TPTP Problem File: SLH0769^1.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 : Fishers_Inequality/0036_Rank_Argument_General/prob_00276_012255__28162260_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1393 ( 566 unt; 123 typ; 0 def)
% Number of atoms : 3844 (1376 equ; 0 cnn)
% Maximal formula atoms : 18 ( 3 avg)
% Number of connectives : 12763 ( 339 ~; 113 |; 245 &;10297 @)
% ( 0 <=>;1769 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 427 ( 427 >; 0 *; 0 +; 0 <<)
% Number of symbols : 114 ( 111 usr; 14 con; 0-6 aty)
% Number of variables : 3706 ( 94 ^;3566 !; 46 ?;3706 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:49:23.677
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
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__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__Matrix__Omat_It__Int__Oint_J,type,
mat_int: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Matrix__Omat_Itf__a_J,type,
mat_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (111)
thf(sy_c_Column__Operations_Omat__addcol_001t__Int__Oint,type,
column5439950039029753374ol_int: int > nat > nat > mat_int > mat_int ).
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__addcol_001tf__a,type,
column_mat_addcol_a: a > nat > nat > mat_a > mat_a ).
thf(sy_c_Column__Operations_Omat__multcol_001t__Int__Oint,type,
column382118079982894795ol_int: nat > int > mat_int > mat_int ).
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__multcol_001tf__a,type,
column_mat_multcol_a: nat > a > mat_a > mat_a ).
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_001tf__a,type,
column5129559316938501008rows_a: nat > nat > mat_a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Omat__addrow__gen_001t__Int__Oint,type,
gauss_8882552878057600758en_int: ( int > int > int ) > ( int > int > int ) > int > nat > nat > mat_int > mat_int ).
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__swaprows_001tf__a,type,
gauss_2482569599970757219rows_a: nat > nat > mat_a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Omultrow__mat_001t__Int__Oint,type,
gauss_3192586071676587637at_int: nat > nat > int > mat_int ).
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_Omultrow__mat_001tf__a,type,
gauss_5015385051186949877_mat_a: nat > nat > a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Oswaprows__mat_001t__Int__Oint,type,
gauss_4917416859360123759at_int: nat > nat > nat > mat_int ).
thf(sy_c_Gauss__Jordan__Elimination_Oswaprows__mat_001t__Nat__Onat,type,
gauss_4919907329869174035at_nat: nat > nat > nat > mat_nat ).
thf(sy_c_Gauss__Jordan__Elimination_Oswaprows__mat_001t__Real__Oreal,type,
gauss_1271566072679876207t_real: nat > nat > nat > mat_real ).
thf(sy_c_Gauss__Jordan__Elimination_Oswaprows__mat_001tf__a,type,
gauss_110929411057020027_mat_a: nat > nat > nat > mat_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oone__class_Oone_001tf__a,type,
one_one_a: a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Int__Oint_J,type,
plus_plus_mat_int: mat_int > mat_int > mat_int ).
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__Matrix__Omat_Itf__a_J,type,
plus_plus_mat_a: mat_a > mat_a > mat_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a,type,
plus_plus_a: a > a > a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_Itf__a_J,type,
times_times_mat_a: mat_a > mat_a > mat_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a,type,
times_times_a: a > a > a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Int__Oint,type,
diagonal_mat_int: mat_int > $o ).
thf(sy_c_Matrix_Odiagonal__mat_001t__Nat__Onat,type,
diagonal_mat_nat: mat_nat > $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__Int__Oint,type,
dim_col_int: mat_int > nat ).
thf(sy_c_Matrix_Odim__col_001t__Nat__Onat,type,
dim_col_nat: mat_nat > 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__Int__Oint,type,
dim_row_int: mat_int > nat ).
thf(sy_c_Matrix_Odim__row_001t__Nat__Onat,type,
dim_row_nat: mat_nat > 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_Oindex__mat_001t__Int__Oint,type,
index_mat_int: mat_int > product_prod_nat_nat > int ).
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__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_Osmult__mat_001t__Int__Oint,type,
smult_mat_int: int > mat_int > mat_int ).
thf(sy_c_Matrix_Osmult__mat_001t__Nat__Onat,type,
smult_mat_nat: nat > mat_nat > mat_nat ).
thf(sy_c_Matrix_Osmult__mat_001t__Real__Oreal,type,
smult_mat_real: real > mat_real > mat_real ).
thf(sy_c_Matrix_Osmult__mat_001tf__a,type,
smult_mat_a: a > mat_a > mat_a ).
thf(sy_c_Matrix_Oupdate__mat_001tf__a,type,
update_mat_a: mat_a > product_prod_nat_nat > a > mat_a ).
thf(sy_c_Matrix_Oupper__triangular_001t__Int__Oint,type,
upper_triangular_int: mat_int > $o ).
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_Matrix__Vector__Extras_Oall__ones__mat_001t__Int__Oint,type,
matrix8485685120660989714at_int: nat > mat_int ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__mat_001t__Nat__Onat,type,
matrix8488175591170039990at_nat: nat > mat_nat ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__mat_001t__Real__Oreal,type,
matrix5418915509738909842t_real: nat > mat_real ).
thf(sy_c_Matrix__Vector__Extras_Oall__ones__mat_001tf__a,type,
matrix7809018878362141592_mat_a: nat > mat_a ).
thf(sy_c_More__Matrix_Okronecker__product_001t__Int__Oint,type,
more_k9102366682204111370ct_int: mat_int > mat_int > mat_int ).
thf(sy_c_More__Matrix_Okronecker__product_001t__Real__Oreal,type,
more_k7431981406657007498t_real: mat_real > mat_real > mat_real ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Omat_It__Int__Oint_J,type,
ord_less_eq_mat_int: mat_int > mat_int > $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_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
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_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_Rank__Argument__General_Oadd__col__to__multiple_001t__Int__Oint,type,
rank_A7394458515849970282le_int: int > list_nat > nat > mat_int > mat_int ).
thf(sy_c_Rank__Argument__General_Oadd__col__to__multiple_001t__Nat__Onat,type,
rank_A7396948986359020558le_nat: nat > list_nat > nat > mat_nat > mat_nat ).
thf(sy_c_Rank__Argument__General_Oadd__col__to__multiple_001t__Real__Oreal,type,
rank_A8462809353401409514e_real: real > list_nat > nat > mat_real > mat_real ).
thf(sy_c_Rank__Argument__General_Oadd__col__to__multiple_001tf__a,type,
rank_A5103328671897079360iple_a: a > list_nat > nat > mat_a > mat_a ).
thf(sy_c_Rank__Argument__General_Oadd__multiple__cols_001tf__a,type,
rank_A1622336257019553452cols_a: a > nat > list_nat > mat_a > mat_a ).
thf(sy_c_Rank__Argument__General_Oadd__multiple__rows_001tf__a,type,
rank_A5086152289722150470rows_a: a > nat > list_nat > mat_a > mat_a ).
thf(sy_c_Rank__Argument__General_Oadd__row__to__multiple_001t__Int__Oint,type,
rank_A6931195264251906052le_int: int > list_nat > nat > mat_int > mat_int ).
thf(sy_c_Rank__Argument__General_Oadd__row__to__multiple_001t__Nat__Onat,type,
rank_A6933685734760956328le_nat: nat > list_nat > nat > mat_nat > mat_nat ).
thf(sy_c_Rank__Argument__General_Oadd__row__to__multiple_001t__Real__Oreal,type,
rank_A3307089894062176388e_real: real > list_nat > nat > mat_real > mat_real ).
thf(sy_c_Rank__Argument__General_Oadd__row__to__multiple_001tf__a,type,
rank_A2757900658775374630iple_a: a > list_nat > nat > mat_a > mat_a ).
thf(sy_v_M,type,
m: mat_a ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_j,type,
j: nat ).
% Relevant facts (1265)
thf(fact_0_assms_I2_J,axiom,
ord_less_nat @ j @ ( dim_col_a @ m ) ).
% assms(2)
thf(fact_1_assms_I1_J,axiom,
ord_less_nat @ i @ ( dim_row_a @ m ) ).
% assms(1)
thf(fact_2_add__row__to__multiple__dim_I1_J,axiom,
! [A: a,Ks: list_nat,L: nat,A2: mat_a] :
( ( dim_row_a @ ( rank_A2757900658775374630iple_a @ A @ Ks @ L @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% add_row_to_multiple_dim(1)
thf(fact_3__092_060open_062_092_060lbrakk_062i_A_060_Adim__row_AM_059_Aj_A_060_Adim__col_AM_059_A1_A_092_060le_062_Ai_059_Ai_A_060_Adim__row_AM_059_A0_A_060_A1_A_092_060or_062_Adim__row_AM_A_092_060le_062_A0_059_A0_A_060_Adim__row_AM_092_060rbrakk_062_A_092_060Longrightarrow_062_Aadd__row__to__multiple_Aa_A_0911_O_O_060dim__row_AM_093_A0_AM_A_E_E_A_Ii_M_Aj_J_A_061_Aa_A_K_AM_A_E_E_A_I0_M_Aj_J_A_L_AM_A_E_E_A_Ii_M_Aj_J_092_060close_062,axiom,
( ( ord_less_nat @ i @ ( dim_row_a @ m ) )
=> ( ( ord_less_nat @ j @ ( dim_col_a @ m ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ i )
=> ( ( ord_less_nat @ i @ ( dim_row_a @ m ) )
=> ( ( ( ord_less_nat @ zero_zero_nat @ one_one_nat )
| ( ord_less_eq_nat @ ( dim_row_a @ m ) @ zero_zero_nat ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( dim_row_a @ m ) )
=> ( ( index_mat_a @ ( rank_A2757900658775374630iple_a @ a2 @ ( upt @ one_one_nat @ ( dim_row_a @ m ) ) @ zero_zero_nat @ m ) @ ( product_Pair_nat_nat @ i @ j ) )
= ( plus_plus_a @ ( times_times_a @ a2 @ ( index_mat_a @ m @ ( product_Pair_nat_nat @ zero_zero_nat @ j ) ) ) @ ( index_mat_a @ m @ ( product_Pair_nat_nat @ i @ j ) ) ) ) ) ) ) ) ) ) ).
% \<open>\<lbrakk>i < dim_row M; j < dim_col M; 1 \<le> i; i < dim_row M; 0 < 1 \<or> dim_row M \<le> 0; 0 < dim_row M\<rbrakk> \<Longrightarrow> add_row_to_multiple a [1..<dim_row M] 0 M $$ (i, j) = a * M $$ (0, j) + M $$ (i, j)\<close>
thf(fact_4_add__row__to__multiple__dim_I2_J,axiom,
! [A: a,Ks: list_nat,L: nat,A2: mat_a] :
( ( dim_col_a @ ( rank_A2757900658775374630iple_a @ A @ Ks @ L @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% add_row_to_multiple_dim(2)
thf(fact_5_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X2 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_6_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A3: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_7_add__row__to__multiple__index__unchanged__bound,axiom,
! [I: nat,A2: mat_a,J: nat,Low: nat,Up: nat,A: a,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( ord_less_nat @ I @ Low )
=> ( ( ord_less_eq_nat @ Up @ I )
=> ( ( index_mat_a @ ( rank_A2757900658775374630iple_a @ A @ ( upt @ Low @ Up ) @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% add_row_to_multiple_index_unchanged_bound
thf(fact_8_class__field_Ozero__not__one,axiom,
zero_zero_real != one_one_real ).
% class_field.zero_not_one
thf(fact_9_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_10_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_11_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_12_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_13_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_14_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_15_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_16_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_17_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_18_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_19_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_20_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_21_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_22_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_23_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_24_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_25_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_26_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_27_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_28_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_29_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_30_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_31_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_32_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_33_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_34_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_35_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_36_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_37_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_38_mult__zero__right,axiom,
! [A: a] :
( ( times_times_a @ A @ zero_zero_a )
= zero_zero_a ) ).
% mult_zero_right
thf(fact_39_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_40_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_41_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_42_mult__zero__left,axiom,
! [A: a] :
( ( times_times_a @ zero_zero_a @ A )
= zero_zero_a ) ).
% mult_zero_left
thf(fact_43_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_44_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_45_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_46_add__0,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% add_0
thf(fact_47_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_48_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_49_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_50_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_51_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_52_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_53_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_54_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_55_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_56_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_57_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_58_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_59_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_60_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_61_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_62_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_63_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_64_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_65_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_66_add_Oright__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.right_neutral
thf(fact_67_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_68_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_69_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_70_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_71_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_72_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_73_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_74_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_75_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_76_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_77_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_78_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_79_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_80_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_81_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_82_mult__1,axiom,
! [A: a] :
( ( times_times_a @ one_one_a @ A )
= A ) ).
% mult_1
thf(fact_83_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_84_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_85_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_86_mult_Oright__neutral,axiom,
! [A: a] :
( ( times_times_a @ A @ one_one_a )
= A ) ).
% mult.right_neutral
thf(fact_87_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_88_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_89_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_90_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_91_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_92_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_93_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_94_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_95_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_96_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_97_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_98_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_99_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_100_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_101_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_102_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_103_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_104_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_105_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_106_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_107_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_108_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_109_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_110_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_111_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_112_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_113_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_114_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_115_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_116_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_117_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_118_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_119_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_120_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_121_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_122_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_123_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_124_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_125_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_126_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_127_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_128_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_129_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_130_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_131_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_132_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_133_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_134_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_135_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_136_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_137_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_138_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_139_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_140_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_141_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_142_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_143_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_144_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_145_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_146_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_147_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_148_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_149_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_150_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_151_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_152_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_153_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_154_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_155_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_156_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_157_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_158_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_159_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_160_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_161_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_162_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_163_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_164_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_165_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_166_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_167_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_168_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_169_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_170_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_171_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_172_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_173_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_174_group__cancel_Oadd1,axiom,
! [A2: a,K: a,A: a,B: a] :
( ( A2
= ( plus_plus_a @ K @ A ) )
=> ( ( plus_plus_a @ A2 @ B )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_175_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_176_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_177_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_178_group__cancel_Oadd2,axiom,
! [B3: a,K: a,B: a,A: a] :
( ( B3
= ( plus_plus_a @ K @ B ) )
=> ( ( plus_plus_a @ A @ B3 )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_179_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_180_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_181_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_182_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_183_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_184_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_185_distrib__left,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ A @ ( plus_plus_a @ B @ C ) )
= ( plus_plus_a @ ( times_times_a @ A @ B ) @ ( times_times_a @ A @ C ) ) ) ).
% distrib_left
thf(fact_186_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_187_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_188_distrib__left,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_189_distrib__right,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ).
% distrib_right
thf(fact_190_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_191_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_192_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_193_add_Oassoc,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_194_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_195_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_196_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_197_class__cring_Ofactors__equal,axiom,
! [A: int,B: int,C: int,D: int] :
( ( A = B )
=> ( ( C = D )
=> ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ D ) ) ) ) ).
% class_cring.factors_equal
thf(fact_198_class__cring_Ofactors__equal,axiom,
! [A: real,B: real,C: real,D: real] :
( ( A = B )
=> ( ( C = D )
=> ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ D ) ) ) ) ).
% class_cring.factors_equal
thf(fact_199_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_200_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_201_mult_Oassoc,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ ( times_times_a @ A @ B ) @ C )
= ( times_times_a @ A @ ( times_times_a @ B @ C ) ) ) ).
% mult.assoc
thf(fact_202_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_203_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_204_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_205_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_206_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_207_add_Ocommute,axiom,
( plus_plus_a
= ( ^ [A4: a,B4: a] : ( plus_plus_a @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_208_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_209_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_210_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_211_combine__common__factor,axiom,
! [A: a,E: a,B: a,C: a] :
( ( plus_plus_a @ ( times_times_a @ A @ E ) @ ( plus_plus_a @ ( times_times_a @ B @ E ) @ C ) )
= ( plus_plus_a @ ( times_times_a @ ( plus_plus_a @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_212_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_213_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_214_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_215_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_216_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_217_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_218_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: a,B: a,C: a,D: a] :
( ( A = B )
=> ( ( C = D )
=> ( ( plus_plus_a @ A @ C )
= ( plus_plus_a @ B @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_219_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( A = B )
=> ( ( C = D )
=> ( ( plus_plus_nat @ A @ C )
= ( plus_plus_nat @ B @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_220_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: real,B: real,C: real,D: real] :
( ( A = B )
=> ( ( C = D )
=> ( ( plus_plus_real @ A @ C )
= ( plus_plus_real @ B @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_221_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: int,B: int,C: int,D: int] :
( ( A = B )
=> ( ( C = D )
=> ( ( plus_plus_int @ A @ C )
= ( plus_plus_int @ B @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_222_add_Oleft__commute,axiom,
! [B: a,A: a,C: a] :
( ( plus_plus_a @ B @ ( plus_plus_a @ A @ C ) )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_223_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_224_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_225_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_226_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_227_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_228_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_229_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_230_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_231_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_232_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_233_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_234_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_235_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_236_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_237_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_238_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_239_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_240_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_241_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_242_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_243_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_244_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_245_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_246_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_247_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_248_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_249_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_250_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_251_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_252_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_253_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_254_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_255_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_256_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_257_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_258_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_259_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_260_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_261_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_262_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_263_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_264_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_265_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_266_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_267_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_268_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_269_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_270_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_271_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_272_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_273_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_274_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_275_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_276_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_277_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_278_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_279_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_280_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_281_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_282_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_283_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_284_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_285_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_286_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_287_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_288_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_289_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_290_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_291_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_292_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_293_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_294_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_295_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_296_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_297_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_298_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_299_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_300_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_301_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_302_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_303_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_304_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_305_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_306_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_307_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_308_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_309_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_310_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_311_add__pos__nonneg,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_pos_nonneg
thf(fact_312_add__pos__nonneg,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_pos_nonneg
thf(fact_313_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_314_add__nonpos__neg,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_nonpos_neg
thf(fact_315_add__nonpos__neg,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_nonpos_neg
thf(fact_316_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_317_add__nonneg__pos,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_nonneg_pos
thf(fact_318_add__nonneg__pos,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_nonneg_pos
thf(fact_319_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_320_add__neg__nonpos,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_neg_nonpos
thf(fact_321_add__neg__nonpos,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_neg_nonpos
thf(fact_322_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_323_not__sum__squares__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_324_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_325_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_326_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_327_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_328_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_329_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_330_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_331_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_332_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_333_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_334_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_335_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_336_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_337_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_338_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_339_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_340_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_341_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_342_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_343_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_344_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_345_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_346_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_347_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_348_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_349_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_350_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_351_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_352_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_353_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_354_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_355_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_356_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_357_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_358_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_359_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_360_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_361_sum__squares__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_362_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_363_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_364_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_365_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_366_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_367_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_368_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_369_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_370_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_371_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_372_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_373_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_374_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_375_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_376_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_377_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_378_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_379_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_380_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_381_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_382_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_383_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_384_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_385_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_386_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_387_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_388_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_389_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_390_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_391_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_392_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_393_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_394_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_395_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_396_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_397_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_398_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_399_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_400_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_401_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_402_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_403_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_404_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_405_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_406_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_407_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_408_linordered__semiring__strict__class_Omult__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 @ ( times_times_real @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_409_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_410_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_411_linordered__semiring__strict__class_Omult__pos__neg,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 ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_412_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_413_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_414_linordered__semiring__strict__class_Omult__neg__pos,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 ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_415_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_416_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_417_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_418_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_419_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_420_mult__neg__neg,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_neg_neg
thf(fact_421_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_422_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_423_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_424_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_425_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_426_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_427_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_428_add__nonpos__nonpos,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_nonpos_nonpos
thf(fact_429_add__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 @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_430_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_431_add__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 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_432_add__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 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_433_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_434_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_435_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_436_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_437_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_438_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_439_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_440_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_441_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_442_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_443_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_444_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_445_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_446_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_447_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_448_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_449_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_450_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_451_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_452_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_453_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_454_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_455_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_456_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_457_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_458_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_459_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_460_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_461_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_462_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_463_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_464_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_465_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_466_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_467_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_468_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_469_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_470_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_471_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_472_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_473_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_474_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_475_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_476_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_477_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_478_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_479_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_480_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_481_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_482_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_483_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_484_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_485_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_486_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_487_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_488_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_489_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_490_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_491_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_492_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_493_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_494_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_495_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_496_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_497_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_498_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_499_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_500_convex__bound__lt,axiom,
! [X: real,A: real,Y: real,U: real,V: real] :
( ( ord_less_real @ X @ A )
=> ( ( ord_less_real @ Y @ 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 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_501_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_502_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_503_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_504_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_505_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_506_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_507_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_508_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_509_add_Ocomm__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.comm_neutral
thf(fact_510_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_511_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_512_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_513_comm__monoid__add__class_Oadd__0,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_514_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_515_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_516_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_517_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_518_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_519_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_520_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_521_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_522_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_523_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_524_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_525_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_526_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_527_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_528_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_529_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_530_mult__not__zero,axiom,
! [A: a,B: a] :
( ( ( times_times_a @ A @ B )
!= zero_zero_a )
=> ( ( A != zero_zero_a )
& ( B != zero_zero_a ) ) ) ).
% mult_not_zero
thf(fact_531_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_532_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_533_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_534_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_535_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_536_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_537_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_538_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_539_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_540_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_541_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_542_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_543_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_544_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_545_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_546_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_547_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_548_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_549_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_550_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_551_mult__less__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_552_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_553_mult__less__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_554_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_555_mult__less__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_556_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_557_mult__less__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_558_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_559_mult__le__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_560_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_561_mult__le__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_562_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_563_mult__le__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_564_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_565_mult__le__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_566_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_567_convex__bound__le,axiom,
! [X: real,A: real,Y: real,U: real,V: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ Y @ 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 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_568_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_569_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_570_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_571_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_572_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_573_mult__left__le,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_574_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_575_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_576_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_577_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_578_mult__right__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_579_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_580_mult__left__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_581_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_582_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_583_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_584_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_585_Pair__inject,axiom,
! [A: nat,B: nat,A3: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_586_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
( ! [A5: nat,B5: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_587_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X3: nat,Y3: nat] :
( P2
= ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_588_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A5: nat,B5: nat] :
( Y
!= ( product_Pair_nat_nat @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_589_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_590_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_591_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_592_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_593_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_594_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_595_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_596_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_597_zero__less__one__class_Ozero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_less_one
thf(fact_598_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_599_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_600_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_601_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_602_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_603_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_604_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_605_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_606_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_607_add__row__to__multiple__index__change__bounds,axiom,
! [I: nat,A2: mat_nat,J: nat,Low: nat,Up: nat,L: nat,A: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A2 ) )
=> ( ( ord_less_eq_nat @ Low @ I )
=> ( ( ord_less_nat @ I @ Up )
=> ( ( ( ord_less_nat @ L @ Low )
| ( ord_less_eq_nat @ Up @ L ) )
=> ( ( ord_less_nat @ L @ ( dim_row_nat @ A2 ) )
=> ( ( index_mat_nat @ ( rank_A6933685734760956328le_nat @ A @ ( upt @ Low @ Up ) @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ) ) ).
% add_row_to_multiple_index_change_bounds
thf(fact_608_add__row__to__multiple__index__change__bounds,axiom,
! [I: nat,A2: mat_int,J: nat,Low: nat,Up: nat,L: nat,A: int] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ( ord_less_eq_nat @ Low @ I )
=> ( ( ord_less_nat @ I @ Up )
=> ( ( ( ord_less_nat @ L @ Low )
| ( ord_less_eq_nat @ Up @ L ) )
=> ( ( ord_less_nat @ L @ ( dim_row_int @ A2 ) )
=> ( ( index_mat_int @ ( rank_A6931195264251906052le_int @ A @ ( upt @ Low @ Up ) @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ) ) ).
% add_row_to_multiple_index_change_bounds
thf(fact_609_add__row__to__multiple__index__change__bounds,axiom,
! [I: nat,A2: mat_real,J: nat,Low: nat,Up: nat,L: nat,A: real] :
( ( ord_less_nat @ I @ ( dim_row_real @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_real @ A2 ) )
=> ( ( ord_less_eq_nat @ Low @ I )
=> ( ( ord_less_nat @ I @ Up )
=> ( ( ( ord_less_nat @ L @ Low )
| ( ord_less_eq_nat @ Up @ L ) )
=> ( ( ord_less_nat @ L @ ( dim_row_real @ A2 ) )
=> ( ( index_mat_real @ ( rank_A3307089894062176388e_real @ A @ ( upt @ Low @ Up ) @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ) ) ).
% add_row_to_multiple_index_change_bounds
thf(fact_610_add__row__to__multiple__index__change__bounds,axiom,
! [I: nat,A2: mat_a,J: nat,Low: nat,Up: nat,L: nat,A: a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( ord_less_eq_nat @ Low @ I )
=> ( ( ord_less_nat @ I @ Up )
=> ( ( ( ord_less_nat @ L @ Low )
| ( ord_less_eq_nat @ Up @ L ) )
=> ( ( ord_less_nat @ L @ ( dim_row_a @ A2 ) )
=> ( ( index_mat_a @ ( rank_A2757900658775374630iple_a @ A @ ( upt @ Low @ Up ) @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_a @ ( times_times_a @ A @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ) ) ).
% add_row_to_multiple_index_change_bounds
thf(fact_611_index__add__mat_I1_J,axiom,
! [I: nat,B3: mat_a,J: nat,A2: mat_a] :
( ( ord_less_nat @ I @ ( dim_row_a @ B3 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ B3 ) )
=> ( ( index_mat_a @ ( plus_plus_mat_a @ A2 @ B3 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_a @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_a @ B3 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_add_mat(1)
thf(fact_612_index__add__mat_I1_J,axiom,
! [I: nat,B3: mat_nat,J: nat,A2: mat_nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ B3 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ B3 ) )
=> ( ( index_mat_nat @ ( plus_plus_mat_nat @ A2 @ B3 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_nat @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_nat @ B3 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_add_mat(1)
thf(fact_613_index__add__mat_I1_J,axiom,
! [I: nat,B3: mat_real,J: nat,A2: mat_real] :
( ( ord_less_nat @ I @ ( dim_row_real @ B3 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_real @ B3 ) )
=> ( ( index_mat_real @ ( plus_plus_mat_real @ A2 @ B3 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_real @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_real @ B3 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_add_mat(1)
thf(fact_614_index__add__mat_I1_J,axiom,
! [I: nat,B3: mat_int,J: nat,A2: mat_int] :
( ( ord_less_nat @ I @ ( dim_row_int @ B3 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ B3 ) )
=> ( ( index_mat_int @ ( plus_plus_mat_int @ A2 @ B3 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_int @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) @ ( index_mat_int @ B3 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_add_mat(1)
thf(fact_615_eq__matI,axiom,
! [B3: mat_a,A2: mat_a] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( dim_row_a @ B3 ) )
=> ( ( ord_less_nat @ J2 @ ( dim_col_a @ B3 ) )
=> ( ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= ( index_mat_a @ B3 @ ( product_Pair_nat_nat @ I2 @ J2 ) ) ) ) )
=> ( ( ( dim_row_a @ A2 )
= ( dim_row_a @ B3 ) )
=> ( ( ( dim_col_a @ A2 )
= ( dim_col_a @ B3 ) )
=> ( A2 = B3 ) ) ) ) ).
% eq_matI
thf(fact_616_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_617_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_618_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_619_less__eq__mat__def,axiom,
( ord_less_eq_mat_nat
= ( ^ [A6: mat_nat,B6: mat_nat] :
( ( ( dim_row_nat @ A6 )
= ( dim_row_nat @ B6 ) )
& ( ( dim_col_nat @ A6 )
= ( dim_col_nat @ B6 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_nat @ B6 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_nat @ B6 ) )
=> ( ord_less_eq_nat @ ( index_mat_nat @ A6 @ ( product_Pair_nat_nat @ I4 @ J3 ) ) @ ( index_mat_nat @ B6 @ ( product_Pair_nat_nat @ I4 @ J3 ) ) ) ) ) ) ) ) ).
% less_eq_mat_def
thf(fact_620_less__eq__mat__def,axiom,
( ord_less_eq_mat_real
= ( ^ [A6: mat_real,B6: mat_real] :
( ( ( dim_row_real @ A6 )
= ( dim_row_real @ B6 ) )
& ( ( dim_col_real @ A6 )
= ( dim_col_real @ B6 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_real @ B6 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_real @ B6 ) )
=> ( ord_less_eq_real @ ( index_mat_real @ A6 @ ( product_Pair_nat_nat @ I4 @ J3 ) ) @ ( index_mat_real @ B6 @ ( product_Pair_nat_nat @ I4 @ J3 ) ) ) ) ) ) ) ) ).
% less_eq_mat_def
thf(fact_621_less__eq__mat__def,axiom,
( ord_less_eq_mat_int
= ( ^ [A6: mat_int,B6: mat_int] :
( ( ( dim_row_int @ A6 )
= ( dim_row_int @ B6 ) )
& ( ( dim_col_int @ A6 )
= ( dim_col_int @ B6 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_int @ B6 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_int @ B6 ) )
=> ( ord_less_eq_int @ ( index_mat_int @ A6 @ ( product_Pair_nat_nat @ I4 @ J3 ) ) @ ( index_mat_int @ B6 @ ( product_Pair_nat_nat @ I4 @ J3 ) ) ) ) ) ) ) ) ).
% less_eq_mat_def
thf(fact_622_mat__eq__iff,axiom,
( ( ^ [Y5: mat_a,Z: mat_a] : ( Y5 = Z ) )
= ( ^ [X4: mat_a,Y6: mat_a] :
( ( ( dim_row_a @ X4 )
= ( dim_row_a @ Y6 ) )
& ( ( dim_col_a @ X4 )
= ( dim_col_a @ Y6 ) )
& ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_a @ Y6 ) )
=> ( ( ord_less_nat @ J3 @ ( dim_col_a @ Y6 ) )
=> ( ( index_mat_a @ X4 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= ( index_mat_a @ Y6 @ ( product_Pair_nat_nat @ I4 @ J3 ) ) ) ) ) ) ) ) ).
% mat_eq_iff
thf(fact_623_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_624_ideal_Oscale__one,axiom,
! [X: int] :
( ( times_times_int @ one_one_int @ X )
= X ) ).
% ideal.scale_one
thf(fact_625_ideal_Oscale__one,axiom,
! [X: real] :
( ( times_times_real @ one_one_real @ X )
= X ) ).
% ideal.scale_one
thf(fact_626_field__le__mult__one__interval,axiom,
! [X: real,Y: real] :
( ! [Z2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ Z2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ Y ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_627_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_628_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_629_ideal_Oscale__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% ideal.scale_zero_right
thf(fact_630_ideal_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% ideal.scale_zero_right
thf(fact_631_ideal_Oscale__zero__left,axiom,
! [X: int] :
( ( times_times_int @ zero_zero_int @ X )
= zero_zero_int ) ).
% ideal.scale_zero_left
thf(fact_632_ideal_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% ideal.scale_zero_left
thf(fact_633_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_634_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_635_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_636_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_637_index__mult__mat_I2_J,axiom,
! [A2: mat_a,B3: mat_a] :
( ( dim_row_a @ ( times_times_mat_a @ A2 @ B3 ) )
= ( dim_row_a @ A2 ) ) ).
% index_mult_mat(2)
thf(fact_638_index__mult__mat_I3_J,axiom,
! [A2: mat_a,B3: mat_a] :
( ( dim_col_a @ ( times_times_mat_a @ A2 @ B3 ) )
= ( dim_col_a @ B3 ) ) ).
% index_mult_mat(3)
thf(fact_639_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_640_index__add__mat_I2_J,axiom,
! [A2: mat_a,B3: mat_a] :
( ( dim_row_a @ ( plus_plus_mat_a @ A2 @ B3 ) )
= ( dim_row_a @ B3 ) ) ).
% index_add_mat(2)
thf(fact_641_index__add__mat_I3_J,axiom,
! [A2: mat_a,B3: mat_a] :
( ( dim_col_a @ ( plus_plus_mat_a @ A2 @ B3 ) )
= ( dim_col_a @ B3 ) ) ).
% index_add_mat(3)
thf(fact_642_add__mult__distrib,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 ) ) ) ).
% add_mult_distrib
thf(fact_643_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_644_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_645_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_646_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_647_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_648_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_649_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_650_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_651_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_652_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_653_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_654_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_655_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_656_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_657_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_658_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_659_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_660_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_661_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_662_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_663_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_664_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_665_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_666_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_667_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_668_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_669_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X5 ) ).
% linordered_field_no_lb
thf(fact_670_ideal_Oscale__left__commute,axiom,
! [A: int,B: int,X: int] :
( ( times_times_int @ A @ ( times_times_int @ B @ X ) )
= ( times_times_int @ B @ ( times_times_int @ A @ X ) ) ) ).
% ideal.scale_left_commute
thf(fact_671_ideal_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 ) ) ) ).
% ideal.scale_left_commute
thf(fact_672_ideal_Oscale__scale,axiom,
! [A: int,B: int,X: int] :
( ( times_times_int @ A @ ( times_times_int @ B @ X ) )
= ( times_times_int @ ( times_times_int @ A @ B ) @ X ) ) ).
% ideal.scale_scale
thf(fact_673_ideal_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 ) ) ).
% ideal.scale_scale
thf(fact_674_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_675_ideal_Oscale__right__distrib,axiom,
! [A: int,X: int,Y: int] :
( ( times_times_int @ A @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( times_times_int @ A @ X ) @ ( times_times_int @ A @ Y ) ) ) ).
% ideal.scale_right_distrib
thf(fact_676_ideal_Oscale__right__distrib,axiom,
! [A: real,X: real,Y: real] :
( ( times_times_real @ A @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ A @ Y ) ) ) ).
% ideal.scale_right_distrib
thf(fact_677_ideal_Oscale__left__distrib,axiom,
! [A: int,B: int,X: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ X )
= ( plus_plus_int @ ( times_times_int @ A @ X ) @ ( times_times_int @ B @ X ) ) ) ).
% ideal.scale_left_distrib
thf(fact_678_ideal_Oscale__left__distrib,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 ) ) ) ).
% ideal.scale_left_distrib
thf(fact_679_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_680_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_681_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_682_field__le__epsilon,axiom,
! [X: real,Y: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_683_sum__squares__le__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_684_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_685_sum__squares__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_686_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_687_index__update__mat1,axiom,
! [I: nat,A2: mat_a,J: nat,A: a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( index_mat_a @ ( update_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= A ) ) ) ).
% index_update_mat1
thf(fact_688_index__update__mat2,axiom,
! [I5: nat,A2: mat_a,J4: nat,Ij: product_prod_nat_nat,A: a] :
( ( ord_less_nat @ I5 @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J4 @ ( dim_col_a @ A2 ) )
=> ( ( ( product_Pair_nat_nat @ I5 @ J4 )
!= Ij )
=> ( ( index_mat_a @ ( update_mat_a @ A2 @ Ij @ A ) @ ( product_Pair_nat_nat @ I5 @ J4 ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I5 @ J4 ) ) ) ) ) ) ).
% index_update_mat2
thf(fact_689_diagonal__mat__def,axiom,
( diagonal_mat_a
= ( ^ [A6: mat_a] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_a @ A6 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_a @ A6 ) )
=> ( ( I4 != J3 )
=> ( ( index_mat_a @ A6 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= zero_zero_a ) ) ) ) ) ) ).
% diagonal_mat_def
thf(fact_690_diagonal__mat__def,axiom,
( diagonal_mat_nat
= ( ^ [A6: mat_nat] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_nat @ A6 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_nat @ A6 ) )
=> ( ( I4 != J3 )
=> ( ( index_mat_nat @ A6 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= zero_zero_nat ) ) ) ) ) ) ).
% diagonal_mat_def
thf(fact_691_diagonal__mat__def,axiom,
( diagonal_mat_real
= ( ^ [A6: mat_real] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_real @ A6 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_real @ A6 ) )
=> ( ( I4 != J3 )
=> ( ( index_mat_real @ A6 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= zero_zero_real ) ) ) ) ) ) ).
% diagonal_mat_def
thf(fact_692_diagonal__mat__def,axiom,
( diagonal_mat_int
= ( ^ [A6: mat_int] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_int @ A6 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( dim_col_int @ A6 ) )
=> ( ( I4 != J3 )
=> ( ( index_mat_int @ A6 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= zero_zero_int ) ) ) ) ) ) ).
% diagonal_mat_def
thf(fact_693_mult__hom_Ohom__zero,axiom,
! [C: a] :
( ( times_times_a @ C @ zero_zero_a )
= zero_zero_a ) ).
% mult_hom.hom_zero
thf(fact_694_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_695_mult__hom_Ohom__zero,axiom,
! [C: int] :
( ( times_times_int @ C @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_696_mult__hom_Ohom__zero,axiom,
! [C: real] :
( ( times_times_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_697_add__col__to__multiple__index__change__bounds,axiom,
! [I: nat,A2: mat_a,J: nat,Low: nat,Up: nat,L: nat,A: a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( ord_less_eq_nat @ Low @ J )
=> ( ( ord_less_nat @ J @ Up )
=> ( ( ( ord_less_nat @ L @ Low )
| ( ord_less_eq_nat @ Up @ L ) )
=> ( ( ord_less_nat @ L @ ( dim_col_a @ A2 ) )
=> ( ( index_mat_a @ ( rank_A5103328671897079360iple_a @ A @ ( upt @ Low @ Up ) @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_a @ ( times_times_a @ A @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ L ) ) ) @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ) ) ).
% add_col_to_multiple_index_change_bounds
thf(fact_698_add__col__to__multiple__index__change__bounds,axiom,
! [I: nat,A2: mat_nat,J: nat,Low: nat,Up: nat,L: nat,A: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A2 ) )
=> ( ( ord_less_eq_nat @ Low @ J )
=> ( ( ord_less_nat @ J @ Up )
=> ( ( ( ord_less_nat @ L @ Low )
| ( ord_less_eq_nat @ Up @ L ) )
=> ( ( ord_less_nat @ L @ ( dim_col_nat @ A2 ) )
=> ( ( index_mat_nat @ ( rank_A7396948986359020558le_nat @ A @ ( upt @ Low @ Up ) @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ L ) ) ) @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ) ) ).
% add_col_to_multiple_index_change_bounds
thf(fact_699_add__col__to__multiple__index__change__bounds,axiom,
! [I: nat,A2: mat_int,J: nat,Low: nat,Up: nat,L: nat,A: int] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ( ord_less_eq_nat @ Low @ J )
=> ( ( ord_less_nat @ J @ Up )
=> ( ( ( ord_less_nat @ L @ Low )
| ( ord_less_eq_nat @ Up @ L ) )
=> ( ( ord_less_nat @ L @ ( dim_col_int @ A2 ) )
=> ( ( index_mat_int @ ( rank_A7394458515849970282le_int @ A @ ( upt @ Low @ Up ) @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ L ) ) ) @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ) ) ).
% add_col_to_multiple_index_change_bounds
thf(fact_700_add__col__to__multiple__index__change__bounds,axiom,
! [I: nat,A2: mat_real,J: nat,Low: nat,Up: nat,L: nat,A: real] :
( ( ord_less_nat @ I @ ( dim_row_real @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_real @ A2 ) )
=> ( ( ord_less_eq_nat @ Low @ J )
=> ( ( ord_less_nat @ J @ Up )
=> ( ( ( ord_less_nat @ L @ Low )
| ( ord_less_eq_nat @ Up @ L ) )
=> ( ( ord_less_nat @ L @ ( dim_col_real @ A2 ) )
=> ( ( index_mat_real @ ( rank_A8462809353401409514e_real @ A @ ( upt @ Low @ Up ) @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ L ) ) ) @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ) ) ).
% add_col_to_multiple_index_change_bounds
thf(fact_701_add__col__to__multiple__dim_I1_J,axiom,
! [A: a,Ks: list_nat,L: nat,A2: mat_a] :
( ( dim_row_a @ ( rank_A5103328671897079360iple_a @ A @ Ks @ L @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% add_col_to_multiple_dim(1)
thf(fact_702_add__col__to__multiple__dim_I2_J,axiom,
! [A: a,Ks: list_nat,L: nat,A2: mat_a] :
( ( dim_col_a @ ( rank_A5103328671897079360iple_a @ A @ Ks @ L @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% add_col_to_multiple_dim(2)
thf(fact_703_dim__update__mat_I1_J,axiom,
! [A2: mat_a,Ij: product_prod_nat_nat,A: a] :
( ( dim_row_a @ ( update_mat_a @ A2 @ Ij @ A ) )
= ( dim_row_a @ A2 ) ) ).
% dim_update_mat(1)
thf(fact_704_dim__update__mat_I2_J,axiom,
! [A2: mat_a,Ij: product_prod_nat_nat,A: a] :
( ( dim_col_a @ ( update_mat_a @ A2 @ Ij @ A ) )
= ( dim_col_a @ A2 ) ) ).
% dim_update_mat(2)
thf(fact_705_gcd_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [A5: nat,B5: nat] :
( X
!= ( product_Pair_nat_nat @ A5 @ B5 ) ) ).
% gcd.cases
thf(fact_706_mult__hom_Ohom__add,axiom,
! [C: a,X: a,Y: a] :
( ( times_times_a @ C @ ( plus_plus_a @ X @ Y ) )
= ( plus_plus_a @ ( times_times_a @ C @ X ) @ ( times_times_a @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_707_mult__hom_Ohom__add,axiom,
! [C: nat,X: nat,Y: nat] :
( ( times_times_nat @ C @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_708_mult__hom_Ohom__add,axiom,
! [C: int,X: int,Y: int] :
( ( times_times_int @ C @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( times_times_int @ C @ X ) @ ( times_times_int @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_709_mult__hom_Ohom__add,axiom,
! [C: real,X: real,Y: real] :
( ( times_times_real @ C @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ C @ X ) @ ( times_times_real @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_710_mult__hom_Ohom__add__eq__zero,axiom,
! [X: a,Y: a,C: a] :
( ( ( plus_plus_a @ X @ Y )
= zero_zero_a )
=> ( ( plus_plus_a @ ( times_times_a @ C @ X ) @ ( times_times_a @ C @ Y ) )
= zero_zero_a ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_711_mult__hom_Ohom__add__eq__zero,axiom,
! [X: nat,Y: nat,C: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_712_mult__hom_Ohom__add__eq__zero,axiom,
! [X: int,Y: int,C: int] :
( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
=> ( ( plus_plus_int @ ( times_times_int @ C @ X ) @ ( times_times_int @ C @ Y ) )
= zero_zero_int ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_713_mult__hom_Ohom__add__eq__zero,axiom,
! [X: real,Y: real,C: real] :
( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
=> ( ( plus_plus_real @ ( times_times_real @ C @ X ) @ ( times_times_real @ C @ Y ) )
= zero_zero_real ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_714_add__col__to__multiple__index__unchanged__bound,axiom,
! [I: nat,A2: mat_a,J: nat,Low: nat,Up: nat,A: a,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( ord_less_nat @ J @ Low )
=> ( ( ord_less_eq_nat @ Up @ J )
=> ( ( index_mat_a @ ( rank_A5103328671897079360iple_a @ A @ ( upt @ Low @ Up ) @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% add_col_to_multiple_index_unchanged_bound
thf(fact_715_mult__le__cancel__iff2,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ ( times_times_real @ Z3 @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_716_mult__le__cancel__iff2,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z3 @ X ) @ ( times_times_int @ Z3 @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_717_mult__le__cancel__iff1,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ Z3 ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_718_mult__le__cancel__iff1,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_719_mult__less__iff1,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ Z3 ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_720_mult__less__iff1,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_721_add__multiple__cols__index__unchanged,axiom,
! [I: nat,A2: mat_a,J: nat,K: nat,A: a,Ls: list_nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( K != J )
=> ( ( index_mat_a @ ( rank_A1622336257019553452cols_a @ A @ K @ Ls @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% add_multiple_cols_index_unchanged
thf(fact_722_add__multiple__rows__index__unchanged,axiom,
! [I: nat,A2: mat_a,J: nat,K: nat,A: a,Ls: list_nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( K != I )
=> ( ( index_mat_a @ ( rank_A5086152289722150470rows_a @ A @ K @ Ls @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% add_multiple_rows_index_unchanged
thf(fact_723_upper__triangularI,axiom,
! [A2: mat_a] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ J2 @ I2 )
=> ( ( ord_less_nat @ I2 @ ( dim_row_a @ A2 ) )
=> ( ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= zero_zero_a ) ) )
=> ( upper_triangular_a @ A2 ) ) ).
% upper_triangularI
thf(fact_724_upper__triangularI,axiom,
! [A2: mat_nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ J2 @ I2 )
=> ( ( ord_less_nat @ I2 @ ( dim_row_nat @ A2 ) )
=> ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= zero_zero_nat ) ) )
=> ( upper_triangular_nat @ A2 ) ) ).
% upper_triangularI
thf(fact_725_upper__triangularI,axiom,
! [A2: mat_real] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ J2 @ I2 )
=> ( ( ord_less_nat @ I2 @ ( dim_row_real @ A2 ) )
=> ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= zero_zero_real ) ) )
=> ( upper_8570057991637822137r_real @ A2 ) ) ).
% upper_triangularI
thf(fact_726_upper__triangularI,axiom,
! [A2: mat_int] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ J2 @ I2 )
=> ( ( ord_less_nat @ I2 @ ( dim_row_int @ A2 ) )
=> ( ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I2 @ J2 ) )
= zero_zero_int ) ) )
=> ( upper_triangular_int @ A2 ) ) ).
% upper_triangularI
thf(fact_727_index__mat__addrow__basic_I3_J,axiom,
! [I: nat,A2: mat_a,J: nat,K: nat,A: a,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( K != I )
=> ( ( index_mat_a @ ( gauss_3441994962245461172_gen_a @ plus_plus_a @ times_times_a @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addrow_basic(3)
thf(fact_728_index__mat__addrow__basic_I3_J,axiom,
! [I: nat,A2: mat_nat,J: nat,K: nat,A: nat,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A2 ) )
=> ( ( K != I )
=> ( ( index_mat_nat @ ( gauss_8885043348566651034en_nat @ plus_plus_nat @ times_times_nat @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addrow_basic(3)
thf(fact_729_index__mat__addrow__basic_I3_J,axiom,
! [I: nat,A2: mat_int,J: nat,K: nat,A: int,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ( K != I )
=> ( ( index_mat_int @ ( gauss_8882552878057600758en_int @ plus_plus_int @ times_times_int @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addrow_basic(3)
thf(fact_730_index__mat__addrow__basic_I3_J,axiom,
! [I: nat,A2: mat_real,J: nat,K: nat,A: real,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_real @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_real @ A2 ) )
=> ( ( K != I )
=> ( ( index_mat_real @ ( gauss_4246877906280926838n_real @ plus_plus_real @ times_times_real @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addrow_basic(3)
thf(fact_731_add__multiple__rows__dim_I1_J,axiom,
! [A: a,K: nat,Ls: list_nat,A2: mat_a] :
( ( dim_row_a @ ( rank_A5086152289722150470rows_a @ A @ K @ Ls @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% add_multiple_rows_dim(1)
thf(fact_732_add__multiple__cols__dim_I1_J,axiom,
! [A: a,K: nat,Ls: list_nat,A2: mat_a] :
( ( dim_row_a @ ( rank_A1622336257019553452cols_a @ A @ K @ Ls @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% add_multiple_cols_dim(1)
thf(fact_733_add__multiple__cols__dim_I2_J,axiom,
! [A: a,K: nat,Ls: list_nat,A2: mat_a] :
( ( dim_col_a @ ( rank_A1622336257019553452cols_a @ A @ K @ Ls @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% add_multiple_cols_dim(2)
thf(fact_734_add__multiple__rows__dim_I2_J,axiom,
! [A: a,K: nat,Ls: list_nat,A2: mat_a] :
( ( dim_col_a @ ( rank_A5086152289722150470rows_a @ A @ K @ Ls @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% add_multiple_rows_dim(2)
thf(fact_735_index__mat__addrow__basic_I4_J,axiom,
! [A: a,K: nat,L: nat,A2: mat_a] :
( ( dim_row_a @ ( gauss_3441994962245461172_gen_a @ plus_plus_a @ times_times_a @ A @ K @ L @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% index_mat_addrow_basic(4)
thf(fact_736_index__mat__addrow__basic_I4_J,axiom,
! [A: nat,K: nat,L: nat,A2: mat_nat] :
( ( dim_row_nat @ ( gauss_8885043348566651034en_nat @ plus_plus_nat @ times_times_nat @ A @ K @ L @ A2 ) )
= ( dim_row_nat @ A2 ) ) ).
% index_mat_addrow_basic(4)
thf(fact_737_index__mat__addrow__basic_I4_J,axiom,
! [A: int,K: nat,L: nat,A2: mat_int] :
( ( dim_row_int @ ( gauss_8882552878057600758en_int @ plus_plus_int @ times_times_int @ A @ K @ L @ A2 ) )
= ( dim_row_int @ A2 ) ) ).
% index_mat_addrow_basic(4)
thf(fact_738_index__mat__addrow__basic_I4_J,axiom,
! [A: real,K: nat,L: nat,A2: mat_real] :
( ( dim_row_real @ ( gauss_4246877906280926838n_real @ plus_plus_real @ times_times_real @ A @ K @ L @ A2 ) )
= ( dim_row_real @ A2 ) ) ).
% index_mat_addrow_basic(4)
thf(fact_739_index__mat__addrow__basic_I5_J,axiom,
! [A: a,K: nat,L: nat,A2: mat_a] :
( ( dim_col_a @ ( gauss_3441994962245461172_gen_a @ plus_plus_a @ times_times_a @ A @ K @ L @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% index_mat_addrow_basic(5)
thf(fact_740_index__mat__addrow__basic_I5_J,axiom,
! [A: nat,K: nat,L: nat,A2: mat_nat] :
( ( dim_col_nat @ ( gauss_8885043348566651034en_nat @ plus_plus_nat @ times_times_nat @ A @ K @ L @ A2 ) )
= ( dim_col_nat @ A2 ) ) ).
% index_mat_addrow_basic(5)
thf(fact_741_index__mat__addrow__basic_I5_J,axiom,
! [A: int,K: nat,L: nat,A2: mat_int] :
( ( dim_col_int @ ( gauss_8882552878057600758en_int @ plus_plus_int @ times_times_int @ A @ K @ L @ A2 ) )
= ( dim_col_int @ A2 ) ) ).
% index_mat_addrow_basic(5)
thf(fact_742_index__mat__addrow__basic_I5_J,axiom,
! [A: real,K: nat,L: nat,A2: mat_real] :
( ( dim_col_real @ ( gauss_4246877906280926838n_real @ plus_plus_real @ times_times_real @ A @ K @ L @ A2 ) )
= ( dim_col_real @ A2 ) ) ).
% index_mat_addrow_basic(5)
thf(fact_743_index__mat__addrow__basic_I1_J,axiom,
! [I: nat,A2: mat_a,J: nat,K: nat,A: a,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( ( K = I )
=> ( ( index_mat_a @ ( gauss_3441994962245461172_gen_a @ plus_plus_a @ times_times_a @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_a @ ( times_times_a @ A @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) )
& ( ( K != I )
=> ( ( index_mat_a @ ( gauss_3441994962245461172_gen_a @ plus_plus_a @ times_times_a @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% index_mat_addrow_basic(1)
thf(fact_744_index__mat__addrow__basic_I1_J,axiom,
! [I: nat,A2: mat_nat,J: nat,K: nat,A: nat,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A2 ) )
=> ( ( ( K = I )
=> ( ( index_mat_nat @ ( gauss_8885043348566651034en_nat @ plus_plus_nat @ times_times_nat @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) )
& ( ( K != I )
=> ( ( index_mat_nat @ ( gauss_8885043348566651034en_nat @ plus_plus_nat @ times_times_nat @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% index_mat_addrow_basic(1)
thf(fact_745_index__mat__addrow__basic_I1_J,axiom,
! [I: nat,A2: mat_int,J: nat,K: nat,A: int,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ( ( K = I )
=> ( ( index_mat_int @ ( gauss_8882552878057600758en_int @ plus_plus_int @ times_times_int @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) )
& ( ( K != I )
=> ( ( index_mat_int @ ( gauss_8882552878057600758en_int @ plus_plus_int @ times_times_int @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% index_mat_addrow_basic(1)
thf(fact_746_index__mat__addrow__basic_I1_J,axiom,
! [I: nat,A2: mat_real,J: nat,K: nat,A: real,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_real @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_real @ A2 ) )
=> ( ( ( K = I )
=> ( ( index_mat_real @ ( gauss_4246877906280926838n_real @ plus_plus_real @ times_times_real @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) )
& ( ( K != I )
=> ( ( index_mat_real @ ( gauss_4246877906280926838n_real @ plus_plus_real @ times_times_real @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% index_mat_addrow_basic(1)
thf(fact_747_index__mat__addrow__basic_I2_J,axiom,
! [I: nat,A2: mat_a,J: nat,A: a,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( index_mat_a @ ( gauss_3441994962245461172_gen_a @ plus_plus_a @ times_times_a @ A @ I @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_a @ ( times_times_a @ A @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addrow_basic(2)
thf(fact_748_index__mat__addrow__basic_I2_J,axiom,
! [I: nat,A2: mat_nat,J: nat,A: nat,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A2 ) )
=> ( ( index_mat_nat @ ( gauss_8885043348566651034en_nat @ plus_plus_nat @ times_times_nat @ A @ I @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addrow_basic(2)
thf(fact_749_index__mat__addrow__basic_I2_J,axiom,
! [I: nat,A2: mat_int,J: nat,A: int,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ( index_mat_int @ ( gauss_8882552878057600758en_int @ plus_plus_int @ times_times_int @ A @ I @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addrow_basic(2)
thf(fact_750_index__mat__addrow__basic_I2_J,axiom,
! [I: nat,A2: mat_real,J: nat,A: real,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_real @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_real @ A2 ) )
=> ( ( index_mat_real @ ( gauss_4246877906280926838n_real @ plus_plus_real @ times_times_real @ A @ I @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addrow_basic(2)
thf(fact_751_upper__triangular__def,axiom,
( upper_triangular_a
= ( ^ [A6: mat_a] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_a @ A6 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ I4 )
=> ( ( index_mat_a @ A6 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= zero_zero_a ) ) ) ) ) ).
% upper_triangular_def
thf(fact_752_upper__triangular__def,axiom,
( upper_triangular_nat
= ( ^ [A6: mat_nat] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_nat @ A6 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ I4 )
=> ( ( index_mat_nat @ A6 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= zero_zero_nat ) ) ) ) ) ).
% upper_triangular_def
thf(fact_753_upper__triangular__def,axiom,
( upper_8570057991637822137r_real
= ( ^ [A6: mat_real] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_real @ A6 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ I4 )
=> ( ( index_mat_real @ A6 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= zero_zero_real ) ) ) ) ) ).
% upper_triangular_def
thf(fact_754_upper__triangular__def,axiom,
( upper_triangular_int
= ( ^ [A6: mat_int] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( dim_row_int @ A6 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ I4 )
=> ( ( index_mat_int @ A6 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= zero_zero_int ) ) ) ) ) ).
% upper_triangular_def
thf(fact_755_upper__triangularD,axiom,
! [A2: mat_a,J: nat,I: nat] :
( ( upper_triangular_a @ A2 )
=> ( ( ord_less_nat @ J @ I )
=> ( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_a ) ) ) ) ).
% upper_triangularD
thf(fact_756_upper__triangularD,axiom,
! [A2: mat_nat,J: nat,I: nat] :
( ( upper_triangular_nat @ A2 )
=> ( ( ord_less_nat @ J @ I )
=> ( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_nat ) ) ) ) ).
% upper_triangularD
thf(fact_757_upper__triangularD,axiom,
! [A2: mat_real,J: nat,I: nat] :
( ( upper_8570057991637822137r_real @ A2 )
=> ( ( ord_less_nat @ J @ I )
=> ( ( ord_less_nat @ I @ ( dim_row_real @ A2 ) )
=> ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_real ) ) ) ) ).
% upper_triangularD
thf(fact_758_upper__triangularD,axiom,
! [A2: mat_int,J: nat,I: nat] :
( ( upper_triangular_int @ A2 )
=> ( ( ord_less_nat @ J @ I )
=> ( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ).
% upper_triangularD
thf(fact_759_index__mat__addrow_I1_J,axiom,
! [I: nat,A2: mat_a,J: nat,K: nat,Ad: a > a > a,Mul: a > a > a,A: a,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( ( K = I )
=> ( ( index_mat_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( Ad @ ( Mul @ A @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) )
& ( ( K != I )
=> ( ( index_mat_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% index_mat_addrow(1)
thf(fact_760_index__mat__addrow_I2_J,axiom,
! [I: nat,A2: mat_a,J: nat,Ad: a > a > a,Mul: a > a > a,A: a,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( index_mat_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A @ I @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( Ad @ ( Mul @ A @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addrow(2)
thf(fact_761_index__mat__addrow_I3_J,axiom,
! [I: nat,A2: mat_a,J: nat,K: nat,Ad: a > a > a,Mul: a > a > a,A: a,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( K != I )
=> ( ( index_mat_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addrow(3)
thf(fact_762_index__mat__addrow_I5_J,axiom,
! [Ad: a > a > a,Mul: a > a > a,A: a,K: nat,L: nat,A2: mat_a] :
( ( dim_col_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A @ K @ L @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% index_mat_addrow(5)
thf(fact_763_index__mat__addrow_I4_J,axiom,
! [Ad: a > a > a,Mul: a > a > a,A: a,K: nat,L: nat,A2: mat_a] :
( ( dim_row_a @ ( gauss_3441994962245461172_gen_a @ Ad @ Mul @ A @ K @ L @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% index_mat_addrow(4)
thf(fact_764_pivot__positions__main__gen_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [I2: nat,J2: nat] :
( X
!= ( product_Pair_nat_nat @ I2 @ J2 ) ) ).
% pivot_positions_main_gen.cases
thf(fact_765_More__Matrix_Oarith,axiom,
! [D: nat,A: nat,C: nat,B: nat] :
( ( ord_less_nat @ D @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ D ) @ C ) @ ( times_times_nat @ A @ B ) ) ) ) ).
% More_Matrix.arith
thf(fact_766_index__mat__multrow__mat_I1_J,axiom,
! [I: nat,N: nat,J: nat,K: nat,A: a] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ( ( K = I )
& ( K = J ) )
=> ( ( index_mat_a @ ( gauss_5015385051186949877_mat_a @ N @ K @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= A ) )
& ( ~ ( ( K = I )
& ( K = J ) )
=> ( ( ( I = J )
=> ( ( index_mat_a @ ( gauss_5015385051186949877_mat_a @ N @ K @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_a ) )
& ( ( I != J )
=> ( ( index_mat_a @ ( gauss_5015385051186949877_mat_a @ N @ K @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_a ) ) ) ) ) ) ) ).
% index_mat_multrow_mat(1)
thf(fact_767_index__mat__multrow__mat_I1_J,axiom,
! [I: nat,N: nat,J: nat,K: nat,A: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ( ( K = I )
& ( K = J ) )
=> ( ( index_mat_nat @ ( gauss_3195076542185637913at_nat @ N @ K @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= A ) )
& ( ~ ( ( K = I )
& ( K = J ) )
=> ( ( ( I = J )
=> ( ( index_mat_nat @ ( gauss_3195076542185637913at_nat @ N @ K @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_nat ) )
& ( ( I != J )
=> ( ( index_mat_nat @ ( gauss_3195076542185637913at_nat @ N @ K @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_nat ) ) ) ) ) ) ) ).
% index_mat_multrow_mat(1)
thf(fact_768_index__mat__multrow__mat_I1_J,axiom,
! [I: nat,N: nat,J: nat,K: nat,A: real] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ( ( K = I )
& ( K = J ) )
=> ( ( index_mat_real @ ( gauss_7241202418770761333t_real @ N @ K @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= A ) )
& ( ~ ( ( K = I )
& ( K = J ) )
=> ( ( ( I = J )
=> ( ( index_mat_real @ ( gauss_7241202418770761333t_real @ N @ K @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_real ) )
& ( ( I != J )
=> ( ( index_mat_real @ ( gauss_7241202418770761333t_real @ N @ K @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_real ) ) ) ) ) ) ) ).
% index_mat_multrow_mat(1)
thf(fact_769_index__mat__multrow__mat_I1_J,axiom,
! [I: nat,N: nat,J: nat,K: nat,A: int] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ( ( K = I )
& ( K = J ) )
=> ( ( index_mat_int @ ( gauss_3192586071676587637at_int @ N @ K @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= A ) )
& ( ~ ( ( K = I )
& ( K = J ) )
=> ( ( ( I = J )
=> ( ( index_mat_int @ ( gauss_3192586071676587637at_int @ N @ K @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int ) )
& ( ( I != J )
=> ( ( index_mat_int @ ( gauss_3192586071676587637at_int @ N @ K @ A ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ) ) ) ).
% index_mat_multrow_mat(1)
thf(fact_770_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_771_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_772_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_773_index__mat__multrow__mat_I2_J,axiom,
! [N: nat,K: nat,A: a] :
( ( dim_row_a @ ( gauss_5015385051186949877_mat_a @ N @ K @ A ) )
= N ) ).
% index_mat_multrow_mat(2)
thf(fact_774_index__mat__multrow__mat_I3_J,axiom,
! [N: nat,K: nat,A: a] :
( ( dim_col_a @ ( gauss_5015385051186949877_mat_a @ N @ K @ A ) )
= N ) ).
% index_mat_multrow_mat(3)
thf(fact_775_helper,axiom,
! [X: int,Y: int,Z3: int] :
( ( times_times_int @ X @ ( times_times_int @ Y @ Z3 ) )
= ( times_times_int @ ( times_times_int @ Y @ X ) @ Z3 ) ) ).
% helper
thf(fact_776_helper,axiom,
! [X: real,Y: real,Z3: real] :
( ( times_times_real @ X @ ( times_times_real @ Y @ Z3 ) )
= ( times_times_real @ ( times_times_real @ Y @ X ) @ Z3 ) ) ).
% helper
thf(fact_777_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_778_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_779_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_780_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc4471711990508489141at_nat] :
~ ! [F2: nat > nat > nat,A5: nat,B5: nat,Acc: nat] :
( X
!= ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_781_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_782_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_783_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_784_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_785_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_786_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_787_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_788_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_789_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_790_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_791_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_792_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_793_kronecker__inverse__index,axiom,
! [R: nat,A2: mat_int,S: nat,V: nat,B3: mat_int,W: nat] :
( ( ord_less_nat @ R @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ S @ ( dim_col_int @ A2 ) )
=> ( ( ord_less_nat @ V @ ( dim_row_int @ B3 ) )
=> ( ( ord_less_nat @ W @ ( dim_col_int @ B3 ) )
=> ( ( index_mat_int @ ( more_k9102366682204111370ct_int @ A2 @ B3 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( dim_row_int @ B3 ) @ R ) @ V ) @ ( plus_plus_nat @ ( times_times_nat @ ( dim_col_int @ B3 ) @ S ) @ W ) ) )
= ( times_times_int @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ R @ S ) ) @ ( index_mat_int @ B3 @ ( product_Pair_nat_nat @ V @ W ) ) ) ) ) ) ) ) ).
% kronecker_inverse_index
thf(fact_794_kronecker__inverse__index,axiom,
! [R: nat,A2: mat_real,S: nat,V: nat,B3: mat_real,W: nat] :
( ( ord_less_nat @ R @ ( dim_row_real @ A2 ) )
=> ( ( ord_less_nat @ S @ ( dim_col_real @ A2 ) )
=> ( ( ord_less_nat @ V @ ( dim_row_real @ B3 ) )
=> ( ( ord_less_nat @ W @ ( dim_col_real @ B3 ) )
=> ( ( index_mat_real @ ( more_k7431981406657007498t_real @ A2 @ B3 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( dim_row_real @ B3 ) @ R ) @ V ) @ ( plus_plus_nat @ ( times_times_nat @ ( dim_col_real @ B3 ) @ S ) @ W ) ) )
= ( times_times_real @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ R @ S ) ) @ ( index_mat_real @ B3 @ ( product_Pair_nat_nat @ V @ W ) ) ) ) ) ) ) ) ).
% kronecker_inverse_index
thf(fact_795_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_796_add__scale__eq__noteq,axiom,
! [R: int,A: int,B: int,C: int,D: int] :
( ( R != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_797_add__scale__eq__noteq,axiom,
! [R: real,A: real,B: real,C: real,D: real] :
( ( R != zero_zero_real )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_798_less__eq__fract__respect,axiom,
! [B: real,B2: real,D: real,D2: real,A: real,A3: real,C: real,C4: real] :
( ( B != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( D != zero_zero_real )
=> ( ( D2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ B2 )
= ( times_times_real @ A3 @ B ) )
=> ( ( ( times_times_real @ C @ D2 )
= ( times_times_real @ C4 @ D ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ D ) ) @ ( times_times_real @ ( times_times_real @ C @ B ) @ ( times_times_real @ B @ D ) ) )
= ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A3 @ D2 ) @ ( times_times_real @ B2 @ D2 ) ) @ ( times_times_real @ ( times_times_real @ C4 @ B2 ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_799_less__eq__fract__respect,axiom,
! [B: int,B2: int,D: int,D2: int,A: int,A3: int,C: int,C4: int] :
( ( B != zero_zero_int )
=> ( ( B2 != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( D2 != zero_zero_int )
=> ( ( ( times_times_int @ A @ B2 )
= ( times_times_int @ A3 @ B ) )
=> ( ( ( times_times_int @ C @ D2 )
= ( times_times_int @ C4 @ D ) )
=> ( ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) )
= ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B2 @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C4 @ B2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_800_all__ones__mat__index,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ ( matrix7809018878362141592_mat_a @ N ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ ( matrix7809018878362141592_mat_a @ N ) ) )
=> ( ( index_mat_a @ ( matrix7809018878362141592_mat_a @ N ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_a ) ) ) ).
% all_ones_mat_index
thf(fact_801_all__ones__mat__index,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ ( matrix7809018878362141592_mat_a @ N ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ ( matrix7809018878362141592_mat_a @ N ) ) )
=> ( ( index_mat_nat @ ( matrix8488175591170039990at_nat @ N ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_nat ) ) ) ).
% all_ones_mat_index
thf(fact_802_all__ones__mat__index,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ ( matrix7809018878362141592_mat_a @ N ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ ( matrix7809018878362141592_mat_a @ N ) ) )
=> ( ( index_mat_real @ ( matrix5418915509738909842t_real @ N ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_real ) ) ) ).
% all_ones_mat_index
thf(fact_803_all__ones__mat__index,axiom,
! [I: nat,N: nat,J: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ ( matrix7809018878362141592_mat_a @ N ) ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ ( matrix7809018878362141592_mat_a @ N ) ) )
=> ( ( index_mat_int @ ( matrix8485685120660989714at_int @ N ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int ) ) ) ).
% all_ones_mat_index
thf(fact_804_index__mat__multcol_I1_J,axiom,
! [I: nat,A2: mat_a,J: nat,K: nat,A: a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( ( K = J )
=> ( ( index_mat_a @ ( column_mat_multcol_a @ K @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_a @ A @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) )
& ( ( K != J )
=> ( ( index_mat_a @ ( column_mat_multcol_a @ K @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% index_mat_multcol(1)
thf(fact_805_index__mat__multcol_I1_J,axiom,
! [I: nat,A2: mat_nat,J: nat,K: nat,A: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A2 ) )
=> ( ( ( K = J )
=> ( ( index_mat_nat @ ( column384608550491945071ol_nat @ K @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_nat @ A @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) )
& ( ( K != J )
=> ( ( index_mat_nat @ ( column384608550491945071ol_nat @ K @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% index_mat_multcol(1)
thf(fact_806_index__mat__multcol_I1_J,axiom,
! [I: nat,A2: mat_int,J: nat,K: nat,A: int] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ( ( K = J )
=> ( ( index_mat_int @ ( column382118079982894795ol_int @ K @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_int @ A @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) )
& ( ( K != J )
=> ( ( index_mat_int @ ( column382118079982894795ol_int @ K @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% index_mat_multcol(1)
thf(fact_807_index__mat__multcol_I1_J,axiom,
! [I: nat,A2: mat_real,J: nat,K: nat,A: real] :
( ( ord_less_nat @ I @ ( dim_row_real @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_real @ A2 ) )
=> ( ( ( K = J )
=> ( ( index_mat_real @ ( column7747928533466807243l_real @ K @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_real @ A @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) )
& ( ( K != J )
=> ( ( index_mat_real @ ( column7747928533466807243l_real @ K @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% index_mat_multcol(1)
thf(fact_808_all__ones__mat__dim__row,axiom,
! [N: nat] :
( ( dim_row_a @ ( matrix7809018878362141592_mat_a @ N ) )
= N ) ).
% all_ones_mat_dim_row
thf(fact_809_all__ones__mat__dim__col,axiom,
! [N: nat] :
( ( dim_col_a @ ( matrix7809018878362141592_mat_a @ N ) )
= N ) ).
% all_ones_mat_dim_col
thf(fact_810_index__mat__multcol_I4_J,axiom,
! [K: nat,A: a,A2: mat_a] :
( ( dim_row_a @ ( column_mat_multcol_a @ K @ A @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% index_mat_multcol(4)
thf(fact_811_index__mat__multcol_I5_J,axiom,
! [K: nat,A: a,A2: mat_a] :
( ( dim_col_a @ ( column_mat_multcol_a @ K @ A @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% index_mat_multcol(5)
thf(fact_812_index__mat__multcol_I3_J,axiom,
! [I: nat,A2: mat_a,J: nat,K: nat,A: a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( K != J )
=> ( ( index_mat_a @ ( column_mat_multcol_a @ K @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_multcol(3)
thf(fact_813_index__mat__multcol_I2_J,axiom,
! [I: nat,A2: mat_a,J: nat,A: a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( index_mat_a @ ( column_mat_multcol_a @ J @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_a @ A @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_multcol(2)
thf(fact_814_index__mat__multcol_I2_J,axiom,
! [I: nat,A2: mat_nat,J: nat,A: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A2 ) )
=> ( ( index_mat_nat @ ( column384608550491945071ol_nat @ J @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_nat @ A @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_multcol(2)
thf(fact_815_index__mat__multcol_I2_J,axiom,
! [I: nat,A2: mat_int,J: nat,A: int] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ( index_mat_int @ ( column382118079982894795ol_int @ J @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_int @ A @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_multcol(2)
thf(fact_816_index__mat__multcol_I2_J,axiom,
! [I: nat,A2: mat_real,J: nat,A: real] :
( ( ord_less_nat @ I @ ( dim_row_real @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_real @ A2 ) )
=> ( ( index_mat_real @ ( column7747928533466807243l_real @ J @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_real @ A @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_multcol(2)
thf(fact_817_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_818_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_819_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_820_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_821_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_822_crossproduct__noteq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_823_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z3: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z3 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z3 ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z3 ) ) ) ).
% crossproduct_eq
thf(fact_824_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z3: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z3 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z3 ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z3 ) ) ) ).
% crossproduct_eq
thf(fact_825_crossproduct__eq,axiom,
! [W: real,Y: real,X: real,Z3: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z3 ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z3 ) @ ( times_times_real @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z3 ) ) ) ).
% crossproduct_eq
thf(fact_826_swap__cols__rows__index,axiom,
! [I: nat,A2: mat_a,J: nat,A: nat,B: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ I @ ( dim_col_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( ord_less_nat @ A @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ B @ ( dim_row_a @ A2 ) )
=> ( ( index_mat_a @ ( column5129559316938501008rows_a @ A @ B @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ ( if_nat @ ( I = A ) @ B @ ( if_nat @ ( I = B ) @ A @ I ) ) @ ( if_nat @ ( J = A ) @ B @ ( if_nat @ ( J = B ) @ A @ J ) ) ) ) ) ) ) ) ) ) ) ).
% swap_cols_rows_index
thf(fact_827_index__mat__addcol_I2_J,axiom,
! [I: nat,A2: mat_a,J: nat,A: a,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( index_mat_a @ ( column_mat_addcol_a @ A @ J @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_a @ ( times_times_a @ A @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ L ) ) ) @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addcol(2)
thf(fact_828_index__mat__addcol_I2_J,axiom,
! [I: nat,A2: mat_nat,J: nat,A: nat,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A2 ) )
=> ( ( index_mat_nat @ ( column5442440509538803650ol_nat @ A @ J @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ L ) ) ) @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addcol(2)
thf(fact_829_index__mat__addcol_I2_J,axiom,
! [I: nat,A2: mat_int,J: nat,A: int,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ( index_mat_int @ ( column5439950039029753374ol_int @ A @ J @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ L ) ) ) @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addcol(2)
thf(fact_830_index__mat__addcol_I2_J,axiom,
! [I: nat,A2: mat_real,J: nat,A: real,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_real @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_real @ A2 ) )
=> ( ( index_mat_real @ ( column5677306341442300318l_real @ A @ J @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ L ) ) ) @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addcol(2)
thf(fact_831_index__mat__addcol_I1_J,axiom,
! [I: nat,A2: mat_a,J: nat,K: nat,A: a,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( ( K = J )
=> ( ( index_mat_a @ ( column_mat_addcol_a @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_a @ ( times_times_a @ A @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ L ) ) ) @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) )
& ( ( K != J )
=> ( ( index_mat_a @ ( column_mat_addcol_a @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% index_mat_addcol(1)
thf(fact_832_index__mat__addcol_I1_J,axiom,
! [I: nat,A2: mat_nat,J: nat,K: nat,A: nat,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A2 ) )
=> ( ( ( K = J )
=> ( ( index_mat_nat @ ( column5442440509538803650ol_nat @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ L ) ) ) @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) )
& ( ( K != J )
=> ( ( index_mat_nat @ ( column5442440509538803650ol_nat @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% index_mat_addcol(1)
thf(fact_833_index__mat__addcol_I1_J,axiom,
! [I: nat,A2: mat_int,J: nat,K: nat,A: int,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ( ( K = J )
=> ( ( index_mat_int @ ( column5439950039029753374ol_int @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ L ) ) ) @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) )
& ( ( K != J )
=> ( ( index_mat_int @ ( column5439950039029753374ol_int @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% index_mat_addcol(1)
thf(fact_834_index__mat__addcol_I1_J,axiom,
! [I: nat,A2: mat_real,J: nat,K: nat,A: real,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_real @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_real @ A2 ) )
=> ( ( ( K = J )
=> ( ( index_mat_real @ ( column5677306341442300318l_real @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ L ) ) ) @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) )
& ( ( K != J )
=> ( ( index_mat_real @ ( column5677306341442300318l_real @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ).
% index_mat_addcol(1)
thf(fact_835_index__mat__swapcols_I1_J,axiom,
! [I: nat,A2: mat_a,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( ( K = J )
=> ( ( index_mat_a @ ( column2528828918332591333cols_a @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ L ) ) ) )
& ( ( K != J )
=> ( ( ( L = J )
=> ( ( index_mat_a @ ( column2528828918332591333cols_a @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ K ) ) ) )
& ( ( L != J )
=> ( ( index_mat_a @ ( column2528828918332591333cols_a @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ) ).
% index_mat_swapcols(1)
thf(fact_836_index__mat__addcol_I3_J,axiom,
! [I: nat,A2: mat_a,J: nat,K: nat,A: a,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( K != J )
=> ( ( index_mat_a @ ( column_mat_addcol_a @ A @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_mat_addcol(3)
thf(fact_837_index__mat__addcol_I4_J,axiom,
! [A: a,K: nat,L: nat,A2: mat_a] :
( ( dim_row_a @ ( column_mat_addcol_a @ A @ K @ L @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% index_mat_addcol(4)
thf(fact_838_index__mat__addcol_I5_J,axiom,
! [A: a,K: nat,L: nat,A2: mat_a] :
( ( dim_col_a @ ( column_mat_addcol_a @ A @ K @ L @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% index_mat_addcol(5)
thf(fact_839_index__mat__swapcols_I2_J,axiom,
! [K: nat,L: nat,A2: mat_a] :
( ( dim_row_a @ ( column2528828918332591333cols_a @ K @ L @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% index_mat_swapcols(2)
thf(fact_840_index__mat__swapcols_I3_J,axiom,
! [K: nat,L: nat,A2: mat_a] :
( ( dim_col_a @ ( column2528828918332591333cols_a @ K @ L @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% index_mat_swapcols(3)
thf(fact_841_swap__cols__rows__carrier_I1_J,axiom,
! [K: nat,L: nat,A2: mat_a] :
( ( dim_row_a @ ( column5129559316938501008rows_a @ K @ L @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% swap_cols_rows_carrier(1)
thf(fact_842_swap__cols__rows__carrier_I2_J,axiom,
! [K: nat,L: nat,A2: mat_a] :
( ( dim_col_a @ ( column5129559316938501008rows_a @ K @ L @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% swap_cols_rows_carrier(2)
thf(fact_843_index__mat__swaprows__mat_I1_J,axiom,
! [I: nat,N: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ( ( ( K = I )
& ( L = J ) )
| ( ( K = J )
& ( L = I ) )
| ( ( I = J )
& ( I != K )
& ( I != L ) ) )
=> ( ( index_mat_a @ ( gauss_110929411057020027_mat_a @ N @ K @ L ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_a ) )
& ( ~ ( ( ( K = I )
& ( L = J ) )
| ( ( K = J )
& ( L = I ) )
| ( ( I = J )
& ( I != K )
& ( I != L ) ) )
=> ( ( index_mat_a @ ( gauss_110929411057020027_mat_a @ N @ K @ L ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_a ) ) ) ) ) ).
% index_mat_swaprows_mat(1)
thf(fact_844_index__mat__swaprows__mat_I1_J,axiom,
! [I: nat,N: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ( ( ( K = I )
& ( L = J ) )
| ( ( K = J )
& ( L = I ) )
| ( ( I = J )
& ( I != K )
& ( I != L ) ) )
=> ( ( index_mat_nat @ ( gauss_4919907329869174035at_nat @ N @ K @ L ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_nat ) )
& ( ~ ( ( ( K = I )
& ( L = J ) )
| ( ( K = J )
& ( L = I ) )
| ( ( I = J )
& ( I != K )
& ( I != L ) ) )
=> ( ( index_mat_nat @ ( gauss_4919907329869174035at_nat @ N @ K @ L ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_nat ) ) ) ) ) ).
% index_mat_swaprows_mat(1)
thf(fact_845_index__mat__swaprows__mat_I1_J,axiom,
! [I: nat,N: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ( ( ( K = I )
& ( L = J ) )
| ( ( K = J )
& ( L = I ) )
| ( ( I = J )
& ( I != K )
& ( I != L ) ) )
=> ( ( index_mat_real @ ( gauss_1271566072679876207t_real @ N @ K @ L ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_real ) )
& ( ~ ( ( ( K = I )
& ( L = J ) )
| ( ( K = J )
& ( L = I ) )
| ( ( I = J )
& ( I != K )
& ( I != L ) ) )
=> ( ( index_mat_real @ ( gauss_1271566072679876207t_real @ N @ K @ L ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_real ) ) ) ) ) ).
% index_mat_swaprows_mat(1)
thf(fact_846_index__mat__swaprows__mat_I1_J,axiom,
! [I: nat,N: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( ( ( ( K = I )
& ( L = J ) )
| ( ( K = J )
& ( L = I ) )
| ( ( I = J )
& ( I != K )
& ( I != L ) ) )
=> ( ( index_mat_int @ ( gauss_4917416859360123759at_int @ N @ K @ L ) @ ( product_Pair_nat_nat @ I @ J ) )
= one_one_int ) )
& ( ~ ( ( ( K = I )
& ( L = J ) )
| ( ( K = J )
& ( L = I ) )
| ( ( I = J )
& ( I != K )
& ( I != L ) ) )
=> ( ( index_mat_int @ ( gauss_4917416859360123759at_int @ N @ K @ L ) @ ( product_Pair_nat_nat @ I @ J ) )
= zero_zero_int ) ) ) ) ) ).
% index_mat_swaprows_mat(1)
thf(fact_847_index__smult__mat_I1_J,axiom,
! [I: nat,A2: mat_a,J: nat,A: a] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( index_mat_a @ ( smult_mat_a @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_a @ A @ ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_smult_mat(1)
thf(fact_848_index__smult__mat_I1_J,axiom,
! [I: nat,A2: mat_nat,J: nat,A: nat] :
( ( ord_less_nat @ I @ ( dim_row_nat @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_nat @ A2 ) )
=> ( ( index_mat_nat @ ( smult_mat_nat @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_nat @ A @ ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_smult_mat(1)
thf(fact_849_index__smult__mat_I1_J,axiom,
! [I: nat,A2: mat_int,J: nat,A: int] :
( ( ord_less_nat @ I @ ( dim_row_int @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_int @ A2 ) )
=> ( ( index_mat_int @ ( smult_mat_int @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_int @ A @ ( index_mat_int @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_smult_mat(1)
thf(fact_850_index__smult__mat_I1_J,axiom,
! [I: nat,A2: mat_real,J: nat,A: real] :
( ( ord_less_nat @ I @ ( dim_row_real @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_real @ A2 ) )
=> ( ( index_mat_real @ ( smult_mat_real @ A @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( times_times_real @ A @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ).
% index_smult_mat(1)
thf(fact_851_index__mat__swaprows_I1_J,axiom,
! [I: nat,A2: mat_a,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ ( dim_row_a @ A2 ) )
=> ( ( ord_less_nat @ J @ ( dim_col_a @ A2 ) )
=> ( ( ( K = I )
=> ( ( index_mat_a @ ( gauss_2482569599970757219rows_a @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ L @ J ) ) ) )
& ( ( K != I )
=> ( ( ( L = I )
=> ( ( index_mat_a @ ( gauss_2482569599970757219rows_a @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ K @ J ) ) ) )
& ( ( L != I )
=> ( ( index_mat_a @ ( gauss_2482569599970757219rows_a @ K @ L @ A2 ) @ ( product_Pair_nat_nat @ I @ J ) )
= ( index_mat_a @ A2 @ ( product_Pair_nat_nat @ I @ J ) ) ) ) ) ) ) ) ) ).
% index_mat_swaprows(1)
thf(fact_852_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_853_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_854_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_855_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_856_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_857_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_858_index__smult__mat_I2_J,axiom,
! [A: a,A2: mat_a] :
( ( dim_row_a @ ( smult_mat_a @ A @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% index_smult_mat(2)
thf(fact_859_index__smult__mat_I3_J,axiom,
! [A: a,A2: mat_a] :
( ( dim_col_a @ ( smult_mat_a @ A @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% index_smult_mat(3)
thf(fact_860_index__mat__swaprows_I2_J,axiom,
! [K: nat,L: nat,A2: mat_a] :
( ( dim_row_a @ ( gauss_2482569599970757219rows_a @ K @ L @ A2 ) )
= ( dim_row_a @ A2 ) ) ).
% index_mat_swaprows(2)
thf(fact_861_index__mat__swaprows_I3_J,axiom,
! [K: nat,L: nat,A2: mat_a] :
( ( dim_col_a @ ( gauss_2482569599970757219rows_a @ K @ L @ A2 ) )
= ( dim_col_a @ A2 ) ) ).
% index_mat_swaprows(3)
thf(fact_862_index__mat__swaprows__mat_I2_J,axiom,
! [N: nat,K: nat,L: nat] :
( ( dim_row_a @ ( gauss_110929411057020027_mat_a @ N @ K @ L ) )
= N ) ).
% index_mat_swaprows_mat(2)
thf(fact_863_index__mat__swaprows__mat_I3_J,axiom,
! [N: nat,K: nat,L: nat] :
( ( dim_col_a @ ( gauss_110929411057020027_mat_a @ N @ K @ L ) )
= N ) ).
% index_mat_swaprows_mat(3)
thf(fact_864_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_865_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_866_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_867_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_868_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_869_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_870_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_871_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_872_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_873_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_874_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_875_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_876_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_877_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_878_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_879_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_880_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_881_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_882_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_883_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_884_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_885_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_886_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_887_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_888_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_889_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_890_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_891_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_892_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_893_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_894_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_895_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_896_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_897_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_898_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_899_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_900_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_901_order__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_902_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_903_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_904_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_905_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_906_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_907_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_908_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_909_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_910_order__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_911_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_912_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_913_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_914_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_915_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_916_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_917_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_918_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_919_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_920_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_921_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_922_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_923_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_924_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_925_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_926_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_927_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_928_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: real,B5: real] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_929_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_930_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_931_order__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_932_order__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_933_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_934_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_935_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_936_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_937_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_938_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_939_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_940_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_941_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_942_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_943_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_944_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_945_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_946_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
& ( ord_less_eq_real @ Y6 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_947_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
= ( ^ [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
& ( ord_less_eq_int @ Y6 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_948_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_949_le__cases3,axiom,
! [X: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_950_le__cases3,axiom,
! [X: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_951_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_952_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_953_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_954_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_955_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_956_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_957_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_958_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_959_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_960_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_961_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_962_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_963_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_964_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_965_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_966_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_967_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_968_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_969_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_970_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_971_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_972_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_973_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_974_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_975_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_976_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_977_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_978_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_979_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_980_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_981_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_982_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_983_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_984_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_985_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_986_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_987_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_988_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_989_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_990_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_991_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_992_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_993_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_994_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_995_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_996_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_997_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_998_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_999_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1000_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1001_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1002_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1003_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1004_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1005_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1006_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1007_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1008_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1009_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1010_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1011_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_1012_order__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_1013_order__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_1014_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_1015_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_1016_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_1017_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_1018_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_1019_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_1020_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_1021_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_1022_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_1023_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_1024_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_1025_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_1026_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1027_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1028_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1029_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_1030_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_1031_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_1032_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_1033_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_1034_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_1035_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1036_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1037_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1038_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_1039_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_1040_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_1041_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_1042_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A5: real,B5: real] :
( ( ord_less_real @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: real] : ( P @ A5 @ A5 )
=> ( ! [A5: real,B5: real] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_1043_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_1044_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_1045_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_1046_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_1047_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_1048_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_1049_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_1050_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_1051_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_1052_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_1053_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_1054_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_1055_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_1056_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_1057_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_1058_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_1059_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_1060_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_1061_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_1062_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_1063_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_1064_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_1065_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_1066_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_1067_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1068_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1069_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1070_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z2: real] :
( ( ord_less_real @ X @ Z2 )
& ( ord_less_real @ Z2 @ Y ) ) ) ).
% dense
thf(fact_1071_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_1072_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_1073_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_1074_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_1075_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_1076_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1077_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1078_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1079_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1080_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1081_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1082_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1083_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1084_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1085_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1086_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1087_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1088_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1089_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1090_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1091_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1092_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1093_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1094_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1095_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1096_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1097_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1098_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1099_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1100_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1101_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1102_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1103_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1104_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1105_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1106_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1107_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1108_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1109_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1110_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1111_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1112_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
= ( P @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1113_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).
% distinct_upt
thf(fact_1114_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1115_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K2: nat] :
? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( F @ K2 @ I3 ) )
=> ? [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_1116_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1117_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1118_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1119_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N )
= A ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1120_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1121_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1122_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N2: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_1123_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1124_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1125_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1126_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1127_zadd__int__left,axiom,
! [M: nat,N: nat,Z3: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z3 ) ) ).
% zadd_int_left
thf(fact_1128_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1129_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1130_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1131_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1132_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1133_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N2: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1134_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1135_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1136_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1137_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z4: int] :
? [N2: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1138_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1139_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_1140_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1141_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1142_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1143_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_1144_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1145_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_1146_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1147_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1148_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1149_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1150_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_1151_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_1152_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_1153_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1154_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1155_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1156_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_1157_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1158_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1159_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1160_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1161_int__cases,axiom,
! [Z3: int] :
( ! [N3: nat] :
( Z3
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_1162_int__cases2,axiom,
! [Z3: int] :
( ! [N3: nat] :
( Z3
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_1163_int__of__nat__induct,axiom,
! [P: int > $o,Z3: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z3 ) ) ) ).
% int_of_nat_induct
thf(fact_1164_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1165_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1166_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1167_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_1168_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1169_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1170_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1171_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1172_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1173_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1174_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1175_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1176_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1177_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1178_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1179_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z2 )
=> ( R3 @ X3 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R3 @ N3 @ ( suc @ N3 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1180_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1181_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1182_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1183_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1184_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1185_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1186_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1187_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1188_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1189_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1190_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1191_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1192_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1193_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_1194_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1195_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1196_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1197_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_1198_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1199_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1200_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1201_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_1202_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1203_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1204_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1205_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1206_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_1207_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1208_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1209_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1210_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_1211_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_1212_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_1213_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_1214_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1215_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1216_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1217_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1218_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1219_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1220_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1221_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_1222_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1223_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1224_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1225_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_1226_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q ) ) ) ) ).
% less_natE
thf(fact_1227_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1228_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1229_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1230_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1231_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_1232_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_1233_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_1234_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1235_unit__vecs__first_Ocases,axiom,
! [X: product_prod_nat_nat] :
( ! [N3: nat] :
( X
!= ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) )
=> ~ ! [N3: nat,I2: nat] :
( X
!= ( product_Pair_nat_nat @ N3 @ ( suc @ I2 ) ) ) ) ).
% unit_vecs_first.cases
thf(fact_1236_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1237_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1238_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1239_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1240_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_1241_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1242_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1243_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1244_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1245_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_1246_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_1247_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_1248_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1249_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1250_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1251_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1252_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1253_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_1254_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z4: int] :
? [N2: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1255_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1256_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_1257_le__Suc__eq_H,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ ( suc @ Y ) )
= ( ( X = zero_zero_nat )
| ? [X7: nat] :
( ( X
= ( suc @ X7 ) )
& ( ord_less_eq_nat @ X7 @ Y ) ) ) ) ).
% le_Suc_eq'
thf(fact_1258_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1259_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1260_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1261_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1262_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1263_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1264_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
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
i = zero_zero_nat ).
thf(conj_1,conjecture,
( ( index_mat_a @ ( rank_A2757900658775374630iple_a @ a2 @ ( upt @ one_one_nat @ ( dim_row_a @ m ) ) @ zero_zero_nat @ m ) @ ( product_Pair_nat_nat @ i @ j ) )
= ( index_mat_a @ m @ ( product_Pair_nat_nat @ i @ j ) ) ) ).
%------------------------------------------------------------------------------