat_a,Y5: mat_a] :
( ( ( dim_row_a @ X3 )
= ( dim_row_a @ Y5 ) )
& ( ( dim_col_a @ X3 )
= ( dim_col_a @ Y5 ) )
& ! [I: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ Y5 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ Y5 ) )
=> ( ( index_mat_a @ X3 @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ Y5 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ).
% mat_eq_iff
thf(fact_67_eq__matI,axiom,
! [B4: mat_real,A4: mat_real] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_real @ B4 ) )
=> ( ( ord_less_nat @ J2 @ ( dim_col_real @ B4 ) )
=> ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= ( index_mat_real @ B4 @ ( product_Pair_nat_nat @ I2 @ J2 ) ) ) ) )
=> ( ( ( dim_row_real @ A4 )
= ( dim_row_real @ B4 ) )
=> ( ( ( dim_col_real @ A4 )
= ( dim_col_real @ B4 ) )
=> ( A4 = B4 ) ) ) ) ).
% eq_matI
thf(fact_68_eq__matI,axiom,
! [B4: mat_nat,A4: mat_nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_nat @ B4 ) )
=> ( ( ord_less_nat @ J2 @ ( dim_col_nat @ B4 ) )
=> ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= ( index_mat_nat @ B4 @ ( product_Pair_nat_nat @ I2 @ J2 ) ) ) ) )
=> ( ( ( dim_row_nat @ A4 )
= ( dim_row_nat @ B4 ) )
=> ( ( ( dim_col_nat @ A4 )
= ( dim_col_nat @ B4 ) )
=> ( A4 = B4 ) ) ) ) ).
% eq_matI
thf(fact_69_eq__matI,axiom,
! [B4: mat_a,A4: mat_a] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_a @ B4 ) )
=> ( ( ord_less_nat @ J2 @ ( dim_col_a @ B4 ) )
=> ( ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= ( index_mat_a @ B4 @ ( product_Pair_nat_nat @ I2 @ J2 ) ) ) ) )
=> ( ( ( dim_row_a @ A4 )
= ( dim_row_a @ B4 ) )
=> ( ( ( dim_col_a @ A4 )
= ( dim_col_a @ B4 ) )
=> ( A4 = B4 ) ) ) ) ).
% eq_matI
thf(fact_70_index__mat__addrow_I1_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,K: nat,Ad: real > real > real,Mul: real > real > real,A: real,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( ( K = I3 )
=> ( ( index_mat_real @ ( gauss_4246877906280926838n_real @ Ad @ Mul @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( Ad @ ( Mul @ A @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ L @ J3 ) ) ) @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) )
& ( ( K != I3 )
=> ( ( index_mat_real @ ( gauss_4246877906280926838n_real @ Ad @ Mul @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ).
% index_mat_addrow(1)
thf(fact_71_index__mat__addrow_I1_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,K: nat,Ad: nat > nat > nat,Mul: nat > nat > nat,A: nat,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( ( K = I3 )
=> ( ( index_mat_nat @ ( gauss_8885043348566651034en_nat @ Ad @ Mul @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( Ad @ ( Mul @ A @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ L @ J3 ) ) ) @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) )
& ( ( K != I3 )
=> ( ( index_mat_nat @ ( gauss_8885043348566651034en_nat @ Ad @ Mul @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ).
% index_mat_addrow(1)
thf(fact_72_index__mat__addrow_I1_J,axiom,
! [I3: nat,A4: mat_a,J3: nat,K: nat,Ad: a > a > a,Mul: a > a > a,A: a,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( ( K = I3 )
=> ( ( index_mat_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( Ad @ ( Mul @ A @ ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ L @ J3 ) ) ) @ ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) )
& ( ( K != I3 )
=> ( ( index_mat_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ).
% index_mat_addrow(1)
thf(fact_73_index__mat__addrow_I2_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,Ad: real > real > real,Mul: real > real > real,A: real,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( index_mat_real @ ( gauss_4246877906280926838n_real @ Ad @ Mul @ A @ I3 @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( Ad @ ( Mul @ A @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ L @ J3 ) ) ) @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_addrow(2)
thf(fact_74_index__mat__addrow_I2_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,Ad: nat > nat > nat,Mul: nat > nat > nat,A: nat,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( index_mat_nat @ ( gauss_8885043348566651034en_nat @ Ad @ Mul @ A @ I3 @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( Ad @ ( Mul @ A @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ L @ J3 ) ) ) @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_addrow(2)
thf(fact_75_index__mat__addrow_I2_J,axiom,
! [I3: nat,A4: mat_a,J3: nat,Ad: a > a > a,Mul: a > a > a,A: a,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( index_mat_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A @ I3 @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( Ad @ ( Mul @ A @ ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ L @ J3 ) ) ) @ ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_addrow(2)
thf(fact_76_index__mat__addrow_I3_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,K: nat,Ad: real > real > real,Mul: real > real > real,A: real,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( K != I3 )
=> ( ( index_mat_real @ ( gauss_4246877906280926838n_real @ Ad @ Mul @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_addrow(3)
thf(fact_77_index__mat__addrow_I3_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,K: nat,Ad: nat > nat > nat,Mul: nat > nat > nat,A: nat,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( K != I3 )
=> ( ( index_mat_nat @ ( gauss_8885043348566651034en_nat @ Ad @ Mul @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_addrow(3)
thf(fact_78_index__mat__addrow_I3_J,axiom,
! [I3: nat,A4: mat_a,J3: nat,K: nat,Ad: a > a > a,Mul: a > a > a,A: a,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( K != I3 )
=> ( ( index_mat_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_addrow(3)
thf(fact_79_index__update__mat2,axiom,
! [I4: nat,A4: mat_real,J4: nat,Ij: product_prod_nat_nat,A: real] :
( ( ord_less_nat @ I4 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J4 @ ( dim_col_real @ A4 ) )
=> ( ( ( product_Pair_nat_nat @ I4 @ J4 )
!= Ij )
=> ( ( index_mat_real @ ( update_mat_real @ A4 @ Ij @ A ) @ ( product_Pair_nat_nat @ I4 @ J4 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I4 @ J4 ) ) ) ) ) ) ).
% index_update_mat2
thf(fact_80_index__update__mat2,axiom,
! [I4: nat,A4: mat_nat,J4: nat,Ij: product_prod_nat_nat,A: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J4 @ ( dim_col_nat @ A4 ) )
=> ( ( ( product_Pair_nat_nat @ I4 @ J4 )
!= Ij )
=> ( ( index_mat_nat @ ( update_mat_nat @ A4 @ Ij @ A ) @ ( product_Pair_nat_nat @ I4 @ J4 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I4 @ J4 ) ) ) ) ) ) ).
% index_update_mat2
thf(fact_81_index__update__mat2,axiom,
! [I4: nat,A4: mat_a,J4: nat,Ij: product_prod_nat_nat,A: a] :
( ( ord_less_nat @ I4 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J4 @ ( dim_col_a @ A4 ) )
=> ( ( ( product_Pair_nat_nat @ I4 @ J4 )
!= Ij )
=> ( ( index_mat_a @ ( update_mat_a @ A4 @ Ij @ A ) @ ( product_Pair_nat_nat @ I4 @ J4 ) )
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I4 @ J4 ) ) ) ) ) ) ).
% index_update_mat2
thf(fact_82_index__update__mat1,axiom,
! [I3: nat,A4: mat_real,J3: nat,A: real] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( index_mat_real @ ( update_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) @ A ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= A ) ) ) ).
% index_update_mat1
thf(fact_83_index__update__mat1,axiom,
! [I3: nat,A4: mat_nat,J3: nat,A: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( index_mat_nat @ ( update_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) @ A ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= A ) ) ) ).
% index_update_mat1
thf(fact_84_index__update__mat1,axiom,
! [I3: nat,A4: mat_a,J3: nat,A: a] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( index_mat_a @ ( update_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) @ A ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= A ) ) ) ).
% index_update_mat1
thf(fact_85_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_86_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A4: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_89_Collect__cong,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X2: real] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_real @ P )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_90_index__mat__swaprows_I1_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( ( K = I3 )
=> ( ( index_mat_real @ ( gauss_821192380332421767s_real @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ L @ J3 ) ) ) )
& ( ( K != I3 )
=> ( ( ( L = I3 )
=> ( ( index_mat_real @ ( gauss_821192380332421767s_real @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ K @ J3 ) ) ) )
& ( ( L != I3 )
=> ( ( index_mat_real @ ( gauss_821192380332421767s_real @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ) ).
% index_mat_swaprows(1)
thf(fact_91_index__mat__swaprows_I1_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( ( K = I3 )
=> ( ( index_mat_nat @ ( gauss_2892196111178452267ws_nat @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ L @ J3 ) ) ) )
& ( ( K != I3 )
=> ( ( ( L = I3 )
=> ( ( index_mat_nat @ ( gauss_2892196111178452267ws_nat @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ K @ J3 ) ) ) )
& ( ( L != I3 )
=> ( ( index_mat_nat @ ( gauss_2892196111178452267ws_nat @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ) ).
% index_mat_swaprows(1)
thf(fact_92_index__mat__swaprows_I1_J,axiom,
! [I3: nat,A4: mat_a,J3: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( ( K = I3 )
=> ( ( index_mat_a @ ( gauss_2482569599970757219rows_a @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ L @ J3 ) ) ) )
& ( ( K != I3 )
=> ( ( ( L = I3 )
=> ( ( index_mat_a @ ( gauss_2482569599970757219rows_a @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ K @ J3 ) ) ) )
& ( ( L != I3 )
=> ( ( index_mat_a @ ( gauss_2482569599970757219rows_a @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ) ).
% index_mat_swaprows(1)
thf(fact_93_index__mat__multrow_I1_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,K: nat,Mul: real > real > real,A: real] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( ( K = I3 )
=> ( ( index_mat_real @ ( gauss_1037889766561479105n_real @ Mul @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( Mul @ A @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) )
& ( ( K != I3 )
=> ( ( index_mat_real @ ( gauss_1037889766561479105n_real @ Mul @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ).
% index_mat_multrow(1)
thf(fact_94_index__mat__multrow_I1_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,K: nat,Mul: nat > nat > nat,A: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( ( K = I3 )
=> ( ( index_mat_nat @ ( gauss_2409696420326117733en_nat @ Mul @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( Mul @ A @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) )
& ( ( K != I3 )
=> ( ( index_mat_nat @ ( gauss_2409696420326117733en_nat @ Mul @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ).
% index_mat_multrow(1)
thf(fact_95_index__mat__multrow_I1_J,axiom,
! [I3: nat,A4: mat_a,J3: nat,K: nat,Mul: a > a > a,A: a] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( ( K = I3 )
=> ( ( index_mat_a @ ( gauss_5154200947219177641_gen_a @ Mul @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( Mul @ A @ ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) )
& ( ( K != I3 )
=> ( ( index_mat_a @ ( gauss_5154200947219177641_gen_a @ Mul @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ).
% index_mat_multrow(1)
thf(fact_96_index__mat__multrow_I2_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,Mul: real > real > real,A: real] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( index_mat_real @ ( gauss_1037889766561479105n_real @ Mul @ I3 @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( Mul @ A @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_multrow(2)
thf(fact_97_index__mat__multrow_I2_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,Mul: nat > nat > nat,A: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( index_mat_nat @ ( gauss_2409696420326117733en_nat @ Mul @ I3 @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( Mul @ A @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_multrow(2)
thf(fact_98_index__mat__multrow_I2_J,axiom,
! [I3: nat,A4: mat_a,J3: nat,Mul: a > a > a,A: a] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( index_mat_a @ ( gauss_5154200947219177641_gen_a @ Mul @ I3 @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( Mul @ A @ ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_multrow(2)
thf(fact_99_index__mat__multrow_I3_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,K: nat,Mul: real > real > real,A: real] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( K != I3 )
=> ( ( index_mat_real @ ( gauss_1037889766561479105n_real @ Mul @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_multrow(3)
thf(fact_100_index__mat__multrow_I3_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,K: nat,Mul: nat > nat > nat,A: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( K != I3 )
=> ( ( index_mat_nat @ ( gauss_2409696420326117733en_nat @ Mul @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_multrow(3)
thf(fact_101_index__mat__multrow_I3_J,axiom,
! [I3: nat,A4: mat_a,J3: nat,K: nat,Mul: a > a > a,A: a] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( K != I3 )
=> ( ( index_mat_a @ ( gauss_5154200947219177641_gen_a @ Mul @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_multrow(3)
thf(fact_102_index__mat__multcol_I3_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,K: nat,A: real] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( K != J3 )
=> ( ( index_mat_real @ ( column7747928533466807243l_real @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_multcol(3)
thf(fact_103_index__mat__multcol_I3_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,K: nat,A: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( K != J3 )
=> ( ( index_mat_nat @ ( column384608550491945071ol_nat @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_multcol(3)
thf(fact_104_index__mat__multcol_I4_J,axiom,
! [K: nat,A: real,A4: mat_real] :
( ( dim_row_real @ ( column7747928533466807243l_real @ K @ A @ A4 ) )
= ( dim_row_real @ A4 ) ) ).
% index_mat_multcol(4)
thf(fact_105_index__mat__multcol_I4_J,axiom,
! [K: nat,A: nat,A4: mat_nat] :
( ( dim_row_nat @ ( column384608550491945071ol_nat @ K @ A @ A4 ) )
= ( dim_row_nat @ A4 ) ) ).
% index_mat_multcol(4)
thf(fact_106_index__mat__multcol_I5_J,axiom,
! [K: nat,A: real,A4: mat_real] :
( ( dim_col_real @ ( column7747928533466807243l_real @ K @ A @ A4 ) )
= ( dim_col_real @ A4 ) ) ).
% index_mat_multcol(5)
thf(fact_107_index__mat__multcol_I5_J,axiom,
! [K: nat,A: nat,A4: mat_nat] :
( ( dim_col_nat @ ( column384608550491945071ol_nat @ K @ A @ A4 ) )
= ( dim_col_nat @ A4 ) ) ).
% index_mat_multcol(5)
thf(fact_108_dim__update__mat_I1_J,axiom,
! [A4: mat_real,Ij: product_prod_nat_nat,A: real] :
( ( dim_row_real @ ( update_mat_real @ A4 @ Ij @ A ) )
= ( dim_row_real @ A4 ) ) ).
% dim_update_mat(1)
thf(fact_109_dim__update__mat_I1_J,axiom,
! [A4: mat_nat,Ij: product_prod_nat_nat,A: nat] :
( ( dim_row_nat @ ( update_mat_nat @ A4 @ Ij @ A ) )
= ( dim_row_nat @ A4 ) ) ).
% dim_update_mat(1)
thf(fact_110_dim__update__mat_I1_J,axiom,
! [A4: mat_a,Ij: product_prod_nat_nat,A: a] :
( ( dim_row_a @ ( update_mat_a @ A4 @ Ij @ A ) )
= ( dim_row_a @ A4 ) ) ).
% dim_update_mat(1)
thf(fact_111_dim__update__mat_I2_J,axiom,
! [A4: mat_real,Ij: product_prod_nat_nat,A: real] :
( ( dim_col_real @ ( update_mat_real @ A4 @ Ij @ A ) )
= ( dim_col_real @ A4 ) ) ).
% dim_update_mat(2)
thf(fact_112_dim__update__mat_I2_J,axiom,
! [A4: mat_nat,Ij: product_prod_nat_nat,A: nat] :
( ( dim_col_nat @ ( update_mat_nat @ A4 @ Ij @ A ) )
= ( dim_col_nat @ A4 ) ) ).
% dim_update_mat(2)
thf(fact_113_dim__update__mat_I2_J,axiom,
! [A4: mat_a,Ij: product_prod_nat_nat,A: a] :
( ( dim_col_a @ ( update_mat_a @ A4 @ Ij @ A ) )
= ( dim_col_a @ A4 ) ) ).
% dim_update_mat(2)
thf(fact_114_index__mat__multrow_I4_J,axiom,
! [Mul: real > real > real,K: nat,A: real,A4: mat_real] :
( ( dim_row_real @ ( gauss_1037889766561479105n_real @ Mul @ K @ A @ A4 ) )
= ( dim_row_real @ A4 ) ) ).
% index_mat_multrow(4)
thf(fact_115_index__mat__multrow_I4_J,axiom,
! [Mul: nat > nat > nat,K: nat,A: nat,A4: mat_nat] :
( ( dim_row_nat @ ( gauss_2409696420326117733en_nat @ Mul @ K @ A @ A4 ) )
= ( dim_row_nat @ A4 ) ) ).
% index_mat_multrow(4)
thf(fact_116_index__mat__multrow_I4_J,axiom,
! [Mul: a > a > a,K: nat,A: a,A4: mat_a] :
( ( dim_row_a @ ( gauss_5154200947219177641_gen_a @ Mul @ K @ A @ A4 ) )
= ( dim_row_a @ A4 ) ) ).
% index_mat_multrow(4)
thf(fact_117_index__mat__multrow_I5_J,axiom,
! [Mul: real > real > real,K: nat,A: real,A4: mat_real] :
( ( dim_col_real @ ( gauss_1037889766561479105n_real @ Mul @ K @ A @ A4 ) )
= ( dim_col_real @ A4 ) ) ).
% index_mat_multrow(5)
thf(fact_118_index__mat__multrow_I5_J,axiom,
! [Mul: nat > nat > nat,K: nat,A: nat,A4: mat_nat] :
( ( dim_col_nat @ ( gauss_2409696420326117733en_nat @ Mul @ K @ A @ A4 ) )
= ( dim_col_nat @ A4 ) ) ).
% index_mat_multrow(5)
thf(fact_119_index__mat__multrow_I5_J,axiom,
! [Mul: a > a > a,K: nat,A: a,A4: mat_a] :
( ( dim_col_a @ ( gauss_5154200947219177641_gen_a @ Mul @ K @ A @ A4 ) )
= ( dim_col_a @ A4 ) ) ).
% index_mat_multrow(5)
thf(fact_120_index__mat__swaprows_I2_J,axiom,
! [K: nat,L: nat,A4: mat_real] :
( ( dim_row_real @ ( gauss_821192380332421767s_real @ K @ L @ A4 ) )
= ( dim_row_real @ A4 ) ) ).
% index_mat_swaprows(2)
thf(fact_121_index__mat__swaprows_I2_J,axiom,
! [K: nat,L: nat,A4: mat_nat] :
( ( dim_row_nat @ ( gauss_2892196111178452267ws_nat @ K @ L @ A4 ) )
= ( dim_row_nat @ A4 ) ) ).
% index_mat_swaprows(2)
thf(fact_122_index__mat__swaprows_I2_J,axiom,
! [K: nat,L: nat,A4: mat_a] :
( ( dim_row_a @ ( gauss_2482569599970757219rows_a @ K @ L @ A4 ) )
= ( dim_row_a @ A4 ) ) ).
% index_mat_swaprows(2)
thf(fact_123_index__mat__swaprows_I3_J,axiom,
! [K: nat,L: nat,A4: mat_real] :
( ( dim_col_real @ ( gauss_821192380332421767s_real @ K @ L @ A4 ) )
= ( dim_col_real @ A4 ) ) ).
% index_mat_swaprows(3)
thf(fact_124_index__mat__swaprows_I3_J,axiom,
! [K: nat,L: nat,A4: mat_nat] :
( ( dim_col_nat @ ( gauss_2892196111178452267ws_nat @ K @ L @ A4 ) )
= ( dim_col_nat @ A4 ) ) ).
% index_mat_swaprows(3)
thf(fact_125_index__mat__swaprows_I3_J,axiom,
! [K: nat,L: nat,A4: mat_a] :
( ( dim_col_a @ ( gauss_2482569599970757219rows_a @ K @ L @ A4 ) )
= ( dim_col_a @ A4 ) ) ).
% index_mat_swaprows(3)
thf(fact_126_index__mat__addrow_I4_J,axiom,
! [Ad: real > real > real,Mul: real > real > real,A: real,K: nat,L: nat,A4: mat_real] :
( ( dim_row_real @ ( gauss_4246877906280926838n_real @ Ad @ Mul @ A @ K @ L @ A4 ) )
= ( dim_row_real @ A4 ) ) ).
% index_mat_addrow(4)
thf(fact_127_index__mat__addrow_I4_J,axiom,
! [Ad: nat > nat > nat,Mul: nat > nat > nat,A: nat,K: nat,L: nat,A4: mat_nat] :
( ( dim_row_nat @ ( gauss_8885043348566651034en_nat @ Ad @ Mul @ A @ K @ L @ A4 ) )
= ( dim_row_nat @ A4 ) ) ).
% index_mat_addrow(4)
thf(fact_128_index__mat__addrow_I4_J,axiom,
! [Ad: a > a > a,Mul: a > a > a,A: a,K: nat,L: nat,A4: mat_a] :
( ( dim_row_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A @ K @ L @ A4 ) )
= ( dim_row_a @ A4 ) ) ).
% index_mat_addrow(4)
thf(fact_129_index__mat__addrow_I5_J,axiom,
! [Ad: real > real > real,Mul: real > real > real,A: real,K: nat,L: nat,A4: mat_real] :
( ( dim_col_real @ ( gauss_4246877906280926838n_real @ Ad @ Mul @ A @ K @ L @ A4 ) )
= ( dim_col_real @ A4 ) ) ).
% index_mat_addrow(5)
thf(fact_130_index__mat__addrow_I5_J,axiom,
! [Ad: nat > nat > nat,Mul: nat > nat > nat,A: nat,K: nat,L: nat,A4: mat_nat] :
( ( dim_col_nat @ ( gauss_8885043348566651034en_nat @ Ad @ Mul @ A @ K @ L @ A4 ) )
= ( dim_col_nat @ A4 ) ) ).
% index_mat_addrow(5)
thf(fact_131_index__mat__addrow_I5_J,axiom,
! [Ad: a > a > a,Mul: a > a > a,A: a,K: nat,L: nat,A4: mat_a] :
( ( dim_col_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A @ K @ L @ A4 ) )
= ( dim_col_a @ A4 ) ) ).
% index_mat_addrow(5)
thf(fact_132_index__mat__addcol_I3_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,K: nat,A: real,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( K != J3 )
=> ( ( index_mat_real @ ( column5677306341442300318l_real @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_addcol(3)
thf(fact_133_index__mat__addcol_I3_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,K: nat,A: nat,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( K != J3 )
=> ( ( index_mat_nat @ ( column5442440509538803650ol_nat @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_addcol(3)
thf(fact_134_swap__cols__rows__index,axiom,
! [I3: nat,A4: mat_real,J3: nat,A: nat,B: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ I3 @ ( dim_col_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( ord_less_nat @ A @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ B @ ( dim_row_real @ A4 ) )
=> ( ( index_mat_real @ ( column2532385344177419930s_real @ A @ B @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ ( if_nat @ ( I3 = A ) @ B @ ( if_nat @ ( I3 = B ) @ A @ I3 ) ) @ ( if_nat @ ( J3 = A ) @ B @ ( if_nat @ ( J3 = B ) @ A @ J3 ) ) ) ) ) ) ) ) ) ) ) ).
% swap_cols_rows_index
thf(fact_135_swap__cols__rows__index,axiom,
! [I3: nat,A4: mat_nat,J3: nat,A: nat,B: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ I3 @ ( dim_col_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( ord_less_nat @ A @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ B @ ( dim_row_nat @ A4 ) )
=> ( ( index_mat_nat @ ( column141131285749525182ws_nat @ A @ B @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ ( if_nat @ ( I3 = A ) @ B @ ( if_nat @ ( I3 = B ) @ A @ I3 ) ) @ ( if_nat @ ( J3 = A ) @ B @ ( if_nat @ ( J3 = B ) @ A @ J3 ) ) ) ) ) ) ) ) ) ) ) ).
% swap_cols_rows_index
thf(fact_136_swap__cols__rows__index,axiom,
! [I3: nat,A4: mat_a,J3: nat,A: nat,B: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ I3 @ ( dim_col_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( ord_less_nat @ A @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ B @ ( dim_row_a @ A4 ) )
=> ( ( index_mat_a @ ( column5129559316938501008rows_a @ A @ B @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ ( if_nat @ ( I3 = A ) @ B @ ( if_nat @ ( I3 = B ) @ A @ I3 ) ) @ ( if_nat @ ( J3 = A ) @ B @ ( if_nat @ ( J3 = B ) @ A @ J3 ) ) ) ) ) ) ) ) ) ) ) ).
% swap_cols_rows_index
thf(fact_137_index__mat__swapcols_I1_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( ( K = J3 )
=> ( ( index_mat_real @ ( column2501654400089035909s_real @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ L ) ) ) )
& ( ( K != J3 )
=> ( ( ( L = J3 )
=> ( ( index_mat_real @ ( column2501654400089035909s_real @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ K ) ) ) )
& ( ( L != J3 )
=> ( ( index_mat_real @ ( column2501654400089035909s_real @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ) ).
% index_mat_swapcols(1)
thf(fact_138_index__mat__swapcols_I1_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( ( K = J3 )
=> ( ( index_mat_nat @ ( column8975334967120514601ls_nat @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ L ) ) ) )
& ( ( K != J3 )
=> ( ( ( L = J3 )
=> ( ( index_mat_nat @ ( column8975334967120514601ls_nat @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ K ) ) ) )
& ( ( L != J3 )
=> ( ( index_mat_nat @ ( column8975334967120514601ls_nat @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ) ).
% index_mat_swapcols(1)
thf(fact_139_index__mat__swapcols_I1_J,axiom,
! [I3: nat,A4: mat_a,J3: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( ( K = J3 )
=> ( ( index_mat_a @ ( column2528828918332591333cols_a @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ L ) ) ) )
& ( ( K != J3 )
=> ( ( ( L = J3 )
=> ( ( index_mat_a @ ( column2528828918332591333cols_a @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ K ) ) ) )
& ( ( L != J3 )
=> ( ( index_mat_a @ ( column2528828918332591333cols_a @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ) ).
% index_mat_swapcols(1)
thf(fact_140_elements__matD,axiom,
! [A: nat,A4: mat_nat] :
( ( member_nat @ A @ ( elements_mat_nat @ A4 ) )
=> ? [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_nat @ A4 ) )
& ( ord_less_nat @ J2 @ ( dim_col_nat @ A4 ) )
& ( A
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I2 @ J2 ) ) ) ) ) ).
% elements_matD
thf(fact_141_elements__matD,axiom,
! [A: real,A4: mat_real] :
( ( member_real @ A @ ( elements_mat_real @ A4 ) )
=> ? [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_real @ A4 ) )
& ( ord_less_nat @ J2 @ ( dim_col_real @ A4 ) )
& ( A
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I2 @ J2 ) ) ) ) ) ).
% elements_matD
thf(fact_142_elements__matD,axiom,
! [A: a,A4: mat_a] :
( ( member_a @ A @ ( elements_mat_a @ A4 ) )
=> ? [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_a @ A4 ) )
& ( ord_less_nat @ J2 @ ( dim_col_a @ A4 ) )
& ( A
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I2 @ J2 ) ) ) ) ) ).
% elements_matD
thf(fact_143_diagonal__mat__def,axiom,
( diagon116145390467335098l_real
= ( ^ [A5: mat_Pr4418187712550559894l_real] :
! [I: nat] :
( ( ord_less_nat @ I @ ( dim_ro6251147279260880953l_real @ A5 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( dim_co5882570131174956883l_real @ A5 ) )
=> ( ( I != J )
=> ( ( index_5873430104901285053l_real @ A5 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_z1365759597461889520l_real ) ) ) ) ) ) ).
% diagonal_mat_def
thf(fact_144_diagonal__mat__def,axiom,
( diagon3374609721740611678al_nat
= ( ^ [A5: mat_Pr441993697091756986al_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( dim_ro3160409105269369949al_nat @ A5 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( dim_co7697896349633685623al_nat @ A5 ) )
=> ( ( I != J )
=> ( ( index_467652507318620897al_nat @ A5 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_z5987101913011988884al_nat ) ) ) ) ) ) ).
% diagonal_mat_def
thf(fact_145_diagonal__mat__def,axiom,
( diagon7349657413217491934t_real
= ( ^ [A5: mat_Pr6475371594752929594t_real] :
! [I: nat] :
( ( ord_less_nat @ I @ ( dim_ro7135456796746250205t_real @ A5 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( dim_co2449572004255790071t_real @ A5 ) )
=> ( ( I != J )
=> ( ( index_4442700198795501153t_real @ A5 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_z738777567634093332t_real ) ) ) ) ) ) ).
% diagonal_mat_def
thf(fact_146_diagonal__mat__def,axiom,
( diagon239625111944231554at_nat
= ( ^ [A5: mat_Pr3994417008679617630at_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( dim_ro1249899285275649537at_nat @ A5 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( dim_co6492538100599295771at_nat @ A5 ) )
=> ( ( I != J )
=> ( ( index_7364406170160795269at_nat @ A5 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_z3979849011205770936at_nat ) ) ) ) ) ) ).
% diagonal_mat_def
thf(fact_147_diagonal__mat__def,axiom,
( diagonal_mat_a
= ( ^ [A5: mat_a] :
! [I: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A5 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( dim_col_a @ A5 ) )
=> ( ( I != J )
=> ( ( index_mat_a @ A5 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_a ) ) ) ) ) ) ).
% diagonal_mat_def
thf(fact_148_diagonal__mat__def,axiom,
( diagonal_mat_nat
= ( ^ [A5: mat_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A5 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( dim_col_nat @ A5 ) )
=> ( ( I != J )
=> ( ( index_mat_nat @ A5 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_nat ) ) ) ) ) ) ).
% diagonal_mat_def
thf(fact_149_diagonal__mat__def,axiom,
( diagonal_mat_real
= ( ^ [A5: mat_real] :
! [I: nat] :
( ( ord_less_nat @ I @ ( dim_row_real @ A5 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( dim_col_real @ A5 ) )
=> ( ( I != J )
=> ( ( index_mat_real @ A5 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_real ) ) ) ) ) ) ).
% diagonal_mat_def
thf(fact_150_index__mat__multcol_I2_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,A: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( index_mat_nat @ ( column384608550491945071ol_nat @ J3 @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( times_times_nat @ A @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_multcol(2)
thf(fact_151_index__mat__multcol_I2_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,A: real] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( index_mat_real @ ( column7747928533466807243l_real @ J3 @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( times_times_real @ A @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_multcol(2)
thf(fact_152_index__mat__multcol_I1_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,K: nat,A: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( ( K = J3 )
=> ( ( index_mat_nat @ ( column384608550491945071ol_nat @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( times_times_nat @ A @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) )
& ( ( K != J3 )
=> ( ( index_mat_nat @ ( column384608550491945071ol_nat @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ).
% index_mat_multcol(1)
thf(fact_153_index__mat__multcol_I1_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,K: nat,A: real] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( ( K = J3 )
=> ( ( index_mat_real @ ( column7747928533466807243l_real @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( times_times_real @ A @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) )
& ( ( K != J3 )
=> ( ( index_mat_real @ ( column7747928533466807243l_real @ K @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ).
% index_mat_multcol(1)
thf(fact_154_index__map__mat_I1_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,F: real > a] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( index_mat_a @ ( map_mat_real_a @ F @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( F @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_map_mat(1)
thf(fact_155_index__map__mat_I1_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,F: nat > a] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( index_mat_a @ ( map_mat_nat_a @ F @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( F @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_map_mat(1)
thf(fact_156_index__map__mat_I1_J,axiom,
! [I3: nat,A4: mat_a,J3: nat,F: a > real] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( index_mat_real @ ( map_mat_a_real @ F @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( F @ ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_map_mat(1)
thf(fact_157_index__map__mat_I1_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,F: real > real] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( index_mat_real @ ( map_mat_real_real @ F @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( F @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_map_mat(1)
thf(fact_158_index__map__mat_I1_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,F: nat > real] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( index_mat_real @ ( map_mat_nat_real @ F @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( F @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_map_mat(1)
thf(fact_159_index__map__mat_I1_J,axiom,
! [I3: nat,A4: mat_a,J3: nat,F: a > nat] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( index_mat_nat @ ( map_mat_a_nat @ F @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( F @ ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_map_mat(1)
thf(fact_160_index__map__mat_I1_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,F: real > nat] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( index_mat_nat @ ( map_mat_real_nat @ F @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( F @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_map_mat(1)
thf(fact_161_index__map__mat_I1_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,F: nat > nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( index_mat_nat @ ( map_mat_nat_nat @ F @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( F @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_map_mat(1)
thf(fact_162_index__map__mat_I1_J,axiom,
! [I3: nat,A4: mat_a,J3: nat,F: a > a] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( index_mat_a @ ( map_mat_a_a @ F @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( F @ ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_map_mat(1)
thf(fact_163_index__transpose__mat_I1_J,axiom,
! [I3: nat,A4: mat_real,J3: nat] :
( ( ord_less_nat @ I3 @ ( dim_col_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_row_real @ A4 ) )
=> ( ( index_mat_real @ ( transpose_mat_real @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ J3 @ I3 ) ) ) ) ) ).
% index_transpose_mat(1)
thf(fact_164_index__transpose__mat_I1_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat] :
( ( ord_less_nat @ I3 @ ( dim_col_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_row_nat @ A4 ) )
=> ( ( index_mat_nat @ ( transpose_mat_nat @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ J3 @ I3 ) ) ) ) ) ).
% index_transpose_mat(1)
thf(fact_165_index__transpose__mat_I1_J,axiom,
! [I3: nat,A4: mat_a,J3: nat] :
( ( ord_less_nat @ I3 @ ( dim_col_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_row_a @ A4 ) )
=> ( ( index_mat_a @ ( transpose_mat_a @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ J3 @ I3 ) ) ) ) ) ).
% index_transpose_mat(1)
thf(fact_166_Matrix_Otranspose__transpose,axiom,
! [A4: mat_a] :
( ( transpose_mat_a @ ( transpose_mat_a @ A4 ) )
= A4 ) ).
% Matrix.transpose_transpose
thf(fact_167_map__mat__transpose,axiom,
! [F: a > a,A4: mat_a] :
( ( transpose_mat_a @ ( map_mat_a_a @ F @ A4 ) )
= ( map_mat_a_a @ F @ ( transpose_mat_a @ A4 ) ) ) ).
% map_mat_transpose
thf(fact_168_transpose__mat__eq,axiom,
! [A4: mat_a,B4: mat_a] :
( ( ( transpose_mat_a @ A4 )
= ( transpose_mat_a @ B4 ) )
= ( A4 = B4 ) ) ).
% transpose_mat_eq
thf(fact_169_swap__cols__rows__def,axiom,
( column5129559316938501008rows_a
= ( ^ [K2: nat,L2: nat,A5: mat_a] : ( gauss_2482569599970757219rows_a @ K2 @ L2 @ ( column2528828918332591333cols_a @ K2 @ L2 @ A5 ) ) ) ) ).
% swap_cols_rows_def
thf(fact_170_index__mult__mat_I2_J,axiom,
! [A4: mat_real,B4: mat_real] :
( ( dim_row_real @ ( times_times_mat_real @ A4 @ B4 ) )
= ( dim_row_real @ A4 ) ) ).
% index_mult_mat(2)
thf(fact_171_index__mult__mat_I2_J,axiom,
! [A4: mat_nat,B4: mat_nat] :
( ( dim_row_nat @ ( times_times_mat_nat @ A4 @ B4 ) )
= ( dim_row_nat @ A4 ) ) ).
% index_mult_mat(2)
thf(fact_172_index__mult__mat_I3_J,axiom,
! [A4: mat_real,B4: mat_real] :
( ( dim_col_real @ ( times_times_mat_real @ A4 @ B4 ) )
= ( dim_col_real @ B4 ) ) ).
% index_mult_mat(3)
thf(fact_173_index__mult__mat_I3_J,axiom,
! [A4: mat_nat,B4: mat_nat] :
( ( dim_col_nat @ ( times_times_mat_nat @ A4 @ B4 ) )
= ( dim_col_nat @ B4 ) ) ).
% index_mult_mat(3)
thf(fact_174_index__map__mat_I2_J,axiom,
! [F: real > a,A4: mat_real] :
( ( dim_row_a @ ( map_mat_real_a @ F @ A4 ) )
= ( dim_row_real @ A4 ) ) ).
% index_map_mat(2)
thf(fact_175_index__map__mat_I2_J,axiom,
! [F: nat > a,A4: mat_nat] :
( ( dim_row_a @ ( map_mat_nat_a @ F @ A4 ) )
= ( dim_row_nat @ A4 ) ) ).
% index_map_mat(2)
thf(fact_176_index__map__mat_I2_J,axiom,
! [F: a > real,A4: mat_a] :
( ( dim_row_real @ ( map_mat_a_real @ F @ A4 ) )
= ( dim_row_a @ A4 ) ) ).
% index_map_mat(2)
thf(fact_177_index__map__mat_I2_J,axiom,
! [F: real > real,A4: mat_real] :
( ( dim_row_real @ ( map_mat_real_real @ F @ A4 ) )
= ( dim_row_real @ A4 ) ) ).
% index_map_mat(2)
thf(fact_178_index__map__mat_I2_J,axiom,
! [F: nat > real,A4: mat_nat] :
( ( dim_row_real @ ( map_mat_nat_real @ F @ A4 ) )
= ( dim_row_nat @ A4 ) ) ).
% index_map_mat(2)
thf(fact_179_index__map__mat_I2_J,axiom,
! [F: a > nat,A4: mat_a] :
( ( dim_row_nat @ ( map_mat_a_nat @ F @ A4 ) )
= ( dim_row_a @ A4 ) ) ).
% index_map_mat(2)
thf(fact_180_index__map__mat_I2_J,axiom,
! [F: real > nat,A4: mat_real] :
( ( dim_row_nat @ ( map_mat_real_nat @ F @ A4 ) )
= ( dim_row_real @ A4 ) ) ).
% index_map_mat(2)
thf(fact_181_index__map__mat_I2_J,axiom,
! [F: nat > nat,A4: mat_nat] :
( ( dim_row_nat @ ( map_mat_nat_nat @ F @ A4 ) )
= ( dim_row_nat @ A4 ) ) ).
% index_map_mat(2)
thf(fact_182_index__map__mat_I2_J,axiom,
! [F: a > a,A4: mat_a] :
( ( dim_row_a @ ( map_mat_a_a @ F @ A4 ) )
= ( dim_row_a @ A4 ) ) ).
% index_map_mat(2)
thf(fact_183_index__map__mat_I3_J,axiom,
! [F: a > real,A4: mat_a] :
( ( dim_col_real @ ( map_mat_a_real @ F @ A4 ) )
= ( dim_col_a @ A4 ) ) ).
% index_map_mat(3)
thf(fact_184_index__map__mat_I3_J,axiom,
! [F: real > real,A4: mat_real] :
( ( dim_col_real @ ( map_mat_real_real @ F @ A4 ) )
= ( dim_col_real @ A4 ) ) ).
% index_map_mat(3)
thf(fact_185_index__map__mat_I3_J,axiom,
! [F: nat > real,A4: mat_nat] :
( ( dim_col_real @ ( map_mat_nat_real @ F @ A4 ) )
= ( dim_col_nat @ A4 ) ) ).
% index_map_mat(3)
thf(fact_186_index__map__mat_I3_J,axiom,
! [F: a > nat,A4: mat_a] :
( ( dim_col_nat @ ( map_mat_a_nat @ F @ A4 ) )
= ( dim_col_a @ A4 ) ) ).
% index_map_mat(3)
thf(fact_187_index__map__mat_I3_J,axiom,
! [F: real > nat,A4: mat_real] :
( ( dim_col_nat @ ( map_mat_real_nat @ F @ A4 ) )
= ( dim_col_real @ A4 ) ) ).
% index_map_mat(3)
thf(fact_188_index__map__mat_I3_J,axiom,
! [F: nat > nat,A4: mat_nat] :
( ( dim_col_nat @ ( map_mat_nat_nat @ F @ A4 ) )
= ( dim_col_nat @ A4 ) ) ).
% index_map_mat(3)
thf(fact_189_index__map__mat_I3_J,axiom,
! [F: a > a,A4: mat_a] :
( ( dim_col_a @ ( map_mat_a_a @ F @ A4 ) )
= ( dim_col_a @ A4 ) ) ).
% index_map_mat(3)
thf(fact_190_index__mat__swapcols_I2_J,axiom,
! [K: nat,L: nat,A4: mat_a] :
( ( dim_row_a @ ( column2528828918332591333cols_a @ K @ L @ A4 ) )
= ( dim_row_a @ A4 ) ) ).
% index_mat_swapcols(2)
thf(fact_191_index__mat__swapcols_I3_J,axiom,
! [K: nat,L: nat,A4: mat_a] :
( ( dim_col_a @ ( column2528828918332591333cols_a @ K @ L @ A4 ) )
= ( dim_col_a @ A4 ) ) ).
% index_mat_swapcols(3)
thf(fact_192_swap__cols__rows__carrier_I1_J,axiom,
! [K: nat,L: nat,A4: mat_a] :
( ( dim_row_a @ ( column5129559316938501008rows_a @ K @ L @ A4 ) )
= ( dim_row_a @ A4 ) ) ).
% swap_cols_rows_carrier(1)
thf(fact_193_swap__cols__rows__carrier_I2_J,axiom,
! [K: nat,L: nat,A4: mat_a] :
( ( dim_col_a @ ( column5129559316938501008rows_a @ K @ L @ A4 ) )
= ( dim_col_a @ A4 ) ) ).
% swap_cols_rows_carrier(2)
thf(fact_194_index__transpose__mat_I3_J,axiom,
! [A4: mat_a] :
( ( dim_col_a @ ( transpose_mat_a @ A4 ) )
= ( dim_row_a @ A4 ) ) ).
% index_transpose_mat(3)
thf(fact_195_index__transpose__mat_I2_J,axiom,
! [A4: mat_a] :
( ( dim_row_a @ ( transpose_mat_a @ A4 ) )
= ( dim_col_a @ A4 ) ) ).
% index_transpose_mat(2)
thf(fact_196_gcd_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [A3: nat,B3: nat] :
( X
!= ( product_Pair_nat_nat @ A3 @ B3 ) ) ).
% gcd.cases
thf(fact_197_diag__mat__transpose,axiom,
! [A4: mat_a] :
( ( ( dim_row_a @ A4 )
= ( dim_col_a @ A4 ) )
=> ( ( diag_mat_a @ ( transpose_mat_a @ A4 ) )
= ( diag_mat_a @ A4 ) ) ) ).
% diag_mat_transpose
thf(fact_198_mult__less__iff1,axiom,
! [Z2: real,X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y3 @ Z2 ) )
= ( ord_less_real @ X @ Y3 ) ) ) ).
% mult_less_iff1
thf(fact_199_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_200_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_201_mult__less__cancel__right__disj,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_202_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_203_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_204_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_205_mult__less__cancel__left__disj,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_206_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_207_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_208_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_209_linorder__neqE__linordered__idom,axiom,
! [X: real,Y3: real] :
( ( X != Y3 )
=> ( ~ ( ord_less_real @ X @ Y3 )
=> ( ord_less_real @ Y3 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_210_mult__right__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_211_mult__right__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_212_mult__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_213_mult__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_214_mult__left__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_215_mult__left__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_216_mult__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_217_mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_218_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_219_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_220_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_221_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_222_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_223_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_224_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_225_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_226_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_227_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_228_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_229_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_230_mult__sign__intros_I8_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_sign_intros(8)
thf(fact_231_mult__sign__intros_I7_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(7)
thf(fact_232_mult__sign__intros_I7_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(7)
thf(fact_233_mult__sign__intros_I6_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(6)
thf(fact_234_mult__sign__intros_I6_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(6)
thf(fact_235_mult__sign__intros_I5_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_236_mult__sign__intros_I5_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_237_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_238_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_239_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_240_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_241_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_242_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_243_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_244_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_245_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_246_mult__less__cancel__left__neg,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_247_mult__less__cancel__left__pos,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_248_index__mat__addcol_I1_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,K: nat,A: nat,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( ( K = J3 )
=> ( ( index_mat_nat @ ( column5442440509538803650ol_nat @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ L ) ) ) @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) )
& ( ( K != J3 )
=> ( ( index_mat_nat @ ( column5442440509538803650ol_nat @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ).
% index_mat_addcol(1)
thf(fact_249_index__mat__addcol_I1_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,K: nat,A: real,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( ( K = J3 )
=> ( ( index_mat_real @ ( column5677306341442300318l_real @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ L ) ) ) @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) )
& ( ( K != J3 )
=> ( ( index_mat_real @ ( column5677306341442300318l_real @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ).
% index_mat_addcol(1)
thf(fact_250_index__mat__addcol_I2_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,A: nat,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( index_mat_nat @ ( column5442440509538803650ol_nat @ A @ J3 @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ L ) ) ) @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_addcol(2)
thf(fact_251_index__mat__addcol_I2_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,A: real,L: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( index_mat_real @ ( column5677306341442300318l_real @ A @ J3 @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ L ) ) ) @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_mat_addcol(2)
thf(fact_252_zero__prod__def,axiom,
( zero_z3979849011205770936at_nat
= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_253_zero__prod__def,axiom,
( zero_z738777567634093332t_real
= ( produc7837566107596912789t_real @ zero_zero_nat @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_254_zero__prod__def,axiom,
( zero_z5987101913011988884al_nat
= ( produc3181502643871035669al_nat @ zero_zero_real @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_255_zero__prod__def,axiom,
( zero_z1365759597461889520l_real
= ( produc4511245868158468465l_real @ zero_zero_real @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_256_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_257_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_258_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_259_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_260_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_261_add__Pair,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( plus_p9057090461656269880at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C2 @ D3 ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ).
% add_Pair
thf(fact_262_add__Pair,axiom,
! [A: nat,B: real,C2: nat,D3: real] :
( ( plus_p8900843186509212308t_real @ ( produc7837566107596912789t_real @ A @ B ) @ ( produc7837566107596912789t_real @ C2 @ D3 ) )
= ( produc7837566107596912789t_real @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_real @ B @ D3 ) ) ) ).
% add_Pair
thf(fact_263_add__Pair,axiom,
! [A: real,B: nat,C2: real,D3: nat] :
( ( plus_p4925795495032332052al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc3181502643871035669al_nat @ C2 @ D3 ) )
= ( produc3181502643871035669al_nat @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ).
% add_Pair
thf(fact_264_add__Pair,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( plus_p1196244663705802608l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( produc4511245868158468465l_real @ C2 @ D3 ) )
= ( produc4511245868158468465l_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D3 ) ) ) ).
% add_Pair
thf(fact_265_Groups_Oadd__ac_I3_J,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% Groups.add_ac(3)
thf(fact_266_Groups_Oadd__ac_I3_J,axiom,
! [B: real,A: real,C2: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C2 ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% Groups.add_ac(3)
thf(fact_267_Groups_Oadd__ac_I2_J,axiom,
( plus_plus_nat
= ( ^ [A6: nat,B5: nat] : ( plus_plus_nat @ B5 @ A6 ) ) ) ).
% Groups.add_ac(2)
thf(fact_268_Groups_Oadd__ac_I2_J,axiom,
( plus_plus_real
= ( ^ [A6: real,B5: real] : ( plus_plus_real @ B5 @ A6 ) ) ) ).
% Groups.add_ac(2)
thf(fact_269_Groups_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% Groups.add_ac(1)
thf(fact_270_Groups_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% Groups.add_ac(1)
thf(fact_271_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_272_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_273_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( I3 = J3 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I3 @ K )
= ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_274_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( I3 = J3 )
& ( K = L ) )
=> ( ( plus_plus_real @ I3 @ K )
= ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_275_group__cancel_Oadd1,axiom,
! [A4: nat,K: nat,A: nat,B: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_276_group__cancel_Oadd1,axiom,
! [A4: real,K: real,A: real,B: real] :
( ( A4
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A4 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_277_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_278_group__cancel_Oadd2,axiom,
! [B4: real,K: real,B: real,A: real] :
( ( B4
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B4 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_279_add_Oleft__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_280_add_Oright__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_281_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_282_add__left__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_283_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_284_add__left__imp__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_285_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_286_add__right__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_287_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_288_add__right__imp__eq,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_289_group__cancel_Orule0,axiom,
! [A: nat] :
( A
= ( plus_plus_nat @ A @ zero_zero_nat ) ) ).
% group_cancel.rule0
thf(fact_290_group__cancel_Orule0,axiom,
! [A: real] :
( A
= ( plus_plus_real @ A @ zero_zero_real ) ) ).
% group_cancel.rule0
thf(fact_291_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_292_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_293_add__0__left,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0_left
thf(fact_294_add__0__left,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0_left
thf(fact_295_add__0__right,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add_0_right
thf(fact_296_add__0__right,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add_0_right
thf(fact_297_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_298_double__zero,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_zero
thf(fact_299_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_300_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_301_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_302_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_303_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_304_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_305_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_306_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_307_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_308_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_309_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y3 ) )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_310_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J3 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_311_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( ord_less_real @ I3 @ J3 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_312_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( I3 = J3 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_313_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( I3 = J3 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_314_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J3 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_315_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( ord_less_real @ I3 @ J3 )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_316_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_317_add__strict__mono,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D3 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_318_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_319_add__less__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_320_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_321_add__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_322_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_323_add__less__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_324_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_325_add__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_326_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_327_add__less__imp__less__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_328_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_329_add__less__imp__less__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_330_Rings_Oring__distribs_I2_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% Rings.ring_distribs(2)
thf(fact_331_Rings_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% Rings.ring_distribs(2)
thf(fact_332_Rings_Oring__distribs_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% Rings.ring_distribs(1)
thf(fact_333_Rings_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% Rings.ring_distribs(1)
thf(fact_334_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_335_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_336_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_337_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_338_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_339_combine__common__factor,axiom,
! [A: real,E2: real,B: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_340_sum__squares__eq__zero__iff,axiom,
! [X: real,Y3: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_341_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_342_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_343_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_344_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_345_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_346_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_347_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_348_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_349_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_350_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_351_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_352_pos__add__strict,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_353_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C: nat] :
( ( B
= ( plus_plus_nat @ A @ C ) )
=> ( C = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_354_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_355_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_356_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_357_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_358_sum__squares__gt__zero__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) )
= ( ( X != zero_zero_real )
| ( Y3 != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_359_add__less__zeroD,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y3 ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_360_index__add__mat_I1_J,axiom,
! [I3: nat,B4: mat_nat,J3: nat,A4: mat_nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ B4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ B4 ) )
=> ( ( index_mat_nat @ ( plus_plus_mat_nat @ A4 @ B4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( plus_plus_nat @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) @ ( index_mat_nat @ B4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_add_mat(1)
thf(fact_361_index__add__mat_I1_J,axiom,
! [I3: nat,B4: mat_real,J3: nat,A4: mat_real] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ B4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ B4 ) )
=> ( ( index_mat_real @ ( plus_plus_mat_real @ A4 @ B4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( plus_plus_real @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) @ ( index_mat_real @ B4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_add_mat(1)
thf(fact_362_not__sum__squares__lt__zero,axiom,
! [X: real,Y3: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_363_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_364_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_365_Groups_Omult__ac_I3_J,axiom,
! [B: nat,A: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% Groups.mult_ac(3)
thf(fact_366_Groups_Omult__ac_I3_J,axiom,
! [B: real,A: real,C2: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C2 ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% Groups.mult_ac(3)
thf(fact_367_Groups_Omult__ac_I2_J,axiom,
( times_times_nat
= ( ^ [A6: nat,B5: nat] : ( times_times_nat @ B5 @ A6 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_368_Groups_Omult__ac_I2_J,axiom,
( times_times_real
= ( ^ [A6: real,B5: real] : ( times_times_real @ B5 @ A6 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_369_Groups_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% Groups.mult_ac(1)
thf(fact_370_Groups_Omult__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% Groups.mult_ac(1)
thf(fact_371_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_372_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_373_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_374_mult__hom_Ohom__add__eq__zero,axiom,
! [X: nat,Y3: nat,C2: nat] :
( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C2 @ X ) @ ( times_times_nat @ C2 @ Y3 ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_375_mult__hom_Ohom__add__eq__zero,axiom,
! [X: real,Y3: real,C2: real] :
( ( ( plus_plus_real @ X @ Y3 )
= zero_zero_real )
=> ( ( plus_plus_real @ ( times_times_real @ C2 @ X ) @ ( times_times_real @ C2 @ Y3 ) )
= zero_zero_real ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_376_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C2: nat,D3: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C2 != D3 ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C2 ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D3 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_377_add__scale__eq__noteq,axiom,
! [R: real,A: real,B: real,C2: real,D3: real] :
( ( R != zero_zero_real )
=> ( ( ( A = B )
& ( C2 != D3 ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R @ C2 ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R @ D3 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_378_cross3__simps_I49_J,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ X )
= ( plus_plus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ B @ X ) ) ) ).
% cross3_simps(49)
thf(fact_379_cross3__simps_I48_J,axiom,
! [A: real,X: real,Y3: real] :
( ( times_times_real @ A @ ( plus_plus_real @ X @ Y3 ) )
= ( plus_plus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ A @ Y3 ) ) ) ).
% cross3_simps(48)
thf(fact_380_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_381_vector__space__over__itself_Oscale__scale,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_382_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_383_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_384_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_385_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_386_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_387_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_388_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_389_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_390_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_391_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_392_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_393_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_394_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_395_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_396_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_397_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_398_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_399_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_400_mult__less__mono1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J3 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_401_mult__less__mono2,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J3 ) ) ) ) ).
% mult_less_mono2
thf(fact_402_mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel1
thf(fact_403_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_404_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_405_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_406_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_407_trans__less__add2,axiom,
! [I3: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_less_add2
thf(fact_408_trans__less__add1,axiom,
! [I3: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_less_add1
thf(fact_409_add__less__mono1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% add_less_mono1
thf(fact_410_not__add__less2,axiom,
! [J3: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J3 @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_411_not__add__less1,axiom,
! [I3: nat,J3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ I3 ) ).
% not_add_less1
thf(fact_412_add__less__mono,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_less_mono
thf(fact_413_add__lessD1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ K )
=> ( ord_less_nat @ I3 @ K ) ) ).
% add_lessD1
thf(fact_414_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_415_less__imp__add__positive,axiom,
! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I3 @ K3 )
= J3 ) ) ) ).
% less_imp_add_positive
thf(fact_416_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A: real,B: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_417_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X: real,B: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_418_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X: real,Y3: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y3 ) )
=> ( X = Y3 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_419_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X: real,Y3: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y3 ) )
= ( ( X = Y3 )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_420_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_421_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_422_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( times_times_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_423_mult__hom_Ohom__zero,axiom,
! [C2: nat] :
( ( times_times_nat @ C2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_424_mult__hom_Ohom__zero,axiom,
! [C2: real] :
( ( times_times_real @ C2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_425_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_426_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_427_crossproduct__noteq,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ( A != B )
& ( C2 != D3 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D3 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D3 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_428_crossproduct__noteq,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ( A != B )
& ( C2 != D3 ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D3 ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D3 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_429_crossproduct__eq,axiom,
! [W: nat,Y3: nat,X: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X @ Y3 ) ) )
= ( ( W = X )
| ( Y3 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_430_crossproduct__eq,axiom,
! [W: real,Y3: real,X: real,Z2: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X @ Z2 ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z2 ) @ ( times_times_real @ X @ Y3 ) ) )
= ( ( W = X )
| ( Y3 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_431_mult__hom_Ohom__add,axiom,
! [C2: nat,X: nat,Y3: nat] :
( ( times_times_nat @ C2 @ ( plus_plus_nat @ X @ Y3 ) )
= ( plus_plus_nat @ ( times_times_nat @ C2 @ X ) @ ( times_times_nat @ C2 @ Y3 ) ) ) ).
% mult_hom.hom_add
thf(fact_432_mult__hom_Ohom__add,axiom,
! [C2: real,X: real,Y3: real] :
( ( times_times_real @ C2 @ ( plus_plus_real @ X @ Y3 ) )
= ( plus_plus_real @ ( times_times_real @ C2 @ X ) @ ( times_times_real @ C2 @ Y3 ) ) ) ).
% mult_hom.hom_add
thf(fact_433_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_434_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_435_pth__d,axiom,
! [X: real] :
( ( plus_plus_real @ X @ zero_zero_real )
= X ) ).
% pth_d
thf(fact_436_eq__add__iff,axiom,
! [X: real,Y3: real] :
( ( X
= ( plus_plus_real @ X @ Y3 ) )
= ( Y3 = zero_zero_real ) ) ).
% eq_add_iff
thf(fact_437_nat__distrib_I1_J,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% nat_distrib(1)
thf(fact_438_times__nat_Osimps_I1_J,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% times_nat.simps(1)
thf(fact_439_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_440_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_441_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_442_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_443_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_444_left__add__mult__distrib,axiom,
! [I3: nat,U: nat,J3: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I3 @ J3 ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_445_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
= ( P @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_446_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_447_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_448_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_449_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_450_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_451_verit__comp__simplify_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify(1)
thf(fact_452_verit__comp__simplify_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify(1)
thf(fact_453_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_454_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_455_semiring__norm_I137_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% semiring_norm(137)
thf(fact_456_semiring__norm_I137_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% semiring_norm(137)
thf(fact_457_pth__7_I1_J,axiom,
! [X: real] :
( ( plus_plus_real @ zero_zero_real @ X )
= X ) ).
% pth_7(1)
thf(fact_458_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_459_mult__delta__right,axiom,
! [B: $o,X: nat,Y3: nat] :
( ( B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y3 @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y3 @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_460_mult__delta__right,axiom,
! [B: $o,X: real,Y3: real] :
( ( B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y3 @ zero_zero_real ) )
= ( times_times_real @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y3 @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_461_mult__delta__left,axiom,
! [B: $o,X: nat,Y3: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y3 )
= ( times_times_nat @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y3 )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_462_mult__delta__left,axiom,
! [B: $o,X: real,Y3: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y3 )
= ( times_times_real @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y3 )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_463_upper__triangularD,axiom,
! [A4: mat_a,J3: nat,I3: nat] :
( ( upper_triangular_a @ A4 )
=> ( ( ord_less_nat @ J3 @ I3 )
=> ( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= zero_zero_a ) ) ) ) ).
% upper_triangularD
thf(fact_464_upper__triangularD,axiom,
! [A4: mat_nat,J3: nat,I3: nat] :
( ( upper_triangular_nat @ A4 )
=> ( ( ord_less_nat @ J3 @ I3 )
=> ( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= zero_zero_nat ) ) ) ) ).
% upper_triangularD
thf(fact_465_upper__triangularD,axiom,
! [A4: mat_real,J3: nat,I3: nat] :
( ( upper_8570057991637822137r_real @ A4 )
=> ( ( ord_less_nat @ J3 @ I3 )
=> ( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= zero_zero_real ) ) ) ) ).
% upper_triangularD
thf(fact_466_upper__triangularI,axiom,
! [A4: mat_a] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ J2 @ I2 )
=> ( ( ord_less_nat @ I2 @ ( dim_row_a @ A4 ) )
=> ( ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= zero_zero_a ) ) )
=> ( upper_triangular_a @ A4 ) ) ).
% upper_triangularI
thf(fact_467_upper__triangularI,axiom,
! [A4: mat_nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ J2 @ I2 )
=> ( ( ord_less_nat @ I2 @ ( dim_row_nat @ A4 ) )
=> ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= zero_zero_nat ) ) )
=> ( upper_triangular_nat @ A4 ) ) ).
% upper_triangularI
thf(fact_468_upper__triangularI,axiom,
! [A4: mat_real] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ J2 @ I2 )
=> ( ( ord_less_nat @ I2 @ ( dim_row_real @ A4 ) )
=> ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= zero_zero_real ) ) )
=> ( upper_8570057991637822137r_real @ A4 ) ) ).
% upper_triangularI
thf(fact_469_upper__triangular__def,axiom,
( upper_triangular_a
= ( ^ [A5: mat_a] :
! [I: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A5 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ I )
=> ( ( index_mat_a @ A5 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_a ) ) ) ) ) ).
% upper_triangular_def
thf(fact_470_upper__triangular__def,axiom,
( upper_triangular_nat
= ( ^ [A5: mat_nat] :
! [I: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A5 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ I )
=> ( ( index_mat_nat @ A5 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_nat ) ) ) ) ) ).
% upper_triangular_def
thf(fact_471_upper__triangular__def,axiom,
( upper_8570057991637822137r_real
= ( ^ [A5: mat_real] :
! [I: nat] :
( ( ord_less_nat @ I @ ( dim_row_real @ A5 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ I )
=> ( ( index_mat_real @ A5 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_real ) ) ) ) ) ).
% upper_triangular_def
thf(fact_472_pivot__funD_I5_J,axiom,
! [A4: mat_nat,Nr: nat,F: nat > nat,Nc: nat,I3: nat,I4: nat] :
( ( ( dim_row_nat @ A4 )
= Nr )
=> ( ( gauss_8416567519840421984un_nat @ A4 @ F @ Nc )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ ( F @ I3 ) @ Nc )
=> ( ( ord_less_nat @ I4 @ Nr )
=> ( ( I4 != I3 )
=> ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I4 @ ( F @ I3 ) ) )
= zero_zero_nat ) ) ) ) ) ) ) ).
% pivot_funD(5)
thf(fact_473_pivot__funD_I5_J,axiom,
! [A4: mat_real,Nr: nat,F: nat > nat,Nc: nat,I3: nat,I4: nat] :
( ( ( dim_row_real @ A4 )
= Nr )
=> ( ( gauss_5041415250090615612n_real @ A4 @ F @ Nc )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ ( F @ I3 ) @ Nc )
=> ( ( ord_less_nat @ I4 @ Nr )
=> ( ( I4 != I3 )
=> ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I4 @ ( F @ I3 ) ) )
= zero_zero_real ) ) ) ) ) ) ) ).
% pivot_funD(5)
thf(fact_474_pivot__funD_I2_J,axiom,
! [A4: mat_nat,Nr: nat,F: nat > nat,Nc: nat,I3: nat,J3: nat] :
( ( ( dim_row_nat @ A4 )
= Nr )
=> ( ( gauss_8416567519840421984un_nat @ A4 @ F @ Nc )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J3 @ ( F @ I3 ) )
=> ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= zero_zero_nat ) ) ) ) ) ).
% pivot_funD(2)
thf(fact_475_pivot__funD_I2_J,axiom,
! [A4: mat_real,Nr: nat,F: nat > nat,Nc: nat,I3: nat,J3: nat] :
( ( ( dim_row_real @ A4 )
= Nr )
=> ( ( gauss_5041415250090615612n_real @ A4 @ F @ Nc )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J3 @ ( F @ I3 ) )
=> ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= zero_zero_real ) ) ) ) ) ).
% pivot_funD(2)
thf(fact_476_index__smult__mat_I1_J,axiom,
! [I3: nat,A4: mat_nat,J3: nat,A: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ A4 ) )
=> ( ( index_mat_nat @ ( smult_mat_nat @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( times_times_nat @ A @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_smult_mat(1)
thf(fact_477_index__smult__mat_I1_J,axiom,
! [I3: nat,A4: mat_real,J3: nat,A: real] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( index_mat_real @ ( smult_mat_real @ A @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( times_times_real @ A @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_smult_mat(1)
thf(fact_478_pivot__fun__multrow,axiom,
! [A4: mat_nat,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,I0: nat,A: nat] :
( ( gauss_8416567519840421984un_nat @ A4 @ F @ Jj )
=> ( ( ( dim_row_nat @ A4 )
= Nr )
=> ( ( ( dim_col_nat @ A4 )
= Nc )
=> ( ( ( F @ I0 )
= Jj )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_8416567519840421984un_nat @ ( gauss_2409696420326117733en_nat @ times_times_nat @ I0 @ A @ A4 ) @ F @ Jj ) ) ) ) ) ) ).
% pivot_fun_multrow
thf(fact_479_pivot__fun__multrow,axiom,
! [A4: mat_real,F: nat > nat,Jj: nat,Nr: nat,Nc: nat,I0: nat,A: real] :
( ( gauss_5041415250090615612n_real @ A4 @ F @ Jj )
=> ( ( ( dim_row_real @ A4 )
= Nr )
=> ( ( ( dim_col_real @ A4 )
= Nc )
=> ( ( ( F @ I0 )
= Jj )
=> ( ( ord_less_eq_nat @ Jj @ Nc )
=> ( gauss_5041415250090615612n_real @ ( gauss_1037889766561479105n_real @ times_times_real @ I0 @ A @ A4 ) @ F @ Jj ) ) ) ) ) ) ).
% pivot_fun_multrow
thf(fact_480_pivot__funD_I4_J,axiom,
! [A4: mat_nat,Nr: nat,F: nat > nat,Nc: nat,I3: nat] :
( ( ( dim_row_nat @ A4 )
= Nr )
=> ( ( gauss_8416567519840421984un_nat @ A4 @ F @ Nc )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ ( F @ I3 ) @ Nc )
=> ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ ( F @ I3 ) ) )
= one_one_nat ) ) ) ) ) ).
% pivot_funD(4)
thf(fact_481_pivot__funD_I4_J,axiom,
! [A4: mat_real,Nr: nat,F: nat > nat,Nc: nat,I3: nat] :
( ( ( dim_row_real @ A4 )
= Nr )
=> ( ( gauss_5041415250090615612n_real @ A4 @ F @ Nc )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ ( F @ I3 ) @ Nc )
=> ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ ( F @ I3 ) ) )
= one_one_real ) ) ) ) ) ).
% pivot_funD(4)
thf(fact_482_diff__ev__def,axiom,
( jordan8934236962569034858v_real
= ( ^ [A5: mat_real,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ( index_mat_real @ A5 @ ( product_Pair_nat_nat @ I @ I ) )
!= ( index_mat_real @ A5 @ ( product_Pair_nat_nat @ J @ J ) ) )
=> ( ( index_mat_real @ A5 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_real ) ) ) ) ) ).
% diff_ev_def
thf(fact_483_verit__comp__simplify_I29_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% verit_comp_simplify(29)
thf(fact_484_verit__comp__simplify_I29_J,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% verit_comp_simplify(29)
thf(fact_485_semiring__norm_I112_J,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% semiring_norm(112)
thf(fact_486_semiring__norm_I112_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% semiring_norm(112)
thf(fact_487_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_488_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_489_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_490_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_491_mult__left__le__one__le,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y3 @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_492_mult__right__le__one__le,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y3 ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_493_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_494_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_495_mult__left__le,axiom,
! [C2: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_496_mult__left__le,axiom,
! [C2: real,A: real] :
( ( ord_less_eq_real @ C2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_497_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_498_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_499_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_500_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_501_verit__comp__simplify_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify(3)
thf(fact_502_verit__comp__simplify_I3_J,axiom,
! [B2: real,A2: real] :
( ( ~ ( ord_less_eq_real @ B2 @ A2 ) )
= ( ord_less_real @ A2 @ B2 ) ) ).
% verit_comp_simplify(3)
thf(fact_503_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_504_add__le__imp__le__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_505_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_506_add__le__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_507_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_508_add__le__imp__le__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_509_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_510_add__le__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_511_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B5: nat] :
? [C3: nat] :
( B5
= ( plus_plus_nat @ A6 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_512_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_513_add__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_514_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C: nat] :
( B
!= ( plus_plus_nat @ A @ C ) ) ) ).
% less_eqE
thf(fact_515_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_516_add__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_517_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_518_add__mono,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_519_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J3 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_520_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I3 @ J3 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_521_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( I3 = J3 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_522_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( I3 = J3 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_523_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J3 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_524_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I3 @ J3 )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_525_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_526_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_527_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_528_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_529_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_530_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_531_le__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% le_simps(1)
thf(fact_532_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_533_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_534_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_535_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_536_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J3: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_537_eq__numeral__extra_I1_J,axiom,
zero_zero_nat != one_one_nat ).
% eq_numeral_extra(1)
thf(fact_538_eq__numeral__extra_I1_J,axiom,
zero_zero_real != one_one_real ).
% eq_numeral_extra(1)
thf(fact_539_more__arith__simps_I5_J,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% more_arith_simps(5)
thf(fact_540_more__arith__simps_I5_J,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% more_arith_simps(5)
thf(fact_541_more__arith__simps_I6_J,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% more_arith_simps(6)
thf(fact_542_more__arith__simps_I6_J,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% more_arith_simps(6)
thf(fact_543_vector__space__over__itself_Ovector__space__assms_I4_J,axiom,
! [X: real] :
( ( times_times_real @ one_one_real @ X )
= X ) ).
% vector_space_over_itself.vector_space_assms(4)
thf(fact_544_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_545_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_546_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_547_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_548_rel__simps_I71_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% rel_simps(71)
thf(fact_549_rel__simps_I71_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% rel_simps(71)
thf(fact_550_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_551_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_552_trans__le__add2,axiom,
! [I3: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_le_add2
thf(fact_553_trans__le__add1,axiom,
! [I3: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_le_add1
thf(fact_554_add__le__mono1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% add_le_mono1
thf(fact_555_add__le__mono,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_le_mono
thf(fact_556_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_557_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_558_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_559_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_560_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_561_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_562_mult__le__cancel__left1,axiom,
! [C2: real,B: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_563_mult__le__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_564_mult__le__cancel__right1,axiom,
! [C2: real,B: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_565_mult__le__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_566_mult__less__cancel__left1,axiom,
! [C2: real,B: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_567_mult__less__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_568_mult__less__cancel__right1,axiom,
! [C2: real,B: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_569_mult__less__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_570_convex__bound__le,axiom,
! [X: real,A: real,Y3: real,U: real,V: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ Y3 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_571_convex__bound__lt,axiom,
! [X: real,A: real,Y3: real,U: real,V: real] :
( ( ord_less_real @ X @ A )
=> ( ( ord_less_real @ Y3 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_572_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_573_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_574_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_575_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_576_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_577_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_578_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_579_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_580_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_581_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_582_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_583_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_584_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_585_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_586_mult__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_587_mult__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_588_mult__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_589_mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_590_mult__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_591_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_592_mult__left__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_593_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_594_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_595_mult__mono_H,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_596_mult__mono_H,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_597_mult__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_598_mult__mono,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_599_zero__compare__simps_I3_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% zero_compare_simps(3)
thf(fact_600_zero__compare__simps_I3_J,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% zero_compare_simps(3)
thf(fact_601_add__sign__intros_I8_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(8)
thf(fact_602_add__sign__intros_I8_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_sign_intros(8)
thf(fact_603_add__sign__intros_I4_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_sign_intros(4)
thf(fact_604_add__sign__intros_I4_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_sign_intros(4)
thf(fact_605_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_606_add__decreasing,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_607_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_608_add__decreasing2,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_609_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_610_add__increasing2,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_611_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
=> ( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_612_add__nonneg__eq__0__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ( ( plus_plus_real @ X @ Y3 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_613_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_614_add__nonpos__eq__0__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y3 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_615_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_616_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_617_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_618_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_619_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_620_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_621_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_622_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_623_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_624_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_625_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_626_add__less__le__mono,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D3 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_627_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_628_add__le__less__mono,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D3 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_629_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J3 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_630_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( ord_less_real @ I3 @ J3 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_631_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J3 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_632_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I3 @ J3 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_633_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_634_mult__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ( times_times_real @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_635_mult__cancel__right1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_636_mult__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ( times_times_real @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_637_mult__cancel__left1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_638_rel__simps_I69_J,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% rel_simps(69)
thf(fact_639_rel__simps_I69_J,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% rel_simps(69)
thf(fact_640_rel__simps_I68_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% rel_simps(68)
thf(fact_641_rel__simps_I68_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% rel_simps(68)
thf(fact_642_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_643_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_644_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_645_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_646_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_647_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_648_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_649_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_650_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_651_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_652_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_653_less__eq__mat__def,axiom,
( ord_less_eq_mat_nat
= ( ^ [A5: mat_nat,B6: mat_nat] :
( ( ( dim_row_nat @ A5 )
= ( dim_row_nat @ B6 ) )
& ( ( dim_col_nat @ A5 )
= ( dim_col_nat @ B6 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ B6 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( dim_col_nat @ B6 ) )
=> ( ord_less_eq_nat @ ( index_mat_nat @ A5 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_nat @ B6 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ).
% less_eq_mat_def
thf(fact_654_less__eq__mat__def,axiom,
( ord_less_eq_mat_real
= ( ^ [A5: mat_real,B6: mat_real] :
( ( ( dim_row_real @ A5 )
= ( dim_row_real @ B6 ) )
& ( ( dim_col_real @ A5 )
= ( dim_col_real @ B6 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( dim_row_real @ B6 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( dim_col_real @ B6 ) )
=> ( ord_less_eq_real @ ( index_mat_real @ A5 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_real @ B6 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ).
% less_eq_mat_def
thf(fact_655_mult__le__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_656_mult__le__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_657_mult__left__less__imp__less,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_658_mult__left__less__imp__less,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_659_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_660_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D3 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_661_mult__less__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_662_mult__right__less__imp__less,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_663_mult__right__less__imp__less,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_664_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_665_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_666_mult__less__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_667_mult__le__cancel__left__neg,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_668_mult__le__cancel__left__pos,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_669_mult__left__le__imp__le,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_670_mult__left__le__imp__le,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_671_mult__right__le__imp__le,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_672_mult__right__le__imp__le,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_673_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_674_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D3 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_675_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_676_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_677_mult__le__cancel__iff1,axiom,
! [Z2: real,X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y3 @ Z2 ) )
= ( ord_less_eq_real @ X @ Y3 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_678_mult__le__cancel__iff2,axiom,
! [Z2: real,X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ ( times_times_real @ Z2 @ Y3 ) )
= ( ord_less_eq_real @ X @ Y3 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_679_sum__squares__ge__zero,axiom,
! [X: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% sum_squares_ge_zero
thf(fact_680_sum__squares__le__zero__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_681_zero__compare__simps_I2_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% zero_compare_simps(2)
thf(fact_682_zero__compare__simps_I2_J,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% zero_compare_simps(2)
thf(fact_683_zero__compare__simps_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% zero_compare_simps(1)
thf(fact_684_zero__compare__simps_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% zero_compare_simps(1)
thf(fact_685_add__sign__intros_I7_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(7)
thf(fact_686_add__sign__intros_I7_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_sign_intros(7)
thf(fact_687_add__sign__intros_I5_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(5)
thf(fact_688_add__sign__intros_I5_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_sign_intros(5)
thf(fact_689_add__sign__intros_I3_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_sign_intros(3)
thf(fact_690_add__sign__intros_I3_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_sign_intros(3)
thf(fact_691_add__sign__intros_I1_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_sign_intros(1)
thf(fact_692_add__sign__intros_I1_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_sign_intros(1)
thf(fact_693_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_694_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_695_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_696_mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel1
thf(fact_697_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_698_field__le__mult__one__interval,axiom,
! [X: real,Y3: real] :
( ! [Z3: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ Z3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ Y3 ) ) )
=> ( ord_less_eq_real @ X @ Y3 ) ) ).
% field_le_mult_one_interval
thf(fact_699_kuhn__lemma,axiom,
! [P2: nat,N: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P2 )
=> ( ! [X2: nat > nat] :
( ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( ord_less_eq_nat @ ( X2 @ I5 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ( Label @ X2 @ I2 )
= zero_zero_nat )
| ( ( Label @ X2 @ I2 )
= one_one_nat ) ) ) )
=> ( ! [X2: nat > nat] :
( ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( ord_less_eq_nat @ ( X2 @ I5 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ( X2 @ I2 )
= zero_zero_nat )
=> ( ( Label @ X2 @ I2 )
= zero_zero_nat ) ) ) )
=> ( ! [X2: nat > nat] :
( ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( ord_less_eq_nat @ ( X2 @ I5 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ( X2 @ I2 )
= P2 )
=> ( ( Label @ X2 @ I2 )
= one_one_nat ) ) ) )
=> ~ ! [Q2: nat > nat] :
( ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( ord_less_nat @ ( Q2 @ I5 ) @ P2 ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ? [R2: nat > nat] :
( ! [J5: nat] :
( ( ord_less_nat @ J5 @ N )
=> ( ( ord_less_eq_nat @ ( Q2 @ J5 ) @ ( R2 @ J5 ) )
& ( ord_less_eq_nat @ ( R2 @ J5 ) @ ( plus_plus_nat @ ( Q2 @ J5 ) @ one_one_nat ) ) ) )
& ? [S2: nat > nat] :
( ! [J5: nat] :
( ( ord_less_nat @ J5 @ N )
=> ( ( ord_less_eq_nat @ ( Q2 @ J5 ) @ ( S2 @ J5 ) )
& ( ord_less_eq_nat @ ( S2 @ J5 ) @ ( plus_plus_nat @ ( Q2 @ J5 ) @ one_one_nat ) ) ) )
& ( ( Label @ R2 @ I5 )
!= ( Label @ S2 @ I5 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_700_affine__ineq,axiom,
! [X: real,V: real,U: real] :
( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( ord_less_eq_real @ V @ U )
=> ( ord_less_eq_real @ ( plus_plus_real @ V @ ( times_times_real @ X @ U ) ) @ ( plus_plus_real @ U @ ( times_times_real @ X @ V ) ) ) ) ) ).
% affine_ineq
thf(fact_701_field__le__epsilon,axiom,
! [X: real,Y3: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y3 @ E ) ) )
=> ( ord_less_eq_real @ X @ Y3 ) ) ).
% field_le_epsilon
thf(fact_702_mult__eq__1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= one_one_nat )
= ( ( A = one_one_nat )
& ( B = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_703_mult__eq__1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ( ( times_times_real @ A @ B )
= one_one_real )
= ( ( A = one_one_real )
& ( B = one_one_real ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_704_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 )
=> ! [I2: nat] :
( ( Q @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) )
& ( ord_less_eq_real @ ( X2 @ I2 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X4: nat > real,I5: nat] : ( ord_less_eq_nat @ ( L3 @ X4 @ I5 ) @ one_one_nat )
& ! [X4: nat > real,I5: nat] :
( ( ( P @ X4 )
& ( Q @ I5 )
& ( ( X4 @ I5 )
= zero_zero_real ) )
=> ( ( L3 @ X4 @ I5 )
= zero_zero_nat ) )
& ! [X4: nat > real,I5: nat] :
( ( ( P @ X4 )
& ( Q @ I5 )
& ( ( X4 @ I5 )
= one_one_real ) )
=> ( ( L3 @ X4 @ I5 )
= one_one_nat ) )
& ! [X4: nat > real,I5: nat] :
( ( ( P @ X4 )
& ( Q @ I5 )
& ( ( L3 @ X4 @ I5 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X4 @ I5 ) @ ( F @ X4 @ I5 ) ) )
& ! [X4: nat > real,I5: nat] :
( ( ( P @ X4 )
& ( Q @ I5 )
& ( ( L3 @ X4 @ I5 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X4 @ I5 ) @ ( X4 @ I5 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_705_linordered__field__no__lb,axiom,
! [X4: real] :
? [Y: real] : ( ord_less_real @ Y @ X4 ) ).
% linordered_field_no_lb
thf(fact_706_linordered__field__no__ub,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_707_sum__le__prod1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% sum_le_prod1
thf(fact_708_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_709_complete__real,axiom,
! [S3: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S3 )
=> ( ? [Z4: real] :
! [X2: real] :
( ( member_real @ X2 @ S3 )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ? [Y: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S3 )
=> ( ord_less_eq_real @ X4 @ Y ) )
& ! [Z4: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S3 )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_710_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc4471711990508489141at_nat] :
~ ! [F2: nat > nat > nat,A3: nat,B3: nat,Acc: nat] :
( X
!= ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_711_mult__if__delta,axiom,
! [P: $o,Q3: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q3 )
= Q3 ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q3 )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_712_mult__if__delta,axiom,
! [P: $o,Q3: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q3 )
= Q3 ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q3 )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_713_square__bound__lemma,axiom,
! [X: real] : ( ord_less_real @ X @ ( times_times_real @ ( plus_plus_real @ one_one_real @ X ) @ ( plus_plus_real @ one_one_real @ X ) ) ) ).
% square_bound_lemma
thf(fact_714_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_715_less__eq__fract__respect,axiom,
! [B: real,B2: real,D3: real,D4: real,A: real,A2: real,C2: real,C4: real] :
( ( B != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( D3 != zero_zero_real )
=> ( ( D4 != zero_zero_real )
=> ( ( ( times_times_real @ A @ B2 )
= ( times_times_real @ A2 @ B ) )
=> ( ( ( times_times_real @ C2 @ D4 )
= ( times_times_real @ C4 @ D3 ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A @ D3 ) @ ( times_times_real @ B @ D3 ) ) @ ( times_times_real @ ( times_times_real @ C2 @ B ) @ ( times_times_real @ B @ D3 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A2 @ D4 ) @ ( times_times_real @ B2 @ D4 ) ) @ ( times_times_real @ ( times_times_real @ C4 @ B2 ) @ ( times_times_real @ B2 @ D4 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_716_pivot__funI,axiom,
! [A4: mat_nat,Nr: nat,F: nat > nat,Nc: nat] :
( ( ( dim_row_nat @ A4 )
= Nr )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ Nr )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ Nc ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ Nr )
=> ( ( ord_less_nat @ J2 @ ( F @ I2 ) )
=> ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= zero_zero_nat ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ Nr )
=> ( ( ord_less_nat @ ( suc @ I2 ) @ Nr )
=> ( ( ord_less_nat @ ( F @ I2 ) @ ( F @ ( suc @ I2 ) ) )
| ( ( F @ ( suc @ I2 ) )
= Nc ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ Nr )
=> ( ( ord_less_nat @ ( F @ I2 ) @ Nc )
=> ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I2 @ ( F @ I2 ) ) )
= one_one_nat ) ) )
=> ( ! [I2: nat,I6: nat] :
( ( ord_less_nat @ I2 @ Nr )
=> ( ( ord_less_nat @ ( F @ I2 ) @ Nc )
=> ( ( ord_less_nat @ I6 @ Nr )
=> ( ( I6 != I2 )
=> ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I6 @ ( F @ I2 ) ) )
= zero_zero_nat ) ) ) ) )
=> ( gauss_8416567519840421984un_nat @ A4 @ F @ Nc ) ) ) ) ) ) ) ).
% pivot_funI
thf(fact_717_pivot__funI,axiom,
! [A4: mat_real,Nr: nat,F: nat > nat,Nc: nat] :
( ( ( dim_row_real @ A4 )
= Nr )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ Nr )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ Nc ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ Nr )
=> ( ( ord_less_nat @ J2 @ ( F @ I2 ) )
=> ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= zero_zero_real ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ Nr )
=> ( ( ord_less_nat @ ( suc @ I2 ) @ Nr )
=> ( ( ord_less_nat @ ( F @ I2 ) @ ( F @ ( suc @ I2 ) ) )
| ( ( F @ ( suc @ I2 ) )
= Nc ) ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ Nr )
=> ( ( ord_less_nat @ ( F @ I2 ) @ Nc )
=> ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I2 @ ( F @ I2 ) ) )
= one_one_real ) ) )
=> ( ! [I2: nat,I6: nat] :
( ( ord_less_nat @ I2 @ Nr )
=> ( ( ord_less_nat @ ( F @ I2 ) @ Nc )
=> ( ( ord_less_nat @ I6 @ Nr )
=> ( ( I6 != I2 )
=> ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I6 @ ( F @ I2 ) ) )
= zero_zero_real ) ) ) ) )
=> ( gauss_5041415250090615612n_real @ A4 @ F @ Nc ) ) ) ) ) ) ) ).
% pivot_funI
thf(fact_718_uppert__def,axiom,
( jordan3508124462612338182t_real
= ( ^ [A5: mat_real,I: nat,J: nat] :
( ( ord_less_nat @ J @ I )
=> ( ( index_mat_real @ A5 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_real ) ) ) ) ).
% uppert_def
thf(fact_719_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_720_strict__inc__induct,axiom,
! [I3: nat,J3: nat,P: nat > $o] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ! [I2: nat] :
( ( J3
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_721_less__Suc__induct,axiom,
! [I3: nat,J3: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I3 @ J3 ) ) ) ) ).
% less_Suc_induct
thf(fact_722_less__trans__Suc,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K )
=> ( ord_less_nat @ ( suc @ I3 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_723_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_724_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_725_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_726_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_727_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_728_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_729_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_730_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_731_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_732_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_733_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_734_Suc__lessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_735_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_736_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_737_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_738_Nat_OlessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( ( K
!= ( suc @ I3 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_739_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_740_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_741_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_742_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_743_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_744_unit__vecs__last_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [N2: nat] : ( P @ N2 @ zero_zero_nat )
=> ( ! [N2: nat,I2: nat] :
( ( P @ N2 @ I2 )
=> ( P @ N2 @ ( suc @ I2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% unit_vecs_last.induct
thf(fact_745_nat_Osimps_I3_J,axiom,
! [X22: nat] :
( ( suc @ X22 )
!= zero_zero_nat ) ).
% nat.simps(3)
thf(fact_746_old_Onat_Osimps_I3_J,axiom,
! [Nat: nat] :
( ( suc @ Nat )
!= zero_zero_nat ) ).
% old.nat.simps(3)
thf(fact_747_old_Onat_Osimps_I2_J,axiom,
! [Nat: nat] :
( zero_zero_nat
!= ( suc @ Nat ) ) ).
% old.nat.simps(2)
thf(fact_748_nat_OdiscI,axiom,
! [Nat2: nat,X22: nat] :
( ( Nat2
= ( suc @ X22 ) )
=> ( Nat2 != zero_zero_nat ) ) ).
% nat.discI
thf(fact_749_nat_Oinduct,axiom,
! [P: nat > $o,Nat2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat2 ) ) ) ).
% nat.induct
thf(fact_750_nat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [X23: nat] :
( Y3
!= ( suc @ X23 ) ) ) ).
% nat.exhaust
thf(fact_751_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_752_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X2: nat,Y: nat] :
( ( P @ X2 @ Y )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_753_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_754_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_755_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_not_Zero
thf(fact_756_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_757_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_758_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_759_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_760_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_761_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_762_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J: nat] :
( ( M
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_763_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_764_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_765_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_766_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_767_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_768_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_769_le__simps_I3_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% le_simps(3)
thf(fact_770_le__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% le_simps(2)
thf(fact_771_not__less__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_simps(2)
thf(fact_772_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_773_dec__induct,axiom,
! [I3: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( P @ I3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J3 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J3 ) ) ) ) ).
% dec_induct
thf(fact_774_inc__induct,axiom,
! [I3: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( P @ J3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J3 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% inc_induct
thf(fact_775_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_776_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_777_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_778_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_779_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_780_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_781_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_782_less__add__Suc1,axiom,
! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ I3 @ M ) ) ) ).
% less_add_Suc1
thf(fact_783_less__add__Suc2,axiom,
! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ M @ I3 ) ) ) ).
% less_add_Suc2
thf(fact_784_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_785_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_786_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_787_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_788_unit__vecs__last_Ocases,axiom,
! [X: product_prod_nat_nat] :
( ! [N2: nat] :
( X
!= ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) )
=> ~ ! [N2: nat,I2: nat] :
( X
!= ( product_Pair_nat_nat @ N2 @ ( suc @ I2 ) ) ) ) ).
% unit_vecs_last.cases
thf(fact_789_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_790_semiring__norm_I174_J,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% semiring_norm(174)
thf(fact_791_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_792_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_793_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_794_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_795_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_796_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_797_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_798_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_799_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_800_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_801_lookup__ev_Oinduct,axiom,
! [P: a > nat > mat_a > $o,A0: a,A1: nat,A22: mat_a] :
( ! [Ev: a,X_1: mat_a] : ( P @ Ev @ zero_zero_nat @ X_1 )
=> ( ! [Ev: a,I2: nat,A7: mat_a] :
( ( ( ( index_mat_a @ A7 @ ( product_Pair_nat_nat @ I2 @ I2 ) )
!= Ev )
=> ( P @ Ev @ I2 @ A7 ) )
=> ( P @ Ev @ ( suc @ I2 ) @ A7 ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% lookup_ev.induct
thf(fact_802_lookup__other__ev_Oinduct,axiom,
! [P: a > nat > mat_a > $o,A0: a,A1: nat,A22: mat_a] :
( ! [Ev: a,X_1: mat_a] : ( P @ Ev @ zero_zero_nat @ X_1 )
=> ( ! [Ev: a,I2: nat,A7: mat_a] :
( ( ( ( index_mat_a @ A7 @ ( product_Pair_nat_nat @ I2 @ I2 ) )
= Ev )
=> ( P @ Ev @ I2 @ A7 ) )
=> ( P @ Ev @ ( suc @ I2 ) @ A7 ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% lookup_other_ev.induct
thf(fact_803_pivot__positions__main__gen_Oinduct,axiom,
! [Nr: nat,Nc: nat,A4: mat_a,Zero: a,P: nat > nat > $o,A0: nat,A1: nat] :
( ! [I2: nat,J2: nat] :
( ( ( ord_less_nat @ I2 @ Nr )
=> ( ( ord_less_nat @ J2 @ Nc )
=> ( ( ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= Zero )
=> ( P @ I2 @ ( suc @ J2 ) ) ) ) )
=> ( ( ( ord_less_nat @ I2 @ Nr )
=> ( ( ord_less_nat @ J2 @ Nc )
=> ( ( ( index_mat_a @ A4 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
!= Zero )
=> ( P @ ( suc @ I2 ) @ ( suc @ J2 ) ) ) ) )
=> ( P @ I2 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ).
% pivot_positions_main_gen.induct
thf(fact_804_identify__block_Oinduct,axiom,
! [P: mat_nat > nat > $o,A0: mat_nat,A1: nat] :
( ! [A7: mat_nat] : ( P @ A7 @ zero_zero_nat )
=> ( ! [A7: mat_nat,I2: nat] :
( ( ( ( index_mat_nat @ A7 @ ( product_Pair_nat_nat @ I2 @ ( suc @ I2 ) ) )
= one_one_nat )
=> ( P @ A7 @ I2 ) )
=> ( P @ A7 @ ( suc @ I2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% identify_block.induct
thf(fact_805_identify__block_Oinduct,axiom,
! [P: mat_real > nat > $o,A0: mat_real,A1: nat] :
( ! [A7: mat_real] : ( P @ A7 @ zero_zero_nat )
=> ( ! [A7: mat_real,I2: nat] :
( ( ( ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ I2 @ ( suc @ I2 ) ) )
= one_one_real )
=> ( P @ A7 @ I2 ) )
=> ( P @ A7 @ ( suc @ I2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% identify_block.induct
thf(fact_806_inf__concat_Oinduct,axiom,
! [P: ( nat > nat ) > nat > $o,A0: nat > nat,A1: nat] :
( ! [N2: nat > nat] : ( P @ N2 @ zero_zero_nat )
=> ( ! [N2: nat > nat,K3: nat] :
( ( P @ N2 @ K3 )
=> ( P @ N2 @ ( suc @ K3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% inf_concat.induct
thf(fact_807_inf__concat__simple_Ocases,axiom,
! [X: produc8199716216217303280at_nat] :
( ! [F2: nat > nat] :
( X
!= ( produc72220940542539688at_nat @ F2 @ zero_zero_nat ) )
=> ~ ! [F2: nat > nat,N2: nat] :
( X
!= ( produc72220940542539688at_nat @ F2 @ ( suc @ N2 ) ) ) ) ).
% inf_concat_simple.cases
thf(fact_808_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_809_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K3: nat] :
? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( F @ K3 @ I5 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ! [K4: nat] :
? [K5: nat] :
( ( ord_less_eq_nat @ K4 @ K5 )
& ( F @ K5 @ I2 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_810_identify__block_I3_J,axiom,
! [A4: mat_nat,J3: nat,I3: nat,K: nat] :
( ( ( jordan8923406848002823307ck_nat @ A4 @ J3 )
= I3 )
=> ( ( ord_less_eq_nat @ I3 @ K )
=> ( ( ord_less_nat @ K @ J3 )
=> ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ K @ ( suc @ K ) ) )
= one_one_nat ) ) ) ) ).
% identify_block(3)
thf(fact_811_identify__block_I3_J,axiom,
! [A4: mat_real,J3: nat,I3: nat,K: nat] :
( ( ( jordan6672758942465739239k_real @ A4 @ J3 )
= I3 )
=> ( ( ord_less_eq_nat @ I3 @ K )
=> ( ( ord_less_nat @ K @ J3 )
=> ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ K @ ( suc @ K ) ) )
= one_one_real ) ) ) ) ).
% identify_block(3)
thf(fact_812_index__mat__multrow__mat_I1_J,axiom,
! [I3: nat,N: nat,J3: nat,K: nat,A: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ( ( ( K = I3 )
& ( K = J3 ) )
=> ( ( index_mat_nat @ ( gauss_3195076542185637913at_nat @ N @ K @ A ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= A ) )
& ( ~ ( ( K = I3 )
& ( K = J3 ) )
=> ( ( ( I3 = J3 )
=> ( ( index_mat_nat @ ( gauss_3195076542185637913at_nat @ N @ K @ A ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= one_one_nat ) )
& ( ( I3 != J3 )
=> ( ( index_mat_nat @ ( gauss_3195076542185637913at_nat @ N @ K @ A ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= zero_zero_nat ) ) ) ) ) ) ) ).
% index_mat_multrow_mat(1)
thf(fact_813_index__mat__multrow__mat_I1_J,axiom,
! [I3: nat,N: nat,J3: nat,K: nat,A: real] :
( ( ord_less_nat @ I3 @ N )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ( ( ( K = I3 )
& ( K = J3 ) )
=> ( ( index_mat_real @ ( gauss_7241202418770761333t_real @ N @ K @ A ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= A ) )
& ( ~ ( ( K = I3 )
& ( K = J3 ) )
=> ( ( ( I3 = J3 )
=> ( ( index_mat_real @ ( gauss_7241202418770761333t_real @ N @ K @ A ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= one_one_real ) )
& ( ( I3 != J3 )
=> ( ( index_mat_real @ ( gauss_7241202418770761333t_real @ N @ K @ A ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= zero_zero_real ) ) ) ) ) ) ) ).
% index_mat_multrow_mat(1)
thf(fact_814_identify__block_Osimps_I2_J,axiom,
! [A4: mat_nat,I3: nat] :
( ( ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ ( suc @ I3 ) ) )
= one_one_nat )
=> ( ( jordan8923406848002823307ck_nat @ A4 @ ( suc @ I3 ) )
= ( jordan8923406848002823307ck_nat @ A4 @ I3 ) ) )
& ( ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ ( suc @ I3 ) ) )
!= one_one_nat )
=> ( ( jordan8923406848002823307ck_nat @ A4 @ ( suc @ I3 ) )
= ( suc @ I3 ) ) ) ) ).
% identify_block.simps(2)
thf(fact_815_identify__block_Osimps_I2_J,axiom,
! [A4: mat_real,I3: nat] :
( ( ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ ( suc @ I3 ) ) )
= one_one_real )
=> ( ( jordan6672758942465739239k_real @ A4 @ ( suc @ I3 ) )
= ( jordan6672758942465739239k_real @ A4 @ I3 ) ) )
& ( ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ ( suc @ I3 ) ) )
!= one_one_real )
=> ( ( jordan6672758942465739239k_real @ A4 @ ( suc @ I3 ) )
= ( suc @ I3 ) ) ) ) ).
% identify_block.simps(2)
thf(fact_816_identify__block_Oelims,axiom,
! [X: mat_nat,Xa: nat,Y3: nat] :
( ( ( jordan8923406848002823307ck_nat @ X @ Xa )
= Y3 )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y3 != zero_zero_nat ) )
=> ~ ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ~ ( ( ( ( index_mat_nat @ X @ ( product_Pair_nat_nat @ I2 @ ( suc @ I2 ) ) )
= one_one_nat )
=> ( Y3
= ( jordan8923406848002823307ck_nat @ X @ I2 ) ) )
& ( ( ( index_mat_nat @ X @ ( product_Pair_nat_nat @ I2 @ ( suc @ I2 ) ) )
!= one_one_nat )
=> ( Y3
= ( suc @ I2 ) ) ) ) ) ) ) ).
% identify_block.elims
thf(fact_817_identify__block_Oelims,axiom,
! [X: mat_real,Xa: nat,Y3: nat] :
( ( ( jordan6672758942465739239k_real @ X @ Xa )
= Y3 )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y3 != zero_zero_nat ) )
=> ~ ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ~ ( ( ( ( index_mat_real @ X @ ( product_Pair_nat_nat @ I2 @ ( suc @ I2 ) ) )
= one_one_real )
=> ( Y3
= ( jordan6672758942465739239k_real @ X @ I2 ) ) )
& ( ( ( index_mat_real @ X @ ( product_Pair_nat_nat @ I2 @ ( suc @ I2 ) ) )
!= one_one_real )
=> ( Y3
= ( suc @ I2 ) ) ) ) ) ) ) ).
% identify_block.elims
thf(fact_818_identify__block__main,axiom,
! [A4: mat_nat,J3: nat,I3: nat] :
( ( ( jordan8923406848002823307ck_nat @ A4 @ J3 )
= I3 )
=> ( ( ord_less_eq_nat @ I3 @ J3 )
& ( ( I3 = zero_zero_nat )
| ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I3 @ one_one_nat ) @ I3 ) )
!= one_one_nat ) )
& ! [K4: nat] :
( ( ord_less_eq_nat @ I3 @ K4 )
=> ( ( ord_less_nat @ K4 @ J3 )
=> ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ K4 @ ( suc @ K4 ) ) )
= one_one_nat ) ) ) ) ) ).
% identify_block_main
thf(fact_819_identify__block__main,axiom,
! [A4: mat_real,J3: nat,I3: nat] :
( ( ( jordan6672758942465739239k_real @ A4 @ J3 )
= I3 )
=> ( ( ord_less_eq_nat @ I3 @ J3 )
& ( ( I3 = zero_zero_nat )
| ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I3 @ one_one_nat ) @ I3 ) )
!= one_one_real ) )
& ! [K4: nat] :
( ( ord_less_eq_nat @ I3 @ K4 )
=> ( ( ord_less_nat @ K4 @ J3 )
=> ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ K4 @ ( suc @ K4 ) ) )
= one_one_real ) ) ) ) ) ).
% identify_block_main
thf(fact_820_smult__smult__times,axiom,
! [A: nat,K: nat,A4: mat_nat] :
( ( smult_mat_nat @ A @ ( smult_mat_nat @ K @ A4 ) )
= ( smult_mat_nat @ ( times_times_nat @ A @ K ) @ A4 ) ) ).
% smult_smult_times
thf(fact_821_smult__smult__times,axiom,
! [A: real,K: real,A4: mat_real] :
( ( smult_mat_real @ A @ ( smult_mat_real @ K @ A4 ) )
= ( smult_mat_real @ ( times_times_real @ A @ K ) @ A4 ) ) ).
% smult_smult_times
thf(fact_822_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A6: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A6 @ B5 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_823_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A6: real,B5: real] : ( ord_less_eq_real @ ( minus_minus_real @ A6 @ B5 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_824_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_825_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A6: real,B5: real] : ( ord_less_real @ ( minus_minus_real @ A6 @ B5 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_826_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_827_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_828_arith__special_I21_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% arith_special(21)
thf(fact_829_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_830_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_831_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_832_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_833_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_834_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_835_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_836_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_837_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_838_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_839_le__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C2 @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_840_diff__le__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_841_add__le__imp__le__diff,axiom,
! [I3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_842_add__le__imp__le__diff,axiom,
! [I3: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ N )
=> ( ord_less_eq_real @ I3 @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_843_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_844_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_845_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_846_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_847_add__le__add__imp__diff__le,axiom,
! [I3: nat,K: nat,N: nat,J3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J3 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_848_add__le__add__imp__diff__le,axiom,
! [I3: real,K: real,N: real,J3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J3 @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J3 @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J3 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_849_eq__add__iff1,axiom,
! [A: real,E2: real,C2: real,B: real,D3: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D3 ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C2 )
= D3 ) ) ).
% eq_add_iff1
thf(fact_850_eq__add__iff2,axiom,
! [A: real,E2: real,C2: real,B: real,D3: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D3 ) )
= ( C2
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D3 ) ) ) ).
% eq_add_iff2
thf(fact_851_square__diff__square__factored,axiom,
! [X: real,Y3: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y3 ) @ ( minus_minus_real @ X @ Y3 ) ) ) ).
% square_diff_square_factored
thf(fact_852_mult__diff__mult,axiom,
! [X: real,Y3: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X @ Y3 ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y3 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_853_diff__less__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( ord_less_real @ A @ ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_854_less__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C2 @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_855_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_856_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_857_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_858_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_859_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_860_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_861_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_862_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_863_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_864_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_865_prod__decode__aux_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [K3: nat,M4: nat] :
( ( ~ ( ord_less_eq_nat @ M4 @ K3 )
=> ( P @ ( suc @ K3 ) @ ( minus_minus_nat @ M4 @ ( suc @ K3 ) ) ) )
=> ( P @ K3 @ M4 ) )
=> ( P @ A0 @ A1 ) ) ).
% prod_decode_aux.induct
thf(fact_866_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_867_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_868_less__diff__conv,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J3 ) ) ).
% less_diff_conv
thf(fact_869_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ( minus_minus_nat @ J3 @ I3 )
= K )
= ( J3
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_870_Nat_Odiff__diff__right,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J3 ) ) ) ).
% Nat.diff_diff_right
thf(fact_871_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_872_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_873_Nat_Odiff__add__assoc,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J3 ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_874_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J3 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_875_Nat_Ole__diff__conv2,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J3 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_876_le__diff__conv,axiom,
! [J3: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J3 @ K ) @ I3 )
= ( ord_less_eq_nat @ J3 @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_877_diff__strict__mono,axiom,
! [A: real,B: real,D3: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D3 @ C2 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D3 ) ) ) ) ).
% diff_strict_mono
thf(fact_878_diff__eq__diff__less,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D3 ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C2 @ D3 ) ) ) ).
% diff_eq_diff_less
thf(fact_879_diff__strict__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_880_diff__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_881_less__imp__diff__less,axiom,
! [J3: nat,K: nat,N: nat] :
( ( ord_less_nat @ J3 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J3 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_882_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_883_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_884_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_885_diff__diff__left,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% diff_diff_left
thf(fact_886_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_887_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_888_cross3__simps_I18_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).
% cross3_simps(18)
thf(fact_889_cross3__simps_I17_J,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).
% cross3_simps(17)
thf(fact_890_cross3__simps_I16_J,axiom,
! [A: real,C2: real,B: real] :
( ( A
= ( minus_minus_real @ C2 @ B ) )
= ( ( plus_plus_real @ A @ B )
= C2 ) ) ).
% cross3_simps(16)
thf(fact_891_cross3__simps_I15_J,axiom,
! [A: real,B: real,C2: real] :
( ( ( minus_minus_real @ A @ B )
= C2 )
= ( A
= ( plus_plus_real @ C2 @ B ) ) ) ).
% cross3_simps(15)
thf(fact_892_cross3__simps_I14_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% cross3_simps(14)
thf(fact_893_cross3__simps_I13_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% cross3_simps(13)
thf(fact_894_cross3__simps_I13_J,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% cross3_simps(13)
thf(fact_895_add__diff__add,axiom,
! [A: real,C2: real,B: real,D3: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D3 ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C2 @ D3 ) ) ) ).
% add_diff_add
thf(fact_896_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_897_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_898_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_899_add__diff__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_900_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_901_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_902_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_903_add__diff__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_904_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_905_add__implies__diff,axiom,
! [C2: real,B: real,A: real] :
( ( ( plus_plus_real @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_906_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_907_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_908_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_909_group__cancel_Osub1,axiom,
! [A4: real,K: real,A: real,B: real] :
( ( A4
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A4 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_910_cross3__simps_I51_J,axiom,
! [A: real,X: real,Y3: real] :
( ( times_times_real @ A @ ( minus_minus_real @ X @ Y3 ) )
= ( minus_minus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ A @ Y3 ) ) ) ).
% cross3_simps(51)
thf(fact_911_cross3__simps_I50_J,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ X )
= ( minus_minus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ B @ X ) ) ) ).
% cross3_simps(50)
thf(fact_912_cross3__simps_I28_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% cross3_simps(28)
thf(fact_913_cross3__simps_I27_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% cross3_simps(27)
thf(fact_914_cross3__simps_I26_J,axiom,
! [B: nat,C2: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C2 ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C2 @ A ) ) ) ).
% cross3_simps(26)
thf(fact_915_cross3__simps_I26_J,axiom,
! [B: real,C2: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C2 ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C2 @ A ) ) ) ).
% cross3_simps(26)
thf(fact_916_cross3__simps_I25_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% cross3_simps(25)
thf(fact_917_cross3__simps_I25_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% cross3_simps(25)
thf(fact_918_diff__eq__diff__eq,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D3 ) )
=> ( ( A = B )
= ( C2 = D3 ) ) ) ).
% diff_eq_diff_eq
thf(fact_919_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_920_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C2: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_921_minus__nat_Osimps_I1_J,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.simps(1)
thf(fact_922_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_923_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_924_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_925_diff__mono,axiom,
! [A: real,B: real,D3: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D3 @ C2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D3 ) ) ) ) ).
% diff_mono
thf(fact_926_diff__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_927_diff__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_928_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D3 ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C2 @ D3 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_929_arith__extra__simps_I13_J,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% arith_extra_simps(13)
thf(fact_930_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_931_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_932_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_933_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_934_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A6: real,B5: real] :
( ( minus_minus_real @ A6 @ B5 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_935_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_936_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_937_diff__Pair,axiom,
! [A: nat > nat,B: nat,C2: nat > nat,D3: nat] :
( ( minus_9067931446424981591at_nat @ ( produc72220940542539688at_nat @ A @ B ) @ ( produc72220940542539688at_nat @ C2 @ D3 ) )
= ( produc72220940542539688at_nat @ ( minus_minus_nat_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ D3 ) ) ) ).
% diff_Pair
thf(fact_938_diff__Pair,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( minus_4365393887724441320at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C2 @ D3 ) )
= ( product_Pair_nat_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ D3 ) ) ) ).
% diff_Pair
thf(fact_939_diff__Pair,axiom,
! [A: nat,B: real,C2: nat,D3: real] :
( ( minus_5557628854490389828t_real @ ( produc7837566107596912789t_real @ A @ B ) @ ( produc7837566107596912789t_real @ C2 @ D3 ) )
= ( produc7837566107596912789t_real @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_real @ B @ D3 ) ) ) ).
% diff_Pair
thf(fact_940_diff__Pair,axiom,
! [A: real,B: nat,C2: real,D3: nat] :
( ( minus_1582581163013509572al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc3181502643871035669al_nat @ C2 @ D3 ) )
= ( produc3181502643871035669al_nat @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_nat @ B @ D3 ) ) ) ).
% diff_Pair
thf(fact_941_diff__Pair,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( minus_885040589139849760l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( produc4511245868158468465l_real @ C2 @ D3 ) )
= ( produc4511245868158468465l_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D3 ) ) ) ).
% diff_Pair
thf(fact_942_index__minus__mat_I1_J,axiom,
! [I3: nat,B4: mat_nat,J3: nat,A4: mat_nat] :
( ( ord_less_nat @ I3 @ ( dim_row_nat @ B4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_nat @ B4 ) )
=> ( ( index_mat_nat @ ( minus_minus_mat_nat @ A4 @ B4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( minus_minus_nat @ ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) @ ( index_mat_nat @ B4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_minus_mat(1)
thf(fact_943_index__minus__mat_I1_J,axiom,
! [I3: nat,B4: mat_real,J3: nat,A4: mat_real] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ B4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ B4 ) )
=> ( ( index_mat_real @ ( minus_minus_mat_real @ A4 @ B4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( minus_minus_real @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) @ ( index_mat_real @ B4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ).
% index_minus_mat(1)
thf(fact_944_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E2: real,C2: real,B: real,D3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D3 ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C2 ) @ D3 ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_945_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E2: real,C2: real,B: real,D3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D3 ) )
= ( ord_less_eq_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D3 ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_946_less__add__iff1,axiom,
! [A: real,E2: real,C2: real,B: real,D3: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D3 ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C2 ) @ D3 ) ) ).
% less_add_iff1
thf(fact_947_less__add__iff2,axiom,
! [A: real,E2: real,C2: real,B: real,D3: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D3 ) )
= ( ord_less_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D3 ) ) ) ).
% less_add_iff2
thf(fact_948_square__diff__one__factored,axiom,
! [X: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_949_diff__Suc__less,axiom,
! [N: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_950_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_951_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_952_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_953_less__diff__conv2,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J3 @ K ) @ I3 )
= ( ord_less_nat @ J3 @ ( plus_plus_nat @ I3 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_954_diff__Suc__diff__eq1,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ I3 @ ( suc @ ( minus_minus_nat @ J3 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ ( suc @ J3 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_955_diff__Suc__diff__eq2,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J3 @ K ) ) @ I3 )
= ( minus_minus_nat @ ( suc @ J3 ) @ ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_956_nat__eq__add__iff1,axiom,
! [J3: nat,I3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J3 @ I3 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J3 ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_957_nat__eq__add__iff2,axiom,
! [I3: nat,J3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J3 @ I3 ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_958_nat__le__add__iff1,axiom,
! [J3: nat,I3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J3 @ I3 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J3 ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_959_nat__le__add__iff2,axiom,
! [I3: nat,J3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J3 @ I3 ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_960_nat__diff__add__eq1,axiom,
! [J3: nat,I3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J3 @ I3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J3 ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_961_nat__diff__add__eq2,axiom,
! [I3: nat,J3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J3 @ I3 ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_962_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_963_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_964_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_965_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_966_nat__less__add__iff1,axiom,
! [J3: nat,I3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J3 @ I3 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J3 ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_967_nat__less__add__iff2,axiom,
! [I3: nat,J3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J3 @ I3 ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_968_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_969_linepath__le__1,axiom,
! [A: real,B: real,U: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ U @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( times_times_real @ U @ B ) ) @ one_one_real ) ) ) ) ) ).
% linepath_le_1
thf(fact_970_identify__block_I2_J,axiom,
! [A4: mat_nat,J3: nat,I3: nat] :
( ( ( jordan8923406848002823307ck_nat @ A4 @ J3 )
= I3 )
=> ( ( I3 = zero_zero_nat )
| ( ( index_mat_nat @ A4 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I3 @ one_one_nat ) @ I3 ) )
!= one_one_nat ) ) ) ).
% identify_block(2)
thf(fact_971_identify__block_I2_J,axiom,
! [A4: mat_real,J3: nat,I3: nat] :
( ( ( jordan6672758942465739239k_real @ A4 @ J3 )
= I3 )
=> ( ( I3 = zero_zero_nat )
| ( ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I3 @ one_one_nat ) @ I3 ) )
!= one_one_real ) ) ) ).
% identify_block(2)
thf(fact_972_add__col__sub__index__row,axiom,
! [I3: nat,A4: mat_real,J3: nat,L: nat,K: nat,A: real] :
( ( ord_less_nat @ I3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ I3 @ ( dim_col_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_row_real @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_real @ A4 ) )
=> ( ( ord_less_nat @ L @ ( dim_row_real @ A4 ) )
=> ( ( ( ( I3 = K )
& ( J3 = L ) )
=> ( ( index_mat_real @ ( column3494657893274022100w_real @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( minus_minus_real @ ( minus_minus_real @ ( plus_plus_real @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) @ ( times_times_real @ A @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ I3 ) ) ) ) @ ( times_times_real @ ( times_times_real @ A @ A ) @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ J3 @ I3 ) ) ) ) @ ( times_times_real @ A @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ J3 @ J3 ) ) ) ) ) )
& ( ~ ( ( I3 = K )
& ( J3 = L ) )
=> ( ( ( ( I3 = K )
& ( J3 != L ) )
=> ( ( index_mat_real @ ( column3494657893274022100w_real @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( minus_minus_real @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) @ ( times_times_real @ A @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ L @ J3 ) ) ) ) ) )
& ( ~ ( ( I3 = K )
& ( J3 != L ) )
=> ( ( ( ( I3 != K )
& ( J3 = L ) )
=> ( ( index_mat_real @ ( column3494657893274022100w_real @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( plus_plus_real @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) @ ( times_times_real @ A @ ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ K ) ) ) ) ) )
& ( ~ ( ( I3 != K )
& ( J3 = L ) )
=> ( ( index_mat_real @ ( column3494657893274022100w_real @ A @ K @ L @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_real @ A4 @ ( product_Pair_nat_nat @ I3 @ J3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% add_col_sub_index_row
thf(fact_973_swap__cols__rows__block__index,axiom,
! [I3: nat,A4: mat_a,J3: nat,Low: nat,High: nat] :
( ( ord_less_nat @ I3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ I3 @ ( dim_col_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ A4 ) )
=> ( ( ord_less_eq_nat @ Low @ High )
=> ( ( ord_less_nat @ High @ ( dim_row_a @ A4 ) )
=> ( ( ord_less_nat @ High @ ( dim_col_a @ A4 ) )
=> ( ( index_mat_a @ ( jordan7507754584721484182lock_a @ Low @ High @ A4 ) @ ( product_Pair_nat_nat @ I3 @ J3 ) )
= ( index_mat_a @ A4
@ ( product_Pair_nat_nat
@ ( if_nat @ ( I3 = Low ) @ High
@ ( if_nat
@ ( ( ord_less_nat @ Low @ I3 )
& ( ord_less_eq_nat @ I3 @ High ) )
@ ( minus_minus_nat @ I3 @ one_one_nat )
@ I3 ) )
@ ( if_nat @ ( J3 = Low ) @ High
@ ( if_nat
@ ( ( ord_less_nat @ Low @ J3 )
& ( ord_less_eq_nat @ J3 @ High ) )
@ ( minus_minus_nat @ J3 @ one_one_nat )
@ J3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% swap_cols_rows_block_index
thf(fact_974_swap__cols__rows__block__dims_I2_J,axiom,
! [I3: nat,J3: nat,A4: mat_a] :
( ( dim_col_a @ ( jordan7507754584721484182lock_a @ I3 @ J3 @ A4 ) )
= ( dim_col_a @ A4 ) ) ).
% swap_cols_rows_block_dims(2)
thf(fact_975_swap__cols__rows__block__dims_I1_J,axiom,
! [I3: nat,J3: nat,A4: mat_a] :
( ( dim_row_a @ ( jordan7507754584721484182lock_a @ I3 @ J3 @ A4 ) )
= ( dim_row_a @ A4 ) ) ).
% swap_cols_rows_block_dims(1)
thf(fact_976_swap__cols__rows__block__dims__main,axiom,
! [I3: nat,J3: nat,A4: mat_a] :
( ( ( dim_row_a @ ( jordan7507754584721484182lock_a @ I3 @ J3 @ A4 ) )
= ( dim_row_a @ A4 ) )
& ( ( dim_col_a @ ( jordan7507754584721484182lock_a @ I3 @ J3 @ A4 ) )
= ( dim_col_a @ A4 ) ) ) ).
% swap_cols_rows_block_dims_main
thf(fact_977_segment__bound__lemma,axiom,
! [B4: real,X: real,Y3: real,U: real] :
( ( ord_less_eq_real @ B4 @ X )
=> ( ( ord_less_eq_real @ B4 @ Y3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ U @ one_one_real )
=> ( ord_less_eq_real @ B4 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U ) @ X ) @ ( times_times_real @ U @ Y3 ) ) ) ) ) ) ) ).
% segment_bound_lemma
thf(fact_978_step__1__main_Oinduct,axiom,
! [N: nat,P: nat > nat > mat_real > $o,A0: nat,A1: nat,A22: mat_real] :
( ! [I2: nat,J2: nat,A7: mat_real] :
( ( ~ ( ord_less_eq_nat @ N @ J2 )
=> ( ( I2 = zero_zero_nat )
=> ( P @ ( plus_plus_nat @ J2 @ one_one_nat ) @ ( plus_plus_nat @ J2 @ one_one_nat ) @ A7 ) ) )
=> ( ( ~ ( ord_less_eq_nat @ N @ J2 )
=> ( ( I2 != zero_zero_nat )
=> ! [Xd: mat_real] :
( ( ( ( ( ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I2 @ one_one_nat ) @ ( minus_minus_nat @ I2 @ one_one_nat ) ) )
!= ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ J2 @ J2 ) ) )
& ( ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I2 @ one_one_nat ) @ J2 ) )
!= zero_zero_real ) )
=> ( Xd
= ( column3494657893274022100w_real @ ( divide_divide_real @ ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I2 @ one_one_nat ) @ J2 ) ) @ ( minus_minus_real @ ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ J2 @ J2 ) ) @ ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I2 @ one_one_nat ) @ ( minus_minus_nat @ I2 @ one_one_nat ) ) ) ) ) @ ( minus_minus_nat @ I2 @ one_one_nat ) @ J2 @ A7 ) ) )
& ( ~ ( ( ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I2 @ one_one_nat ) @ ( minus_minus_nat @ I2 @ one_one_nat ) ) )
!= ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ J2 @ J2 ) ) )
& ( ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I2 @ one_one_nat ) @ J2 ) )
!= zero_zero_real ) )
=> ( Xd = A7 ) ) )
=> ( P @ ( minus_minus_nat @ I2 @ one_one_nat ) @ J2 @ Xd ) ) ) )
=> ( P @ I2 @ J2 @ A7 ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ).
% step_1_main.induct
thf(fact_979_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A3: real,B3: real,C: real] :
( ( P @ A3 @ B3 )
=> ( ( P @ B3 @ C )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( P @ A3 @ C ) ) ) ) )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B )
=> ? [D6: real] :
( ( ord_less_real @ zero_zero_real @ D6 )
& ! [A3: real,B3: real] :
( ( ( ord_less_eq_real @ A3 @ X2 )
& ( ord_less_eq_real @ X2 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D6 ) )
=> ( P @ A3 @ B3 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_980_divide__nonpos__pos,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_981_divide__nonpos__neg,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y3 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% divide_nonpos_neg
thf(fact_982_divide__nonneg__pos,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% divide_nonneg_pos
thf(fact_983_divide__nonneg__neg,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_984_divide__le__cancel,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_985_frac__less2,axiom,
! [X: real,Y3: real,W: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y3 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_real @ W @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_986_frac__less,axiom,
! [X: real,Y3: real,W: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y3 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_987_frac__le,axiom,
! [Y3: real,X: real,W: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_eq_real @ X @ Y3 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_988_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_989_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_990_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_991_divide__strict__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_992_mult__imp__less__div__pos,axiom,
! [Y3: real,Z2: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_real @ ( times_times_real @ Z2 @ Y3 ) @ X )
=> ( ord_less_real @ Z2 @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_993_mult__imp__div__pos__less,axiom,
! [Y3: real,X: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z2 @ Y3 ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y3 ) @ Z2 ) ) ) ).
% mult_imp_div_pos_less
thf(fact_994_pos__less__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_995_pos__divide__less__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).
% pos_divide_less_eq
thf(fact_996_neg__less__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).
% neg_less_divide_eq
thf(fact_997_neg__divide__less__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_998_less__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_999_divide__less__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1000_add__divide__eq__if__simps_I2_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= B ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1001_add__divide__eq__if__simps_I1_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1002_add__frac__eq,axiom,
! [Y3: real,Z2: real,X: real,W: real] :
( ( Y3 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ W @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) ) ) ) ).
% add_frac_eq
thf(fact_1003_add__frac__num,axiom,
! [Y3: real,X: real,Z2: real] :
( ( Y3 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y3 ) @ Z2 )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% add_frac_num
thf(fact_1004_add__num__frac,axiom,
! [Y3: real,Z2: real,X: real] :
( ( Y3 != zero_zero_real )
=> ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X @ Y3 ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% add_num_frac
thf(fact_1005_add__divide__eq__iff,axiom,
! [Z2: real,X: real,Y3: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y3 @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% add_divide_eq_iff
thf(fact_1006_divide__add__eq__iff,axiom,
! [Z2: real,X: real,Y3: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z2 ) @ Y3 )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% divide_add_eq_iff
thf(fact_1007_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1008_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1009_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_1010_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1011_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1012_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1013_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1014_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_1015_less__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_1016_divide__less__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_1017_divide__diff__eq__iff,axiom,
! [Z2: real,X: real,Y3: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z2 ) @ Y3 )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% divide_diff_eq_iff
thf(fact_1018_diff__divide__eq__iff,axiom,
! [Z2: real,X: real,Y3: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y3 @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% diff_divide_eq_iff
thf(fact_1019_diff__frac__eq,axiom,
! [Y3: real,Z2: real,X: real,W: real] :
( ( Y3 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ W @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1020_add__divide__eq__if__simps_I4_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1021_gt__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_1022_less__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_1023_divide__right__mono__neg,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ ( divide_divide_real @ A @ C2 ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1024_divide__nonpos__nonpos,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1025_divide__nonpos__nonneg,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1026_divide__nonneg__nonpos,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1027_divide__nonneg__nonneg,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1028_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1029_divide__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).
% divide_right_mono
thf(fact_1030_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_1031_frac__eq__eq,axiom,
! [Y3: real,Z2: real,X: real,W: real] :
( ( Y3 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y3 )
= ( divide_divide_real @ W @ Z2 ) )
= ( ( times_times_real @ X @ Z2 )
= ( times_times_real @ W @ Y3 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1032_divide__eq__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ( divide_divide_real @ B @ C2 )
= A )
= ( ( ( C2 != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_1033_eq__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( A
= ( divide_divide_real @ B @ C2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A @ C2 )
= B ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_1034_divide__eq__imp,axiom,
! [C2: real,B: real,A: real] :
( ( C2 != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C2 ) )
=> ( ( divide_divide_real @ B @ C2 )
= A ) ) ) ).
% divide_eq_imp
thf(fact_1035_eq__divide__imp,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= B )
=> ( A
= ( divide_divide_real @ B @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_1036_nonzero__divide__eq__eq,axiom,
! [C2: real,B: real,A: real] :
( ( C2 != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C2 )
= A )
= ( B
= ( times_times_real @ A @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1037_nonzero__eq__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C2 ) )
= ( ( times_times_real @ A @ C2 )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1038_mult__divide__mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% mult_divide_mult_cancel_left
thf(fact_1039_mult__divide__mult__cancel__right,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% mult_divide_mult_cancel_right
thf(fact_1040_mult__divide__mult__cancel__left__if,axiom,
! [C2: real,A: real,B: real] :
( ( ( C2 = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= zero_zero_real ) )
& ( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1041_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ B @ C2 ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1042_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ C2 @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1043_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1044_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1045_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1046_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1047_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_1048_divide__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1049_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1050_divide__less__cancel,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C2 != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_1051_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_1052_divide__pos__pos,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% divide_pos_pos
thf(fact_1053_divide__pos__neg,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y3 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_1054_divide__neg__pos,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_1055_divide__neg__neg,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y3 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% divide_neg_neg
thf(fact_1056_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_1057_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_1058_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_1059_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_1060_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_1061_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_1062_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_1063_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_1064_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_1065_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_1066_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1067_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1068_add__divide__distrib,axiom,
! [A: real,B: real,C2: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ).
% add_divide_distrib
thf(fact_1069_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C2 )
= ( divide_divide_real @ A @ ( times_times_real @ C2 @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_1070_times__divide__eq__right,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C2 ) ) ).
% times_divide_eq_right
thf(fact_1071_divide__divide__times__eq,axiom,
! [X: real,Y3: real,Z2: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y3 @ Z2 ) ) ) ).
% divide_divide_times_eq
thf(fact_1072_divide__divide__eq__right,axiom,
! [A: real,B: real,C2: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_1073_times__divide__times__eq,axiom,
! [X: real,Y3: real,Z2: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y3 @ W ) ) ) ).
% times_divide_times_eq
thf(fact_1074_divide__divide__eq__left,axiom,
! [A: real,B: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C2 )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% divide_divide_eq_left
thf(fact_1075_times__divide__eq__left,axiom,
! [B: real,C2: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C2 ) ) ).
% times_divide_eq_left
thf(fact_1076_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_1077_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1078_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_1079_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $true @ X @ Y3 )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y3: real] :
( ( if_real @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y3: real] :
( ( if_real @ $true @ X @ Y3 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( index_mat_a @ a2 @ ( product_Pair_nat_nat @ i @ j ) )
= ( index_mat_a @ b @ ( product_Pair_nat_nat @ i @ j ) ) ) ).
%------------------------------------------------------------------------------