TPTP Problem File: SLH0002^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : LP_Duality/0001_LP_Duality/prob_00045_001589__28725902_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1429 ( 606 unt; 154 typ; 0 def)
% Number of atoms : 3650 (1348 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10391 ( 377 ~; 149 |; 161 &;8235 @)
% ( 0 <=>;1469 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 314 ( 314 >; 0 *; 0 +; 0 <<)
% Number of symbols : 130 ( 127 usr; 17 con; 0-3 aty)
% Number of variables : 3326 ( 190 ^;3097 !; 39 ?;3326 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:55:35.630
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Real__Oreal_J_J,type,
set_vec_real: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Real__Oreal_J_J,type,
set_mat_real: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Nat__Onat_J_J,type,
set_vec_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_It__Int__Oint_J_J,type,
set_vec_int: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Nat__Onat_J_J,type,
set_mat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Int__Oint_J_J,type,
set_mat_int: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
set_vec_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
set_mat_a: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Int__Oint_J,type,
poly_int: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Real__Oreal_J,type,
vec_real: $tType ).
thf(ty_n_t__Matrix__Omat_It__Real__Oreal_J,type,
mat_real: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Nat__Onat_J,type,
vec_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_It__Int__Oint_J,type,
vec_int: $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__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_Itf__a_J,type,
poly_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Matrix__Ovec_Itf__a_J,type,
vec_a: $tType ).
thf(ty_n_t__Matrix__Omat_Itf__a_J,type,
mat_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (127)
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__Matrix__Ovec_It__Int__Oint_J,type,
plus_plus_vec_int: vec_int > vec_int > vec_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_It__Nat__Onat_J,type,
plus_plus_vec_nat: vec_nat > vec_nat > vec_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_It__Real__Oreal_J,type,
plus_plus_vec_real: vec_real > vec_real > vec_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Ovec_Itf__a_J,type,
plus_plus_vec_a: vec_a > vec_a > vec_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_001t__Set__Oset_It__Int__Oint_J,type,
plus_plus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
plus_plus_set_mat_a: set_mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
plus_plus_set_vec_a: set_vec_a > set_vec_a > set_vec_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_Itf__a_J,type,
plus_plus_set_a: set_a > set_a > set_a ).
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_001t__Set__Oset_It__Int__Oint_J,type,
times_times_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
times_1230744552615602198_mat_a: set_mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
times_times_set_real: set_real > set_real > set_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_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
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_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Int__Oint,type,
carrier_mat_int: nat > nat > set_mat_int ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Nat__Onat,type,
carrier_mat_nat: nat > nat > set_mat_nat ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Real__Oreal,type,
carrier_mat_real: nat > nat > set_mat_real ).
thf(sy_c_Matrix_Ocarrier__mat_001tf__a,type,
carrier_mat_a: nat > nat > set_mat_a ).
thf(sy_c_Matrix_Ocarrier__vec_001t__Int__Oint,type,
carrier_vec_int: nat > set_vec_int ).
thf(sy_c_Matrix_Ocarrier__vec_001t__Nat__Onat,type,
carrier_vec_nat: nat > set_vec_nat ).
thf(sy_c_Matrix_Ocarrier__vec_001t__Real__Oreal,type,
carrier_vec_real: nat > set_vec_real ).
thf(sy_c_Matrix_Ocarrier__vec_001tf__a,type,
carrier_vec_a: nat > set_vec_a ).
thf(sy_c_Matrix_Ocomm__monoid__add__class_Osum__mat_001t__Int__Oint,type,
comm_m4053738856622352096at_int: mat_int > int ).
thf(sy_c_Matrix_Ocomm__monoid__add__class_Osum__mat_001t__Nat__Onat,type,
comm_m4056229327131402372at_nat: mat_nat > nat ).
thf(sy_c_Matrix_Ocomm__monoid__add__class_Osum__mat_001t__Real__Oreal,type,
comm_m8678487124704766304t_real: mat_real > real ).
thf(sy_c_Matrix_Ocomm__monoid__add__class_Osum__mat_001tf__a,type,
comm_m5291664705200495434_mat_a: mat_a > a ).
thf(sy_c_Matrix_Omat__of__row_001tf__a,type,
mat_of_row_a: vec_a > mat_a ).
thf(sy_c_Matrix_Omult__mat__vec_001t__Int__Oint,type,
mult_mat_vec_int: mat_int > vec_int > vec_int ).
thf(sy_c_Matrix_Omult__mat__vec_001t__Nat__Onat,type,
mult_mat_vec_nat: mat_nat > vec_nat > vec_nat ).
thf(sy_c_Matrix_Omult__mat__vec_001t__Real__Oreal,type,
mult_mat_vec_real: mat_real > vec_real > vec_real ).
thf(sy_c_Matrix_Omult__mat__vec_001tf__a,type,
mult_mat_vec_a: mat_a > vec_a > vec_a ).
thf(sy_c_Matrix_Oscalar__prod_001t__Int__Oint,type,
scalar_prod_int: vec_int > vec_int > int ).
thf(sy_c_Matrix_Oscalar__prod_001t__Nat__Onat,type,
scalar_prod_nat: vec_nat > vec_nat > nat ).
thf(sy_c_Matrix_Oscalar__prod_001t__Real__Oreal,type,
scalar_prod_real: vec_real > vec_real > real ).
thf(sy_c_Matrix_Oscalar__prod_001tf__a,type,
scalar_prod_a: vec_a > vec_a > a ).
thf(sy_c_Matrix_Otranspose__mat_001t__Int__Oint,type,
transpose_mat_int: mat_int > mat_int ).
thf(sy_c_Matrix_Otranspose__mat_001t__Nat__Onat,type,
transpose_mat_nat: mat_nat > mat_nat ).
thf(sy_c_Matrix_Otranspose__mat_001t__Real__Oreal,type,
transpose_mat_real: mat_real > mat_real ).
thf(sy_c_Matrix_Otranspose__mat_001tf__a,type,
transpose_mat_a: mat_a > mat_a ).
thf(sy_c_Matrix_OvCons_001t__Int__Oint,type,
vCons_int: int > vec_int > vec_int ).
thf(sy_c_Matrix_OvCons_001t__Nat__Onat,type,
vCons_nat: nat > vec_nat > vec_nat ).
thf(sy_c_Matrix_OvCons_001t__Real__Oreal,type,
vCons_real: real > vec_real > vec_real ).
thf(sy_c_Matrix_OvCons_001tf__a,type,
vCons_a: a > vec_a > vec_a ).
thf(sy_c_Matrix_Ozero__vec_001t__Int__Oint,type,
zero_vec_int: nat > vec_int ).
thf(sy_c_Matrix_Ozero__vec_001t__Nat__Onat,type,
zero_vec_nat: nat > vec_nat ).
thf(sy_c_Matrix_Ozero__vec_001t__Real__Oreal,type,
zero_vec_real: nat > vec_real ).
thf(sy_c_Matrix_Ozero__vec_001tf__a,type,
zero_vec_a: nat > vec_a ).
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_Norms_Olinf__norm__vec_001t__Int__Oint,type,
linf_norm_vec_int: vec_int > int ).
thf(sy_c_Norms_Olinf__norm__vec_001t__Real__Oreal,type,
linf_norm_vec_real: vec_real > real ).
thf(sy_c_Norms_Olinf__norm__vec_001tf__a,type,
linf_norm_vec_a: vec_a > a ).
thf(sy_c_Norms_Onorm1_001t__Int__Oint,type,
norm1_int: poly_int > int ).
thf(sy_c_Norms_Onorm1_001t__Real__Oreal,type,
norm1_real: poly_real > real ).
thf(sy_c_Norms_Onorm1_001tf__a,type,
norm1_a: poly_a > a ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001tf__a,type,
neg_nu6917059380386235053_inc_a: a > a ).
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__Matrix__Ovec_Itf__a_J,type,
ord_less_vec_a: vec_a > vec_a > $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_001tf__a,type,
ord_less_a: a > a > $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__Ovec_It__Int__Oint_J,type,
ord_less_eq_vec_int: vec_int > vec_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Nat__Onat_J,type,
ord_less_eq_vec_nat: vec_nat > vec_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_It__Real__Oreal_J,type,
ord_less_eq_vec_real: vec_real > vec_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Matrix__Ovec_Itf__a_J,type,
ord_less_eq_vec_a: vec_a > vec_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
ord_le3318621148231462513_mat_a: set_mat_a > set_mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
ord_le4791951621262958845_vec_a: set_vec_a > set_vec_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
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_Schur__Decomposition_Ovec__inv_001t__Real__Oreal,type,
schur_vec_inv_real: vec_real > vec_real ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_Itf__a_J,type,
collect_mat_a: ( mat_a > $o ) > set_mat_a ).
thf(sy_c_Set_OCollect_001t__Matrix__Ovec_Itf__a_J,type,
collect_vec_a: ( vec_a > $o ) > set_vec_a ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Int__Oint_J,type,
member_mat_int: mat_int > set_mat_int > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Nat__Onat_J,type,
member_mat_nat: mat_nat > set_mat_nat > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Real__Oreal_J,type,
member_mat_real: mat_real > set_mat_real > $o ).
thf(sy_c_member_001t__Matrix__Omat_Itf__a_J,type,
member_mat_a: mat_a > set_mat_a > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Int__Oint_J,type,
member_vec_int: vec_int > set_vec_int > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Nat__Onat_J,type,
member_vec_nat: vec_nat > set_vec_nat > $o ).
thf(sy_c_member_001t__Matrix__Ovec_It__Real__Oreal_J,type,
member_vec_real: vec_real > set_vec_real > $o ).
thf(sy_c_member_001t__Matrix__Ovec_Itf__a_J,type,
member_vec_a: vec_a > set_vec_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: mat_a ).
thf(sy_v_b,type,
b: vec_a ).
thf(sy_v_c,type,
c: vec_a ).
thf(sy_v_nc,type,
nc: nat ).
thf(sy_v_nr,type,
nr: nat ).
thf(sy_v_x____,type,
x: vec_a ).
thf(sy_v_y____,type,
y: vec_a ).
% Relevant facts (1271)
thf(fact_0__092_060open_062c_A_092_060bullet_062_Ax_A_092_060le_062_Ab_A_092_060bullet_062_Ay_092_060close_062,axiom,
ord_less_eq_a @ ( scalar_prod_a @ c @ x ) @ ( scalar_prod_a @ b @ y ) ).
% \<open>c \<bullet> x \<le> b \<bullet> y\<close>
thf(fact_1_le,axiom,
ord_less_eq_a @ ( plus_plus_a @ ( scalar_prod_a @ b @ y ) @ one_one_a ) @ ( scalar_prod_a @ c @ x ) ).
% le
thf(fact_2_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_3_order__refl,axiom,
! [X: vec_a] : ( ord_less_eq_vec_a @ X @ X ) ).
% order_refl
thf(fact_4_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_5_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_6_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_7_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_8_dual__order_Orefl,axiom,
! [A: vec_a] : ( ord_less_eq_vec_a @ A @ A ) ).
% dual_order.refl
thf(fact_9_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_10_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_11_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_12_x,axiom,
member_vec_a @ x @ ( carrier_vec_a @ nc ) ).
% x
thf(fact_13_b,axiom,
member_vec_a @ b @ ( carrier_vec_a @ nr ) ).
% b
thf(fact_14_c,axiom,
member_vec_a @ c @ ( carrier_vec_a @ nc ) ).
% c
thf(fact_15_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_16_verit__comp__simplify1_I2_J,axiom,
! [A: vec_a] : ( ord_less_eq_vec_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_17_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_18_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_19_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_20_nle__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_eq_a @ A @ B ) )
= ( ( ord_less_eq_a @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_21_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_22_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_23_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_24_le__cases3,axiom,
! [X: a,Y: a,Z: a] :
( ( ( ord_less_eq_a @ X @ Y )
=> ~ ( ord_less_eq_a @ Y @ Z ) )
=> ( ( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_eq_a @ X @ Z ) )
=> ( ( ( ord_less_eq_a @ X @ Z )
=> ~ ( ord_less_eq_a @ Z @ Y ) )
=> ( ( ( ord_less_eq_a @ Z @ Y )
=> ~ ( ord_less_eq_a @ Y @ X ) )
=> ( ( ( ord_less_eq_a @ Y @ Z )
=> ~ ( ord_less_eq_a @ Z @ X ) )
=> ~ ( ( ord_less_eq_a @ Z @ X )
=> ~ ( ord_less_eq_a @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_25_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_26_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_27_le__cases3,axiom,
! [X: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_28_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: a,Z2: a] : ( Y2 = Z2 ) )
= ( ^ [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
& ( ord_less_eq_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_29_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: vec_a,Z2: vec_a] : ( Y2 = Z2 ) )
= ( ^ [X2: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X2 @ Y3 )
& ( ord_less_eq_vec_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_30_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_31_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 ) )
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_32_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
= ( ^ [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_33_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_34_ord__eq__le__trans,axiom,
! [A: vec_a,B: vec_a,C: vec_a] :
( ( A = B )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ord_less_eq_vec_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_35_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_36_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_37_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_38_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_39_ord__le__eq__trans,axiom,
! [A: vec_a,B: vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_vec_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_40_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_41_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_42_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_43_y,axiom,
ord_less_eq_vec_a @ ( zero_vec_a @ nr ) @ y ).
% y
thf(fact_44_order__antisym__conv,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_45_order__antisym__conv,axiom,
! [Y: vec_a,X: vec_a] :
( ( ord_less_eq_vec_a @ Y @ X )
=> ( ( ord_less_eq_vec_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_46_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_47_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_48_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_49_linorder__le__cases,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_50_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_51_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_52_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_53_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_54_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_55_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > int,C: int] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_56_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > real,C: real] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_57_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_58_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_59_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_60_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_61_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > a,C: a] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_62_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,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_63_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_64_ord__eq__le__subst,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_65_ord__eq__le__subst,axiom,
! [A: int,F: a > int,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_66_ord__eq__le__subst,axiom,
! [A: real,F: a > real,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_67_ord__eq__le__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_68_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_69_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_70_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_71_ord__eq__le__subst,axiom,
! [A: a,F: int > a,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_72_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,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_73_linorder__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_linear
thf(fact_74_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_75_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_76_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_77_verit__la__disequality,axiom,
! [A: a,B: a] :
( ( A = B )
| ~ ( ord_less_eq_a @ A @ B )
| ~ ( ord_less_eq_a @ B @ A ) ) ).
% verit_la_disequality
thf(fact_78_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_79_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_80_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_81_order__eq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_82_order__eq__refl,axiom,
! [X: vec_a,Y: vec_a] :
( ( X = Y )
=> ( ord_less_eq_vec_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_83_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_84_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_85_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_86_order__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_87_order__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_88_order__subst2,axiom,
! [A: a,B: a,F: a > int,C: int] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_89_order__subst2,axiom,
! [A: a,B: a,F: a > real,C: real] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_90_order__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_91_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_92_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_93_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_94_order__subst2,axiom,
! [A: int,B: int,F: int > a,C: a] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_95_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,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_96_order__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_97_order__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_98_order__subst1,axiom,
! [A: a,F: int > a,B: int,C: int] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_99_order__subst1,axiom,
! [A: a,F: real > a,B: real,C: real] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_100_order__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_101_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_102_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,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_103_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,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_104_order__subst1,axiom,
! [A: int,F: a > int,B: a,C: a] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_105_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_106_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: a,Z2: a] : ( Y2 = Z2 ) )
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ( ord_less_eq_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_107_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: vec_a,Z2: vec_a] : ( Y2 = Z2 ) )
= ( ^ [A2: vec_a,B2: vec_a] :
( ( ord_less_eq_vec_a @ A2 @ B2 )
& ( ord_less_eq_vec_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_108_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_109_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_110_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_111_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_112_antisym,axiom,
! [A: vec_a,B: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_eq_vec_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_113_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_114_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_115_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_116_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_117_dual__order_Otrans,axiom,
! [B: vec_a,A: vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ B @ A )
=> ( ( ord_less_eq_vec_a @ C @ B )
=> ( ord_less_eq_vec_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_118_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_119_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_120_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_121_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_122_dual__order_Oantisym,axiom,
! [B: vec_a,A: vec_a] :
( ( ord_less_eq_vec_a @ B @ A )
=> ( ( ord_less_eq_vec_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_123_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_124_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_125_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_126_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: a,Z2: a] : ( Y2 = Z2 ) )
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ( ord_less_eq_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_127_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: vec_a,Z2: vec_a] : ( Y2 = Z2 ) )
= ( ^ [A2: vec_a,B2: vec_a] :
( ( ord_less_eq_vec_a @ B2 @ A2 )
& ( ord_less_eq_vec_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_128_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_129_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_130_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_131_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_132_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_133_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_134_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_135_order__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z )
=> ( ord_less_eq_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_136_order__trans,axiom,
! [X: vec_a,Y: vec_a,Z: vec_a] :
( ( ord_less_eq_vec_a @ X @ Y )
=> ( ( ord_less_eq_vec_a @ Y @ Z )
=> ( ord_less_eq_vec_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_137_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_138_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_139_order__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_140_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_141_order_Otrans,axiom,
! [A: vec_a,B: vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ord_less_eq_vec_a @ A @ C ) ) ) ).
% order.trans
thf(fact_142_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_143_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_144_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_145_order__antisym,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_146_order__antisym,axiom,
! [X: vec_a,Y: vec_a] :
( ( ord_less_eq_vec_a @ X @ Y )
=> ( ( ord_less_eq_vec_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_147_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_148_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_149_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_150_add__le__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( ord_less_eq_a @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_151_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_152_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_153_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_154_add__le__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( ord_less_eq_a @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_155_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_156_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_157_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_158_unbounded,axiom,
! [V: a] :
? [X3: vec_a] :
( ( member_vec_a @ X3 @ ( carrier_vec_a @ nc ) )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ a2 @ X3 ) @ b )
& ( ord_less_eq_a @ V @ ( scalar_prod_a @ c @ X3 ) ) ) ).
% unbounded
thf(fact_159__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_A_092_060lbrakk_062x_A_092_060in_062_Acarrier__vec_Anc_059_AA_A_K_092_060_094sub_062v_Ax_A_092_060le_062_Ab_059_Ab_A_092_060bullet_062_Ay_A_L_A_I1_058_058_Ha_J_A_092_060le_062_Ac_A_092_060bullet_062_Ax_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X3: vec_a] :
( ( member_vec_a @ X3 @ ( carrier_vec_a @ nc ) )
=> ( ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ a2 @ X3 ) @ b )
=> ~ ( ord_less_eq_a @ ( plus_plus_a @ ( scalar_prod_a @ b @ y ) @ one_one_a ) @ ( scalar_prod_a @ c @ X3 ) ) ) ) ).
% \<open>\<And>thesis. (\<And>x. \<lbrakk>x \<in> carrier_vec nc; A *\<^sub>v x \<le> b; b \<bullet> y + (1::'a) \<le> c \<bullet> x\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_160__092_060open_062_092_060exists_062x_092_060in_062carrier__vec_Anc_O_AA_A_K_092_060_094sub_062v_Ax_A_092_060le_062_Ab_A_092_060and_062_Ab_A_092_060bullet_062_Ay_A_L_A_I1_058_058_Ha_J_A_092_060le_062_Ac_A_092_060bullet_062_Ax_092_060close_062,axiom,
? [X3: vec_a] :
( ( member_vec_a @ X3 @ ( carrier_vec_a @ nc ) )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ a2 @ X3 ) @ b )
& ( ord_less_eq_a @ ( plus_plus_a @ ( scalar_prod_a @ b @ y ) @ one_one_a ) @ ( scalar_prod_a @ c @ X3 ) ) ) ).
% \<open>\<exists>x\<in>carrier_vec nc. A *\<^sub>v x \<le> b \<and> b \<bullet> y + (1::'a) \<le> c \<bullet> x\<close>
thf(fact_161_add__scalar__prod__distrib,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a,V_3: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_3 @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ ( plus_plus_vec_a @ V_1 @ V_2 ) @ V_3 )
= ( plus_plus_a @ ( scalar_prod_a @ V_1 @ V_3 ) @ ( scalar_prod_a @ V_2 @ V_3 ) ) ) ) ) ) ).
% add_scalar_prod_distrib
thf(fact_162_add__scalar__prod__distrib,axiom,
! [V_1: vec_nat,N: nat,V_2: vec_nat,V_3: vec_nat] :
( ( member_vec_nat @ V_1 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V_2 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V_3 @ ( carrier_vec_nat @ N ) )
=> ( ( scalar_prod_nat @ ( plus_plus_vec_nat @ V_1 @ V_2 ) @ V_3 )
= ( plus_plus_nat @ ( scalar_prod_nat @ V_1 @ V_3 ) @ ( scalar_prod_nat @ V_2 @ V_3 ) ) ) ) ) ) ).
% add_scalar_prod_distrib
thf(fact_163_add__scalar__prod__distrib,axiom,
! [V_1: vec_int,N: nat,V_2: vec_int,V_3: vec_int] :
( ( member_vec_int @ V_1 @ ( carrier_vec_int @ N ) )
=> ( ( member_vec_int @ V_2 @ ( carrier_vec_int @ N ) )
=> ( ( member_vec_int @ V_3 @ ( carrier_vec_int @ N ) )
=> ( ( scalar_prod_int @ ( plus_plus_vec_int @ V_1 @ V_2 ) @ V_3 )
= ( plus_plus_int @ ( scalar_prod_int @ V_1 @ V_3 ) @ ( scalar_prod_int @ V_2 @ V_3 ) ) ) ) ) ) ).
% add_scalar_prod_distrib
thf(fact_164_add__scalar__prod__distrib,axiom,
! [V_1: vec_real,N: nat,V_2: vec_real,V_3: vec_real] :
( ( member_vec_real @ V_1 @ ( carrier_vec_real @ N ) )
=> ( ( member_vec_real @ V_2 @ ( carrier_vec_real @ N ) )
=> ( ( member_vec_real @ V_3 @ ( carrier_vec_real @ N ) )
=> ( ( scalar_prod_real @ ( plus_plus_vec_real @ V_1 @ V_2 ) @ V_3 )
= ( plus_plus_real @ ( scalar_prod_real @ V_1 @ V_3 ) @ ( scalar_prod_real @ V_2 @ V_3 ) ) ) ) ) ) ).
% add_scalar_prod_distrib
thf(fact_165_scalar__prod__add__distrib,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a,V_3: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_3 @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ V_1 @ ( plus_plus_vec_a @ V_2 @ V_3 ) )
= ( plus_plus_a @ ( scalar_prod_a @ V_1 @ V_2 ) @ ( scalar_prod_a @ V_1 @ V_3 ) ) ) ) ) ) ).
% scalar_prod_add_distrib
thf(fact_166_scalar__prod__add__distrib,axiom,
! [V_1: vec_nat,N: nat,V_2: vec_nat,V_3: vec_nat] :
( ( member_vec_nat @ V_1 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V_2 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V_3 @ ( carrier_vec_nat @ N ) )
=> ( ( scalar_prod_nat @ V_1 @ ( plus_plus_vec_nat @ V_2 @ V_3 ) )
= ( plus_plus_nat @ ( scalar_prod_nat @ V_1 @ V_2 ) @ ( scalar_prod_nat @ V_1 @ V_3 ) ) ) ) ) ) ).
% scalar_prod_add_distrib
thf(fact_167_scalar__prod__add__distrib,axiom,
! [V_1: vec_int,N: nat,V_2: vec_int,V_3: vec_int] :
( ( member_vec_int @ V_1 @ ( carrier_vec_int @ N ) )
=> ( ( member_vec_int @ V_2 @ ( carrier_vec_int @ N ) )
=> ( ( member_vec_int @ V_3 @ ( carrier_vec_int @ N ) )
=> ( ( scalar_prod_int @ V_1 @ ( plus_plus_vec_int @ V_2 @ V_3 ) )
= ( plus_plus_int @ ( scalar_prod_int @ V_1 @ V_2 ) @ ( scalar_prod_int @ V_1 @ V_3 ) ) ) ) ) ) ).
% scalar_prod_add_distrib
thf(fact_168_scalar__prod__add__distrib,axiom,
! [V_1: vec_real,N: nat,V_2: vec_real,V_3: vec_real] :
( ( member_vec_real @ V_1 @ ( carrier_vec_real @ N ) )
=> ( ( member_vec_real @ V_2 @ ( carrier_vec_real @ N ) )
=> ( ( member_vec_real @ V_3 @ ( carrier_vec_real @ N ) )
=> ( ( scalar_prod_real @ V_1 @ ( plus_plus_vec_real @ V_2 @ V_3 ) )
= ( plus_plus_real @ ( scalar_prod_real @ V_1 @ V_2 ) @ ( scalar_prod_real @ V_1 @ V_3 ) ) ) ) ) ) ).
% scalar_prod_add_distrib
thf(fact_169_A,axiom,
member_mat_a @ a2 @ ( carrier_mat_a @ nr @ nc ) ).
% A
thf(fact_170_add__left__cancel,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_171_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_172_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_173_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_174_mem__Collect__eq,axiom,
! [A: vec_a,P: vec_a > $o] :
( ( member_vec_a @ A @ ( collect_vec_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_175_mem__Collect__eq,axiom,
! [A: mat_a,P: mat_a > $o] :
( ( member_mat_a @ A @ ( collect_mat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_176_Collect__mem__eq,axiom,
! [A4: set_vec_a] :
( ( collect_vec_a
@ ^ [X2: vec_a] : ( member_vec_a @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_177_Collect__mem__eq,axiom,
! [A4: set_mat_a] :
( ( collect_mat_a
@ ^ [X2: mat_a] : ( member_mat_a @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_178_add__right__cancel,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_179_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_180_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_181_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_182_set__plus__intro,axiom,
! [A: vec_a,C2: set_vec_a,B: vec_a,D: set_vec_a] :
( ( member_vec_a @ A @ C2 )
=> ( ( member_vec_a @ B @ D )
=> ( member_vec_a @ ( plus_plus_vec_a @ A @ B ) @ ( plus_plus_set_vec_a @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_183_set__plus__intro,axiom,
! [A: mat_a,C2: set_mat_a,B: mat_a,D: set_mat_a] :
( ( member_mat_a @ A @ C2 )
=> ( ( member_mat_a @ B @ D )
=> ( member_mat_a @ ( plus_plus_mat_a @ A @ B ) @ ( plus_plus_set_mat_a @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_184_set__plus__intro,axiom,
! [A: a,C2: set_a,B: a,D: set_a] :
( ( member_a @ A @ C2 )
=> ( ( member_a @ B @ D )
=> ( member_a @ ( plus_plus_a @ A @ B ) @ ( plus_plus_set_a @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_185_set__plus__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_186_set__plus__intro,axiom,
! [A: int,C2: set_int,B: int,D: set_int] :
( ( member_int @ A @ C2 )
=> ( ( member_int @ B @ D )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_set_int @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_187_set__plus__intro,axiom,
! [A: real,C2: set_real,B: real,D: set_real] :
( ( member_real @ A @ C2 )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_set_real @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_188_assoc__add__mat,axiom,
! [A4: mat_a,Nr: nat,Nc: nat,B4: mat_a,C2: mat_a] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( plus_plus_mat_a @ A4 @ B4 ) @ C2 )
= ( plus_plus_mat_a @ A4 @ ( plus_plus_mat_a @ B4 @ C2 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_189_Axb,axiom,
ord_less_eq_vec_a @ ( mult_mat_vec_a @ a2 @ x ) @ b ).
% Axb
thf(fact_190_assoc__add__vec,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a,V_3: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_3 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ ( plus_plus_vec_a @ V_1 @ V_2 ) @ V_3 )
= ( plus_plus_vec_a @ V_1 @ ( plus_plus_vec_a @ V_2 @ V_3 ) ) ) ) ) ) ).
% assoc_add_vec
thf(fact_191_left__zero__vec,axiom,
! [V2: vec_a,N: nat] :
( ( member_vec_a @ V2 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ ( zero_vec_a @ N ) @ V2 )
= V2 ) ) ).
% left_zero_vec
thf(fact_192_right__zero__vec,axiom,
! [V2: vec_a,N: nat] :
( ( member_vec_a @ V2 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ V2 @ ( zero_vec_a @ N ) )
= V2 ) ) ).
% right_zero_vec
thf(fact_193_Ayc,axiom,
( ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ y )
= c ) ).
% Ayc
thf(fact_194__092_060open_062_092_060exists_062y_092_060ge_0620_092_060_094sub_062v_Anr_O_AA_092_060_094sup_062T_A_K_092_060_094sub_062v_Ay_A_061_Ac_092_060close_062,axiom,
? [Y4: vec_a] :
( ( ord_less_eq_vec_a @ ( zero_vec_a @ nr ) @ Y4 )
& ( ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ Y4 )
= c ) ) ).
% \<open>\<exists>y\<ge>0\<^sub>v nr. A\<^sup>T *\<^sub>v y = c\<close>
thf(fact_195_comm__add__mat,axiom,
! [A4: mat_a,Nr: nat,Nc: nat,B4: mat_a] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ A4 @ B4 )
= ( plus_plus_mat_a @ B4 @ A4 ) ) ) ) ).
% comm_add_mat
thf(fact_196_add__carrier__mat,axiom,
! [B4: mat_a,Nr: nat,Nc: nat,A4: mat_a] :
( ( member_mat_a @ B4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( plus_plus_mat_a @ A4 @ B4 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% add_carrier_mat
thf(fact_197_add__inv__exists__vec,axiom,
! [V2: vec_a,N: nat] :
( ( member_vec_a @ V2 @ ( carrier_vec_a @ N ) )
=> ? [X3: vec_a] :
( ( member_vec_a @ X3 @ ( carrier_vec_a @ N ) )
& ( ( plus_plus_vec_a @ X3 @ V2 )
= ( zero_vec_a @ N ) )
& ( ( plus_plus_vec_a @ V2 @ X3 )
= ( zero_vec_a @ N ) ) ) ) ).
% add_inv_exists_vec
thf(fact_198_mult__mat__vec__carrier,axiom,
! [A4: mat_a,Nr: nat,N: nat,V2: vec_a] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_vec_a @ V2 @ ( carrier_vec_a @ N ) )
=> ( member_vec_a @ ( mult_mat_vec_a @ A4 @ V2 ) @ ( carrier_vec_a @ Nr ) ) ) ) ).
% mult_mat_vec_carrier
thf(fact_199_add__mult__distrib__mat__vec,axiom,
! [A4: mat_a,Nr: nat,Nc: nat,B4: mat_a,V2: vec_a] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V2 @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ ( plus_plus_mat_a @ A4 @ B4 ) @ V2 )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ A4 @ V2 ) @ ( mult_mat_vec_a @ B4 @ V2 ) ) ) ) ) ) ).
% add_mult_distrib_mat_vec
thf(fact_200_mult__add__distrib__mat__vec,axiom,
! [A4: mat_a,Nr: nat,Nc: nat,V_1: vec_a,V_2: vec_a] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V_1 @ ( carrier_vec_a @ Nc ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ A4 @ ( plus_plus_vec_a @ V_1 @ V_2 ) )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ A4 @ V_1 ) @ ( mult_mat_vec_a @ A4 @ V_2 ) ) ) ) ) ) ).
% mult_add_distrib_mat_vec
thf(fact_201_zero__carrier__vec,axiom,
! [N: nat] : ( member_vec_a @ ( zero_vec_a @ N ) @ ( carrier_vec_a @ N ) ) ).
% zero_carrier_vec
thf(fact_202_add__carrier__vec,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( member_vec_a @ ( plus_plus_vec_a @ V_1 @ V_2 ) @ ( carrier_vec_a @ N ) ) ) ) ).
% add_carrier_vec
thf(fact_203_comm__add__vec,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ V_1 @ V_2 )
= ( plus_plus_vec_a @ V_2 @ V_1 ) ) ) ) ).
% comm_add_vec
thf(fact_204_one__reorient,axiom,
! [X: a] :
( ( one_one_a = X )
= ( X = one_one_a ) ) ).
% one_reorient
thf(fact_205_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_206_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_207_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_208_set__plus__elim,axiom,
! [X: vec_a,A4: set_vec_a,B4: set_vec_a] :
( ( member_vec_a @ X @ ( plus_plus_set_vec_a @ A4 @ B4 ) )
=> ~ ! [A3: vec_a,B3: vec_a] :
( ( X
= ( plus_plus_vec_a @ A3 @ B3 ) )
=> ( ( member_vec_a @ A3 @ A4 )
=> ~ ( member_vec_a @ B3 @ B4 ) ) ) ) ).
% set_plus_elim
thf(fact_209_set__plus__elim,axiom,
! [X: mat_a,A4: set_mat_a,B4: set_mat_a] :
( ( member_mat_a @ X @ ( plus_plus_set_mat_a @ A4 @ B4 ) )
=> ~ ! [A3: mat_a,B3: mat_a] :
( ( X
= ( plus_plus_mat_a @ A3 @ B3 ) )
=> ( ( member_mat_a @ A3 @ A4 )
=> ~ ( member_mat_a @ B3 @ B4 ) ) ) ) ).
% set_plus_elim
thf(fact_210_set__plus__elim,axiom,
! [X: a,A4: set_a,B4: set_a] :
( ( member_a @ X @ ( plus_plus_set_a @ A4 @ B4 ) )
=> ~ ! [A3: a,B3: a] :
( ( X
= ( plus_plus_a @ A3 @ B3 ) )
=> ( ( member_a @ A3 @ A4 )
=> ~ ( member_a @ B3 @ B4 ) ) ) ) ).
% set_plus_elim
thf(fact_211_set__plus__elim,axiom,
! [X: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A4 @ B4 ) )
=> ~ ! [A3: nat,B3: nat] :
( ( X
= ( plus_plus_nat @ A3 @ B3 ) )
=> ( ( member_nat @ A3 @ A4 )
=> ~ ( member_nat @ B3 @ B4 ) ) ) ) ).
% set_plus_elim
thf(fact_212_set__plus__elim,axiom,
! [X: int,A4: set_int,B4: set_int] :
( ( member_int @ X @ ( plus_plus_set_int @ A4 @ B4 ) )
=> ~ ! [A3: int,B3: int] :
( ( X
= ( plus_plus_int @ A3 @ B3 ) )
=> ( ( member_int @ A3 @ A4 )
=> ~ ( member_int @ B3 @ B4 ) ) ) ) ).
% set_plus_elim
thf(fact_213_set__plus__elim,axiom,
! [X: real,A4: set_real,B4: set_real] :
( ( member_real @ X @ ( plus_plus_set_real @ A4 @ B4 ) )
=> ~ ! [A3: real,B3: real] :
( ( X
= ( plus_plus_real @ A3 @ B3 ) )
=> ( ( member_real @ A3 @ A4 )
=> ~ ( member_real @ B3 @ B4 ) ) ) ) ).
% set_plus_elim
thf(fact_214_add__right__imp__eq,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_215_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_216_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_217_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_218_add__left__imp__eq,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_219_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_220_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_221_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_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: 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_225_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_226_add_Ocommute,axiom,
( plus_plus_a
= ( ^ [A2: a,B2: a] : ( plus_plus_a @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_227_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_228_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_229_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_230_add_Oright__cancel,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_231_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_232_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_233_add_Oleft__cancel,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_234_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_235_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_236_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_237_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_238_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_239_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_240_group__cancel_Oadd2,axiom,
! [B4: a,K: a,B: a,A: a] :
( ( B4
= ( plus_plus_a @ K @ B ) )
=> ( ( plus_plus_a @ A @ B4 )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_241_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_242_group__cancel_Oadd2,axiom,
! [B4: int,K: int,B: int,A: int] :
( ( B4
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B4 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_243_group__cancel_Oadd2,axiom,
! [B4: real,K: real,B: real,A: real] :
( ( B4
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B4 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_244_group__cancel_Oadd1,axiom,
! [A4: a,K: a,A: a,B: a] :
( ( A4
= ( plus_plus_a @ K @ A ) )
=> ( ( plus_plus_a @ A4 @ B )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_245_group__cancel_Oadd1,axiom,
! [A4: nat,K: nat,A: nat,B: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_246_group__cancel_Oadd1,axiom,
! [A4: int,K: int,A: int,B: int] :
( ( A4
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A4 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_247_group__cancel_Oadd1,axiom,
! [A4: real,K: real,A: real,B: real] :
( ( A4
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A4 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_248_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_a @ I @ K )
= ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_249_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_250_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_251_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_252_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_253_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_254_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_255_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_256_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_eq_a @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_257_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_258_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_259_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_260_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( I = J )
& ( ord_less_eq_a @ K @ L ) )
=> ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_261_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_262_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_263_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_264_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_eq_a @ I @ J )
& ( ord_less_eq_a @ K @ L ) )
=> ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_265_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_266_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_267_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_268_add__mono,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D2 )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_269_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_270_add__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_271_add__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_272_add__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) ) ) ).
% add_left_mono
thf(fact_273_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_274_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_275_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_276_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_277_add__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) ) ) ).
% add_right_mono
thf(fact_278_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_279_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_280_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_281_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C4: nat] :
( B2
= ( plus_plus_nat @ A2 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_282_add__le__imp__le__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
=> ( ord_less_eq_a @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_283_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_284_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_285_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_286_add__le__imp__le__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
=> ( ord_less_eq_a @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_287_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_288_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_289_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_290_comm__scalar__prod,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ V_1 @ V_2 )
= ( scalar_prod_a @ V_2 @ V_1 ) ) ) ) ).
% comm_scalar_prod
thf(fact_291__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062y_O_A_092_060lbrakk_0620_092_060_094sub_062v_Anr_A_092_060le_062_Ay_059_AA_092_060_094sup_062T_A_K_092_060_094sub_062v_Ay_A_061_Ac_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Y4: vec_a] :
( ( ord_less_eq_vec_a @ ( zero_vec_a @ nr ) @ Y4 )
=> ( ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ Y4 )
!= c ) ) ).
% \<open>\<And>thesis. (\<And>y. \<lbrakk>0\<^sub>v nr \<le> y; A\<^sup>T *\<^sub>v y = c\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_292_sum__carrier__vec,axiom,
! [A4: set_vec_a,N: nat,B4: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A4 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ B4 @ ( carrier_vec_a @ N ) )
=> ( ord_le4791951621262958845_vec_a @ ( plus_plus_set_vec_a @ A4 @ B4 ) @ ( carrier_vec_a @ N ) ) ) ) ).
% sum_carrier_vec
thf(fact_293_weak__duality__theorem,axiom,
! [A4: mat_nat,Nr: nat,Nc: nat,B: vec_nat,C: vec_nat,X: vec_nat,Y: vec_nat] :
( ( member_mat_nat @ A4 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_vec_nat @ B @ ( carrier_vec_nat @ Nr ) )
=> ( ( member_vec_nat @ C @ ( carrier_vec_nat @ Nc ) )
=> ( ( member_vec_nat @ X @ ( carrier_vec_nat @ Nc ) )
=> ( ( ord_less_eq_vec_nat @ ( mult_mat_vec_nat @ A4 @ X ) @ B )
=> ( ( ord_less_eq_vec_nat @ ( zero_vec_nat @ Nr ) @ Y )
=> ( ( ( mult_mat_vec_nat @ ( transpose_mat_nat @ A4 ) @ Y )
= C )
=> ( ord_less_eq_nat @ ( scalar_prod_nat @ C @ X ) @ ( scalar_prod_nat @ B @ Y ) ) ) ) ) ) ) ) ) ).
% weak_duality_theorem
thf(fact_294_weak__duality__theorem,axiom,
! [A4: mat_int,Nr: nat,Nc: nat,B: vec_int,C: vec_int,X: vec_int,Y: vec_int] :
( ( member_mat_int @ A4 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_vec_int @ B @ ( carrier_vec_int @ Nr ) )
=> ( ( member_vec_int @ C @ ( carrier_vec_int @ Nc ) )
=> ( ( member_vec_int @ X @ ( carrier_vec_int @ Nc ) )
=> ( ( ord_less_eq_vec_int @ ( mult_mat_vec_int @ A4 @ X ) @ B )
=> ( ( ord_less_eq_vec_int @ ( zero_vec_int @ Nr ) @ Y )
=> ( ( ( mult_mat_vec_int @ ( transpose_mat_int @ A4 ) @ Y )
= C )
=> ( ord_less_eq_int @ ( scalar_prod_int @ C @ X ) @ ( scalar_prod_int @ B @ Y ) ) ) ) ) ) ) ) ) ).
% weak_duality_theorem
thf(fact_295_weak__duality__theorem,axiom,
! [A4: mat_real,Nr: nat,Nc: nat,B: vec_real,C: vec_real,X: vec_real,Y: vec_real] :
( ( member_mat_real @ A4 @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ( member_vec_real @ B @ ( carrier_vec_real @ Nr ) )
=> ( ( member_vec_real @ C @ ( carrier_vec_real @ Nc ) )
=> ( ( member_vec_real @ X @ ( carrier_vec_real @ Nc ) )
=> ( ( ord_less_eq_vec_real @ ( mult_mat_vec_real @ A4 @ X ) @ B )
=> ( ( ord_less_eq_vec_real @ ( zero_vec_real @ Nr ) @ Y )
=> ( ( ( mult_mat_vec_real @ ( transpose_mat_real @ A4 ) @ Y )
= C )
=> ( ord_less_eq_real @ ( scalar_prod_real @ C @ X ) @ ( scalar_prod_real @ B @ Y ) ) ) ) ) ) ) ) ) ).
% weak_duality_theorem
thf(fact_296_weak__duality__theorem,axiom,
! [A4: mat_a,Nr: nat,Nc: nat,B: vec_a,C: vec_a,X: vec_a,Y: vec_a] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ Nr ) )
=> ( ( member_vec_a @ C @ ( carrier_vec_a @ Nc ) )
=> ( ( member_vec_a @ X @ ( carrier_vec_a @ Nc ) )
=> ( ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A4 @ X ) @ B )
=> ( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ Y )
=> ( ( ( mult_mat_vec_a @ ( transpose_mat_a @ A4 ) @ Y )
= C )
=> ( ord_less_eq_a @ ( scalar_prod_a @ C @ X ) @ ( scalar_prod_a @ B @ Y ) ) ) ) ) ) ) ) ) ).
% weak_duality_theorem
thf(fact_297_vec__inv,axiom,
! [V2: vec_real,N: nat] :
( ( member_vec_real @ V2 @ ( carrier_vec_real @ N ) )
=> ( ( V2
!= ( zero_vec_real @ N ) )
=> ( ( scalar_prod_real @ ( schur_vec_inv_real @ V2 ) @ V2 )
= one_one_real ) ) ) ).
% vec_inv
thf(fact_298_le__numeral__extra_I4_J,axiom,
ord_less_eq_a @ one_one_a @ one_one_a ).
% le_numeral_extra(4)
thf(fact_299_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_300_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_301_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_302_mat__of__row__carrier_I1_J,axiom,
! [Y: vec_a,N: nat] :
( ( member_vec_a @ Y @ ( carrier_vec_a @ N ) )
=> ( member_mat_a @ ( mat_of_row_a @ Y ) @ ( carrier_mat_a @ one_one_nat @ N ) ) ) ).
% mat_of_row_carrier(1)
thf(fact_303_scalar__prod__right__zero,axiom,
! [V2: vec_a,N: nat] :
( ( member_vec_a @ V2 @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ V2 @ ( zero_vec_a @ N ) )
= zero_zero_a ) ) ).
% scalar_prod_right_zero
thf(fact_304_scalar__prod__right__zero,axiom,
! [V2: vec_nat,N: nat] :
( ( member_vec_nat @ V2 @ ( carrier_vec_nat @ N ) )
=> ( ( scalar_prod_nat @ V2 @ ( zero_vec_nat @ N ) )
= zero_zero_nat ) ) ).
% scalar_prod_right_zero
thf(fact_305_scalar__prod__right__zero,axiom,
! [V2: vec_int,N: nat] :
( ( member_vec_int @ V2 @ ( carrier_vec_int @ N ) )
=> ( ( scalar_prod_int @ V2 @ ( zero_vec_int @ N ) )
= zero_zero_int ) ) ).
% scalar_prod_right_zero
thf(fact_306_scalar__prod__right__zero,axiom,
! [V2: vec_real,N: nat] :
( ( member_vec_real @ V2 @ ( carrier_vec_real @ N ) )
=> ( ( scalar_prod_real @ V2 @ ( zero_vec_real @ N ) )
= zero_zero_real ) ) ).
% scalar_prod_right_zero
thf(fact_307_scalar__prod__left__zero,axiom,
! [V2: vec_a,N: nat] :
( ( member_vec_a @ V2 @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ ( zero_vec_a @ N ) @ V2 )
= zero_zero_a ) ) ).
% scalar_prod_left_zero
thf(fact_308_scalar__prod__left__zero,axiom,
! [V2: vec_nat,N: nat] :
( ( member_vec_nat @ V2 @ ( carrier_vec_nat @ N ) )
=> ( ( scalar_prod_nat @ ( zero_vec_nat @ N ) @ V2 )
= zero_zero_nat ) ) ).
% scalar_prod_left_zero
thf(fact_309_scalar__prod__left__zero,axiom,
! [V2: vec_int,N: nat] :
( ( member_vec_int @ V2 @ ( carrier_vec_int @ N ) )
=> ( ( scalar_prod_int @ ( zero_vec_int @ N ) @ V2 )
= zero_zero_int ) ) ).
% scalar_prod_left_zero
thf(fact_310_scalar__prod__left__zero,axiom,
! [V2: vec_real,N: nat] :
( ( member_vec_real @ V2 @ ( carrier_vec_real @ N ) )
=> ( ( scalar_prod_real @ ( zero_vec_real @ N ) @ V2 )
= zero_zero_real ) ) ).
% scalar_prod_left_zero
thf(fact_311_transpose__vec__mult__scalar,axiom,
! [A4: mat_a,Nr: nat,Nc: nat,X: vec_a,Y: vec_a] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ X @ ( carrier_vec_a @ Nc ) )
=> ( ( member_vec_a @ Y @ ( carrier_vec_a @ Nr ) )
=> ( ( scalar_prod_a @ ( mult_mat_vec_a @ ( transpose_mat_a @ A4 ) @ Y ) @ X )
= ( scalar_prod_a @ Y @ ( mult_mat_vec_a @ A4 @ X ) ) ) ) ) ) ).
% transpose_vec_mult_scalar
thf(fact_312_assoc__mult__mat__vec,axiom,
! [A4: mat_a,N_1: nat,N_2: nat,B4: mat_a,N_3: nat,V2: vec_a] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ N_1 @ N_2 ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ N_2 @ N_3 ) )
=> ( ( member_vec_a @ V2 @ ( carrier_vec_a @ N_3 ) )
=> ( ( mult_mat_vec_a @ ( times_times_mat_a @ A4 @ B4 ) @ V2 )
= ( mult_mat_vec_a @ A4 @ ( mult_mat_vec_a @ B4 @ V2 ) ) ) ) ) ) ).
% assoc_mult_mat_vec
thf(fact_313_set__times__intro,axiom,
! [A: mat_a,C2: set_mat_a,B: mat_a,D: set_mat_a] :
( ( member_mat_a @ A @ C2 )
=> ( ( member_mat_a @ B @ D )
=> ( member_mat_a @ ( times_times_mat_a @ A @ B ) @ ( times_1230744552615602198_mat_a @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_314_set__times__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_315_set__times__intro,axiom,
! [A: int,C2: set_int,B: int,D: set_int] :
( ( member_int @ A @ C2 )
=> ( ( member_int @ B @ D )
=> ( member_int @ ( times_times_int @ A @ B ) @ ( times_times_set_int @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_316_set__times__intro,axiom,
! [A: real,C2: set_real,B: real,D: set_real] :
( ( member_real @ A @ C2 )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( times_times_real @ A @ B ) @ ( times_times_set_real @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_317_Matrix_Otranspose__transpose,axiom,
! [A4: mat_a] :
( ( transpose_mat_a @ ( transpose_mat_a @ A4 ) )
= A4 ) ).
% Matrix.transpose_transpose
thf(fact_318_transpose__mat__eq,axiom,
! [A4: mat_a,B4: mat_a] :
( ( ( transpose_mat_a @ A4 )
= ( transpose_mat_a @ B4 ) )
= ( A4 = B4 ) ) ).
% transpose_mat_eq
thf(fact_319_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_320_add_Oright__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.right_neutral
thf(fact_321_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_322_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_323_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_324_double__zero__sym,axiom,
! [A: a] :
( ( zero_zero_a
= ( plus_plus_a @ A @ A ) )
= ( A = zero_zero_a ) ) ).
% double_zero_sym
thf(fact_325_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_326_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_327_add__cancel__left__left,axiom,
! [B: a,A: a] :
( ( ( plus_plus_a @ B @ A )
= A )
= ( B = zero_zero_a ) ) ).
% add_cancel_left_left
thf(fact_328_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_329_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_330_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_331_add__cancel__left__right,axiom,
! [A: a,B: a] :
( ( ( plus_plus_a @ A @ B )
= A )
= ( B = zero_zero_a ) ) ).
% add_cancel_left_right
thf(fact_332_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_333_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_334_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_335_add__cancel__right__left,axiom,
! [A: a,B: a] :
( ( A
= ( plus_plus_a @ B @ A ) )
= ( B = zero_zero_a ) ) ).
% add_cancel_right_left
thf(fact_336_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_337_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_338_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_339_add__cancel__right__right,axiom,
! [A: a,B: a] :
( ( A
= ( plus_plus_a @ A @ B ) )
= ( B = zero_zero_a ) ) ).
% add_cancel_right_right
thf(fact_340_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_341_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_342_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_343_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_344_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_345_add__0,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% add_0
thf(fact_346_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_347_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_348_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_349_mult_Oright__neutral,axiom,
! [A: a] :
( ( times_times_a @ A @ one_one_a )
= A ) ).
% mult.right_neutral
thf(fact_350_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_351_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_352_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_353_mult__1,axiom,
! [A: a] :
( ( times_times_a @ one_one_a @ A )
= A ) ).
% mult_1
thf(fact_354_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_355_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_356_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_357_assoc__mult__mat,axiom,
! [A4: mat_a,N_1: nat,N_2: nat,B4: mat_a,N_3: nat,C2: mat_a,N_4: nat] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ N_1 @ N_2 ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ N_2 @ N_3 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N_3 @ N_4 ) )
=> ( ( times_times_mat_a @ ( times_times_mat_a @ A4 @ B4 ) @ C2 )
= ( times_times_mat_a @ A4 @ ( times_times_mat_a @ B4 @ C2 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_358_transpose__carrier__mat,axiom,
! [A4: mat_a,Nc: nat,Nr: nat] :
( ( member_mat_a @ ( transpose_mat_a @ A4 ) @ ( carrier_mat_a @ Nc @ Nr ) )
= ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_359_add__le__same__cancel1,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ B @ A ) @ B )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% add_le_same_cancel1
thf(fact_360_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_361_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_362_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_363_add__le__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ B )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% add_le_same_cancel2
thf(fact_364_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_365_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_366_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_367_le__add__same__cancel1,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ ( plus_plus_a @ A @ B ) )
= ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% le_add_same_cancel1
thf(fact_368_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_369_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_370_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_371_le__add__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ ( plus_plus_a @ B @ A ) )
= ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% le_add_same_cancel2
thf(fact_372_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_373_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_374_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_375_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ A ) @ zero_zero_a )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_376_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_377_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_378_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ A @ A ) )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_379_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_380_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_381_set__times__elim,axiom,
! [X: mat_a,A4: set_mat_a,B4: set_mat_a] :
( ( member_mat_a @ X @ ( times_1230744552615602198_mat_a @ A4 @ B4 ) )
=> ~ ! [A3: mat_a,B3: mat_a] :
( ( X
= ( times_times_mat_a @ A3 @ B3 ) )
=> ( ( member_mat_a @ A3 @ A4 )
=> ~ ( member_mat_a @ B3 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_382_set__times__elim,axiom,
! [X: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ X @ ( times_times_set_nat @ A4 @ B4 ) )
=> ~ ! [A3: nat,B3: nat] :
( ( X
= ( times_times_nat @ A3 @ B3 ) )
=> ( ( member_nat @ A3 @ A4 )
=> ~ ( member_nat @ B3 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_383_set__times__elim,axiom,
! [X: int,A4: set_int,B4: set_int] :
( ( member_int @ X @ ( times_times_set_int @ A4 @ B4 ) )
=> ~ ! [A3: int,B3: int] :
( ( X
= ( times_times_int @ A3 @ B3 ) )
=> ( ( member_int @ A3 @ A4 )
=> ~ ( member_int @ B3 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_384_set__times__elim,axiom,
! [X: real,A4: set_real,B4: set_real] :
( ( member_real @ X @ ( times_times_set_real @ A4 @ B4 ) )
=> ~ ! [A3: real,B3: real] :
( ( X
= ( times_times_real @ A3 @ B3 ) )
=> ( ( member_real @ A3 @ A4 )
=> ~ ( member_real @ B3 @ B4 ) ) ) ) ).
% set_times_elim
thf(fact_385_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_386_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_387_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_388_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_389_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_390_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_391_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_392_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_393_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_394_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_395_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_396_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_397_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_398_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_399_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_400_transpose__mult,axiom,
! [A4: mat_a,Nr: nat,N: nat,B4: mat_a,Nc: nat] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( transpose_mat_a @ ( times_times_mat_a @ A4 @ B4 ) )
= ( times_times_mat_a @ ( transpose_mat_a @ B4 ) @ ( transpose_mat_a @ A4 ) ) ) ) ) ).
% transpose_mult
thf(fact_401_le__numeral__extra_I3_J,axiom,
ord_less_eq_a @ zero_zero_a @ zero_zero_a ).
% le_numeral_extra(3)
thf(fact_402_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_403_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_404_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_405_transpose__add,axiom,
! [A4: mat_a,Nr: nat,Nc: nat,B4: mat_a] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( transpose_mat_a @ ( plus_plus_mat_a @ A4 @ B4 ) )
= ( plus_plus_mat_a @ ( transpose_mat_a @ A4 ) @ ( transpose_mat_a @ B4 ) ) ) ) ) ).
% transpose_add
thf(fact_406_comm__monoid__mult__class_Omult__1,axiom,
! [A: a] :
( ( times_times_a @ one_one_a @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_407_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_408_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_409_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_410_mult_Ocomm__neutral,axiom,
! [A: a] :
( ( times_times_a @ A @ one_one_a )
= A ) ).
% mult.comm_neutral
thf(fact_411_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_412_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_413_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_414_mult__carrier__mat,axiom,
! [A4: mat_a,Nr: nat,N: nat,B4: mat_a,Nc: nat] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ N @ Nc ) )
=> ( member_mat_a @ ( times_times_mat_a @ A4 @ B4 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_415_mult__add__distrib__mat,axiom,
! [A4: mat_a,Nr: nat,N: nat,B4: mat_a,Nc: nat,C2: mat_a] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ A4 @ ( plus_plus_mat_a @ B4 @ C2 ) )
= ( plus_plus_mat_a @ ( times_times_mat_a @ A4 @ B4 ) @ ( times_times_mat_a @ A4 @ C2 ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_416_add__mult__distrib__mat,axiom,
! [A4: mat_a,Nr: nat,N: nat,B4: mat_a,C2: mat_a,Nc: nat] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( plus_plus_mat_a @ A4 @ B4 ) @ C2 )
= ( plus_plus_mat_a @ ( times_times_mat_a @ A4 @ C2 ) @ ( times_times_mat_a @ B4 @ C2 ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_417_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_418_verit__sum__simplify,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% verit_sum_simplify
thf(fact_419_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_420_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_421_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_422_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_423_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_424_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_425_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_426_add_Ocomm__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.comm_neutral
thf(fact_427_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_428_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_429_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_430_add_Ogroup__left__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% add.group_left_neutral
thf(fact_431_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_432_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_433_set__zero__plus2,axiom,
! [A4: set_nat,B4: set_nat] :
( ( member_nat @ zero_zero_nat @ A4 )
=> ( ord_less_eq_set_nat @ B4 @ ( plus_plus_set_nat @ A4 @ B4 ) ) ) ).
% set_zero_plus2
thf(fact_434_set__zero__plus2,axiom,
! [A4: set_int,B4: set_int] :
( ( member_int @ zero_zero_int @ A4 )
=> ( ord_less_eq_set_int @ B4 @ ( plus_plus_set_int @ A4 @ B4 ) ) ) ).
% set_zero_plus2
thf(fact_435_set__zero__plus2,axiom,
! [A4: set_real,B4: set_real] :
( ( member_real @ zero_zero_real @ A4 )
=> ( ord_less_eq_set_real @ B4 @ ( plus_plus_set_real @ A4 @ B4 ) ) ) ).
% set_zero_plus2
thf(fact_436_add__decreasing,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_437_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_438_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_439_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_440_add__increasing,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_441_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_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__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_444_add__decreasing2,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_445_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_446_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_447_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_448_add__increasing2,axiom,
! [C: a,B: a,A: a] :
( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ( ord_less_eq_a @ B @ A )
=> ( ord_less_eq_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_449_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_450_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_451_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_452_add__nonneg__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_453_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_454_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_455_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_456_add__nonpos__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_nonpos_nonpos
thf(fact_457_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_458_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_459_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_460_add__nonneg__eq__0__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ( ( plus_plus_a @ X @ Y )
= zero_zero_a )
= ( ( X = zero_zero_a )
& ( Y = zero_zero_a ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_461_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_462_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_463_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_464_add__nonpos__eq__0__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ zero_zero_a )
=> ( ( ord_less_eq_a @ Y @ zero_zero_a )
=> ( ( ( plus_plus_a @ X @ Y )
= zero_zero_a )
= ( ( X = zero_zero_a )
& ( Y = zero_zero_a ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_465_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_466_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_467_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_468_is__num__normalize_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 ) ) ) ).
% is_num_normalize(1)
thf(fact_469_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_470_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_471_comm__add__vecset,axiom,
! [A4: set_vec_a,N: nat,B4: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A4 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ B4 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_set_vec_a @ A4 @ B4 )
= ( plus_plus_set_vec_a @ B4 @ A4 ) ) ) ) ).
% comm_add_vecset
thf(fact_472_assoc__add__vecset,axiom,
! [A4: set_vec_a,N: nat,B4: set_vec_a,C2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A4 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ B4 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ C2 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_set_vec_a @ A4 @ ( plus_plus_set_vec_a @ B4 @ C2 ) )
= ( plus_plus_set_vec_a @ ( plus_plus_set_vec_a @ A4 @ B4 ) @ C2 ) ) ) ) ) ).
% assoc_add_vecset
thf(fact_473_gram__schmidt_OFarkas__Lemma_H,axiom,
! [A4: mat_real,Nr: nat,Nc: nat,B: vec_real] :
( ( member_mat_real @ A4 @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ( member_vec_real @ B @ ( carrier_vec_real @ Nr ) )
=> ( ( ? [X2: vec_real] :
( ( member_vec_real @ X2 @ ( carrier_vec_real @ Nc ) )
& ( ord_less_eq_vec_real @ ( mult_mat_vec_real @ A4 @ X2 ) @ B ) ) )
= ( ! [Y3: vec_real] :
( ( ( ord_less_eq_vec_real @ ( zero_vec_real @ Nr ) @ Y3 )
& ( ( mult_mat_vec_real @ ( transpose_mat_real @ A4 ) @ Y3 )
= ( zero_vec_real @ Nc ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( scalar_prod_real @ Y3 @ B ) ) ) ) ) ) ) ).
% gram_schmidt.Farkas_Lemma'
thf(fact_474_gram__schmidt_OFarkas__Lemma,axiom,
! [A4: mat_real,N: nat,Nr: nat,B: vec_real] :
( ( member_mat_real @ A4 @ ( carrier_mat_real @ N @ Nr ) )
=> ( ( member_vec_real @ B @ ( carrier_vec_real @ N ) )
=> ( ( ? [X2: vec_real] :
( ( ord_less_eq_vec_real @ ( zero_vec_real @ Nr ) @ X2 )
& ( ( mult_mat_vec_real @ A4 @ X2 )
= B ) ) )
= ( ! [Y3: vec_real] :
( ( member_vec_real @ Y3 @ ( carrier_vec_real @ N ) )
=> ( ( ord_less_eq_vec_real @ ( zero_vec_real @ Nr ) @ ( mult_mat_vec_real @ ( transpose_mat_real @ A4 ) @ Y3 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( scalar_prod_real @ Y3 @ B ) ) ) ) ) ) ) ) ).
% gram_schmidt.Farkas_Lemma
thf(fact_475_sum__squares__eq__zero__iff,axiom,
! [X: a,Y: a] :
( ( ( plus_plus_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) )
= zero_zero_a )
= ( ( X = zero_zero_a )
& ( Y = zero_zero_a ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_476_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_477_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_478_mult__cancel__left1,axiom,
! [C: a,B: a] :
( ( C
= ( times_times_a @ C @ B ) )
= ( ( C = zero_zero_a )
| ( B = one_one_a ) ) ) ).
% mult_cancel_left1
thf(fact_479_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_480_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_481_mult__cancel__left2,axiom,
! [C: a,A: a] :
( ( ( times_times_a @ C @ A )
= C )
= ( ( C = zero_zero_a )
| ( A = one_one_a ) ) ) ).
% mult_cancel_left2
thf(fact_482_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_483_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_484_mult__cancel__right1,axiom,
! [C: a,B: a] :
( ( C
= ( times_times_a @ B @ C ) )
= ( ( C = zero_zero_a )
| ( B = one_one_a ) ) ) ).
% mult_cancel_right1
thf(fact_485_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_486_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_487_mult__cancel__right2,axiom,
! [A: a,C: a] :
( ( ( times_times_a @ A @ C )
= C )
= ( ( C = zero_zero_a )
| ( A = one_one_a ) ) ) ).
% mult_cancel_right2
thf(fact_488_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_489_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_490_convex__bound__le,axiom,
! [X: a,A: a,Y: a,U: a,V2: a] :
( ( ord_less_eq_a @ X @ A )
=> ( ( ord_less_eq_a @ Y @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ U )
=> ( ( ord_less_eq_a @ zero_zero_a @ V2 )
=> ( ( ( plus_plus_a @ U @ V2 )
= one_one_a )
=> ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ U @ X ) @ ( times_times_a @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_491_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V2: 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 @ V2 )
=> ( ( ( plus_plus_int @ U @ V2 )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_492_convex__bound__le,axiom,
! [X: real,A: real,Y: real,U: real,V2: 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 @ V2 )
=> ( ( ( plus_plus_real @ U @ V2 )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_493_double__eq__0__iff,axiom,
! [A: a] :
( ( ( plus_plus_a @ A @ A )
= zero_zero_a )
= ( A = zero_zero_a ) ) ).
% double_eq_0_iff
thf(fact_494_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_495_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_496_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_497_mult__hom_Ohom__zero,axiom,
! [C: int] :
( ( times_times_int @ C @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_498_mult__hom_Ohom__zero,axiom,
! [C: real] :
( ( times_times_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_499_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_500_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_501_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_502_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_503_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_504_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_505_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_506_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_507_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_508_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_509_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_510_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_511_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_512_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_513_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_514_zero__neq__one,axiom,
zero_zero_a != one_one_a ).
% zero_neq_one
thf(fact_515_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_516_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_517_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
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: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_531_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_532_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_533_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_534_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_535_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_536_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_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_comm__semiring__class_Odistrib,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 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_546_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_547_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_548_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_549_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_550_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_551_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_552_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_553_ring__class_Oring__distribs_I1_J,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 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_554_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_555_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_556_ring__class_Oring__distribs_I2_J,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 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_557_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_558_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_559_not__one__le__zero,axiom,
~ ( ord_less_eq_a @ one_one_a @ zero_zero_a ) ).
% not_one_le_zero
thf(fact_560_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_561_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_562_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_563_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_a @ zero_zero_a @ one_one_a ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_564_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_565_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_566_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_567_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_a @ zero_zero_a @ one_one_a ).
% zero_less_one_class.zero_le_one
thf(fact_568_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_569_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_570_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_571_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_572_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_573_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_574_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_575_zero__le__mult__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ B ) )
= ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ zero_zero_a @ B ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ B @ zero_zero_a ) ) ) ) ).
% zero_le_mult_iff
thf(fact_576_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_577_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_578_mult__nonneg__nonpos2,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ B @ A ) @ zero_zero_a ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_579_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_580_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_581_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_582_mult__nonpos__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).
% mult_nonpos_nonneg
thf(fact_583_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_584_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_585_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_586_mult__nonneg__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).
% mult_nonneg_nonpos
thf(fact_587_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_588_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_589_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_590_mult__nonneg__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_591_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_592_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_593_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_594_split__mult__neg__le,axiom,
! [A: a,B: a] :
( ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ B @ zero_zero_a ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ zero_zero_a @ B ) ) )
=> ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ).
% split_mult_neg_le
thf(fact_595_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_596_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_597_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_598_mult__le__0__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a )
= ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ B @ zero_zero_a ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ zero_zero_a @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_599_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_600_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_601_mult__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_602_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_603_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_604_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_605_mult__right__mono__neg,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_606_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_607_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_608_mult__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_609_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_610_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_611_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_612_mult__nonpos__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_613_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_614_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_615_mult__left__mono__neg,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_616_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_617_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_618_split__mult__pos__le,axiom,
! [A: a,B: a] :
( ( ( ( ord_less_eq_a @ zero_zero_a @ A )
& ( ord_less_eq_a @ zero_zero_a @ B ) )
| ( ( ord_less_eq_a @ A @ zero_zero_a )
& ( ord_less_eq_a @ B @ zero_zero_a ) ) )
=> ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_619_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_620_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_621_zero__le__square,axiom,
! [A: a] : ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ A ) ) ).
% zero_le_square
thf(fact_622_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_623_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_624_mult__mono_H,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D2 )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_625_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_626_mult__mono_H,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_627_mult__mono_H,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_628_mult__mono,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D2 )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_629_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_630_mult__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_631_mult__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_632_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_633_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_634_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_635_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_636_mult__left__le,axiom,
! [C: a,A: a] :
( ( ord_less_eq_a @ C @ one_one_a )
=> ( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_637_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_638_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_639_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_640_mult__le__one,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ one_one_a )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ B @ one_one_a )
=> ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ one_one_a ) ) ) ) ).
% mult_le_one
thf(fact_641_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_642_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_643_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_644_mult__right__le__one__le,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ( ord_less_eq_a @ Y @ one_one_a )
=> ( ord_less_eq_a @ ( times_times_a @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_645_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_646_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_647_mult__left__le__one__le,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ zero_zero_a @ X )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y )
=> ( ( ord_less_eq_a @ Y @ one_one_a )
=> ( ord_less_eq_a @ ( times_times_a @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_648_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_649_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_650_sum__squares__le__zero__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) ) @ zero_zero_a )
= ( ( X = zero_zero_a )
& ( Y = zero_zero_a ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_651_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_652_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_653_sum__squares__ge__zero,axiom,
! [X: a,Y: a] : ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ ( times_times_a @ X @ X ) @ ( times_times_a @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_654_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_655_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_656_scalar__prod__ge__0,axiom,
! [X: vec_a] : ( ord_less_eq_a @ zero_zero_a @ ( scalar_prod_a @ X @ X ) ) ).
% scalar_prod_ge_0
thf(fact_657_scalar__prod__ge__0,axiom,
! [X: vec_int] : ( ord_less_eq_int @ zero_zero_int @ ( scalar_prod_int @ X @ X ) ) ).
% scalar_prod_ge_0
thf(fact_658_scalar__prod__ge__0,axiom,
! [X: vec_real] : ( ord_less_eq_real @ zero_zero_real @ ( scalar_prod_real @ X @ X ) ) ).
% scalar_prod_ge_0
thf(fact_659_add__scale__eq__noteq,axiom,
! [R: a,A: a,B: a,C: a,D2: a] :
( ( R != zero_zero_a )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( ( plus_plus_a @ A @ ( times_times_a @ R @ C ) )
!= ( plus_plus_a @ B @ ( times_times_a @ R @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_660_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D2: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_661_add__scale__eq__noteq,axiom,
! [R: int,A: int,B: int,C: int,D2: int] :
( ( R != zero_zero_int )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_662_add__scale__eq__noteq,axiom,
! [R: real,A: real,B: real,C: real,D2: real] :
( ( R != zero_zero_real )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_663_square__lesseq__square,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 @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% square_lesseq_square
thf(fact_664_less__eq__fract__respect,axiom,
! [B: a,B5: a,D2: a,D3: a,A: a,A5: a,C: a,C5: a] :
( ( B != zero_zero_a )
=> ( ( B5 != zero_zero_a )
=> ( ( D2 != zero_zero_a )
=> ( ( D3 != zero_zero_a )
=> ( ( ( times_times_a @ A @ B5 )
= ( times_times_a @ A5 @ B ) )
=> ( ( ( times_times_a @ C @ D3 )
= ( times_times_a @ C5 @ D2 ) )
=> ( ( ord_less_eq_a @ ( times_times_a @ ( times_times_a @ A @ D2 ) @ ( times_times_a @ B @ D2 ) ) @ ( times_times_a @ ( times_times_a @ C @ B ) @ ( times_times_a @ B @ D2 ) ) )
= ( ord_less_eq_a @ ( times_times_a @ ( times_times_a @ A5 @ D3 ) @ ( times_times_a @ B5 @ D3 ) ) @ ( times_times_a @ ( times_times_a @ C5 @ B5 ) @ ( times_times_a @ B5 @ D3 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_665_less__eq__fract__respect,axiom,
! [B: int,B5: int,D2: int,D3: int,A: int,A5: int,C: int,C5: int] :
( ( B != zero_zero_int )
=> ( ( B5 != zero_zero_int )
=> ( ( D2 != zero_zero_int )
=> ( ( D3 != zero_zero_int )
=> ( ( ( times_times_int @ A @ B5 )
= ( times_times_int @ A5 @ B ) )
=> ( ( ( times_times_int @ C @ D3 )
= ( times_times_int @ C5 @ D2 ) )
=> ( ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D2 ) ) )
= ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A5 @ D3 ) @ ( times_times_int @ B5 @ D3 ) ) @ ( times_times_int @ ( times_times_int @ C5 @ B5 ) @ ( times_times_int @ B5 @ D3 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_666_less__eq__fract__respect,axiom,
! [B: real,B5: real,D2: real,D3: real,A: real,A5: real,C: real,C5: real] :
( ( B != zero_zero_real )
=> ( ( B5 != zero_zero_real )
=> ( ( D2 != zero_zero_real )
=> ( ( D3 != zero_zero_real )
=> ( ( ( times_times_real @ A @ B5 )
= ( times_times_real @ A5 @ B ) )
=> ( ( ( times_times_real @ C @ D3 )
= ( times_times_real @ C5 @ D2 ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A @ D2 ) @ ( times_times_real @ B @ D2 ) ) @ ( times_times_real @ ( times_times_real @ C @ B ) @ ( times_times_real @ B @ D2 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A5 @ D3 ) @ ( times_times_real @ B5 @ D3 ) ) @ ( times_times_real @ ( times_times_real @ C5 @ B5 ) @ ( times_times_real @ B5 @ D3 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_667_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_668_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_669_subsetI,axiom,
! [A4: set_vec_a,B4: set_vec_a] :
( ! [X3: vec_a] :
( ( member_vec_a @ X3 @ A4 )
=> ( member_vec_a @ X3 @ B4 ) )
=> ( ord_le4791951621262958845_vec_a @ A4 @ B4 ) ) ).
% subsetI
thf(fact_670_subsetI,axiom,
! [A4: set_mat_a,B4: set_mat_a] :
( ! [X3: mat_a] :
( ( member_mat_a @ X3 @ A4 )
=> ( member_mat_a @ X3 @ B4 ) )
=> ( ord_le3318621148231462513_mat_a @ A4 @ B4 ) ) ).
% subsetI
thf(fact_671_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_672_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_673_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_674_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_675_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_676_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_677_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_678_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_679_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_680_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_681_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_682_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_683_in__mono,axiom,
! [A4: set_vec_a,B4: set_vec_a,X: vec_a] :
( ( ord_le4791951621262958845_vec_a @ A4 @ B4 )
=> ( ( member_vec_a @ X @ A4 )
=> ( member_vec_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_684_in__mono,axiom,
! [A4: set_mat_a,B4: set_mat_a,X: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A4 @ B4 )
=> ( ( member_mat_a @ X @ A4 )
=> ( member_mat_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_685_subsetD,axiom,
! [A4: set_vec_a,B4: set_vec_a,C: vec_a] :
( ( ord_le4791951621262958845_vec_a @ A4 @ B4 )
=> ( ( member_vec_a @ C @ A4 )
=> ( member_vec_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_686_subsetD,axiom,
! [A4: set_mat_a,B4: set_mat_a,C: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A4 @ B4 )
=> ( ( member_mat_a @ C @ A4 )
=> ( member_mat_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_687_subset__eq,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [A6: set_vec_a,B6: set_vec_a] :
! [X2: vec_a] :
( ( member_vec_a @ X2 @ A6 )
=> ( member_vec_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_688_subset__eq,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A6: set_mat_a,B6: set_mat_a] :
! [X2: mat_a] :
( ( member_mat_a @ X2 @ A6 )
=> ( member_mat_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_689_subset__iff,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [A6: set_vec_a,B6: set_vec_a] :
! [T: vec_a] :
( ( member_vec_a @ T @ A6 )
=> ( member_vec_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_690_subset__iff,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A6: set_mat_a,B6: set_mat_a] :
! [T: mat_a] :
( ( member_mat_a @ T @ A6 )
=> ( member_mat_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_691_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_692_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_693_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_694_add__0__iff,axiom,
! [B: a,A: a] :
( ( B
= ( plus_plus_a @ B @ A ) )
= ( A = zero_zero_a ) ) ).
% add_0_iff
thf(fact_695_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_696_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_697_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_698_crossproduct__eq,axiom,
! [W: a,Y: a,X: a,Z: a] :
( ( ( plus_plus_a @ ( times_times_a @ W @ Y ) @ ( times_times_a @ X @ Z ) )
= ( plus_plus_a @ ( times_times_a @ W @ Z ) @ ( times_times_a @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_699_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_700_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_701_crossproduct__eq,axiom,
! [W: real,Y: real,X: real,Z: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_702_crossproduct__noteq,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_plus_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D2 ) )
!= ( plus_plus_a @ ( times_times_a @ A @ D2 ) @ ( times_times_a @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_703_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_704_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_705_crossproduct__noteq,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D2 ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D2 ) @ ( times_times_real @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_706_norm1__ge__0,axiom,
! [F: poly_a] : ( ord_less_eq_a @ zero_zero_a @ ( norm1_a @ F ) ) ).
% norm1_ge_0
thf(fact_707_norm1__ge__0,axiom,
! [F: poly_int] : ( ord_less_eq_int @ zero_zero_int @ ( norm1_int @ F ) ) ).
% norm1_ge_0
thf(fact_708_norm1__ge__0,axiom,
! [F: poly_real] : ( ord_less_eq_real @ zero_zero_real @ ( norm1_real @ F ) ) ).
% norm1_ge_0
thf(fact_709_linf__norm__vec__eq__0,axiom,
! [V2: vec_a,N: nat] :
( ( member_vec_a @ V2 @ ( carrier_vec_a @ N ) )
=> ( ( ( linf_norm_vec_a @ V2 )
= zero_zero_a )
= ( V2
= ( zero_vec_a @ N ) ) ) ) ).
% linf_norm_vec_eq_0
thf(fact_710_linf__norm__vec__eq__0,axiom,
! [V2: vec_int,N: nat] :
( ( member_vec_int @ V2 @ ( carrier_vec_int @ N ) )
=> ( ( ( linf_norm_vec_int @ V2 )
= zero_zero_int )
= ( V2
= ( zero_vec_int @ N ) ) ) ) ).
% linf_norm_vec_eq_0
thf(fact_711_linf__norm__vec__eq__0,axiom,
! [V2: vec_real,N: nat] :
( ( member_vec_real @ V2 @ ( carrier_vec_real @ N ) )
=> ( ( ( linf_norm_vec_real @ V2 )
= zero_zero_real )
= ( V2
= ( zero_vec_real @ N ) ) ) ) ).
% linf_norm_vec_eq_0
thf(fact_712_class__field_Ozero__not__one,axiom,
zero_zero_real != one_one_real ).
% class_field.zero_not_one
thf(fact_713_dbl__inc__simps_I2_J,axiom,
( ( neg_nu6917059380386235053_inc_a @ zero_zero_a )
= one_one_a ) ).
% dbl_inc_simps(2)
thf(fact_714_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_715_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_716_sum__mat__add,axiom,
! [A4: mat_a,Nr: nat,Nc: nat,B4: mat_a] :
( ( member_mat_a @ A4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( comm_m5291664705200495434_mat_a @ ( plus_plus_mat_a @ A4 @ B4 ) )
= ( plus_plus_a @ ( comm_m5291664705200495434_mat_a @ A4 ) @ ( comm_m5291664705200495434_mat_a @ B4 ) ) ) ) ) ).
% sum_mat_add
thf(fact_717_sum__mat__add,axiom,
! [A4: mat_nat,Nr: nat,Nc: nat,B4: mat_nat] :
( ( member_mat_nat @ A4 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ B4 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( comm_m4056229327131402372at_nat @ ( plus_plus_mat_nat @ A4 @ B4 ) )
= ( plus_plus_nat @ ( comm_m4056229327131402372at_nat @ A4 ) @ ( comm_m4056229327131402372at_nat @ B4 ) ) ) ) ) ).
% sum_mat_add
thf(fact_718_sum__mat__add,axiom,
! [A4: mat_int,Nr: nat,Nc: nat,B4: mat_int] :
( ( member_mat_int @ A4 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( member_mat_int @ B4 @ ( carrier_mat_int @ Nr @ Nc ) )
=> ( ( comm_m4053738856622352096at_int @ ( plus_plus_mat_int @ A4 @ B4 ) )
= ( plus_plus_int @ ( comm_m4053738856622352096at_int @ A4 ) @ ( comm_m4053738856622352096at_int @ B4 ) ) ) ) ) ).
% sum_mat_add
thf(fact_719_sum__mat__add,axiom,
! [A4: mat_real,Nr: nat,Nc: nat,B4: mat_real] :
( ( member_mat_real @ A4 @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ( member_mat_real @ B4 @ ( carrier_mat_real @ Nr @ Nc ) )
=> ( ( comm_m8678487124704766304t_real @ ( plus_plus_mat_real @ A4 @ B4 ) )
= ( plus_plus_real @ ( comm_m8678487124704766304t_real @ A4 ) @ ( comm_m8678487124704766304t_real @ B4 ) ) ) ) ) ).
% sum_mat_add
thf(fact_720_scalar__prod__vCons,axiom,
! [A: a,V2: vec_a,B: a,W: vec_a] :
( ( scalar_prod_a @ ( vCons_a @ A @ V2 ) @ ( vCons_a @ B @ W ) )
= ( plus_plus_a @ ( times_times_a @ A @ B ) @ ( scalar_prod_a @ V2 @ W ) ) ) ).
% scalar_prod_vCons
thf(fact_721_scalar__prod__vCons,axiom,
! [A: nat,V2: vec_nat,B: nat,W: vec_nat] :
( ( scalar_prod_nat @ ( vCons_nat @ A @ V2 ) @ ( vCons_nat @ B @ W ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( scalar_prod_nat @ V2 @ W ) ) ) ).
% scalar_prod_vCons
thf(fact_722_scalar__prod__vCons,axiom,
! [A: int,V2: vec_int,B: int,W: vec_int] :
( ( scalar_prod_int @ ( vCons_int @ A @ V2 ) @ ( vCons_int @ B @ W ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( scalar_prod_int @ V2 @ W ) ) ) ).
% scalar_prod_vCons
thf(fact_723_scalar__prod__vCons,axiom,
! [A: real,V2: vec_real,B: real,W: vec_real] :
( ( scalar_prod_real @ ( vCons_real @ A @ V2 ) @ ( vCons_real @ B @ W ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( scalar_prod_real @ V2 @ W ) ) ) ).
% scalar_prod_vCons
thf(fact_724_convex__bound__lt,axiom,
! [X: a,A: a,Y: a,U: a,V2: a] :
( ( ord_less_a @ X @ A )
=> ( ( ord_less_a @ Y @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ U )
=> ( ( ord_less_eq_a @ zero_zero_a @ V2 )
=> ( ( ( plus_plus_a @ U @ V2 )
= one_one_a )
=> ( ord_less_a @ ( plus_plus_a @ ( times_times_a @ U @ X ) @ ( times_times_a @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_725_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V2: 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 @ V2 )
=> ( ( ( plus_plus_int @ U @ V2 )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_726_convex__bound__lt,axiom,
! [X: real,A: real,Y: real,U: real,V2: 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 @ V2 )
=> ( ( ( plus_plus_real @ U @ V2 )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_727_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_728_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_729_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_730_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_731_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_732_add__less__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( ord_less_a @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_733_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_734_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_735_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_736_add__less__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( ord_less_a @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_737_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_738_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_739_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_740_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_741_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_742_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_743_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_744_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_745_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_746_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_747_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_748_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_749_add__less__same__cancel1,axiom,
! [B: a,A: a] :
( ( ord_less_a @ ( plus_plus_a @ B @ A ) @ B )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% add_less_same_cancel1
thf(fact_750_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_751_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_752_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_753_add__less__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_a @ ( plus_plus_a @ A @ B ) @ B )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% add_less_same_cancel2
thf(fact_754_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_755_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_756_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_757_less__add__same__cancel1,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ ( plus_plus_a @ A @ B ) )
= ( ord_less_a @ zero_zero_a @ B ) ) ).
% less_add_same_cancel1
thf(fact_758_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_759_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_760_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_761_less__add__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ ( plus_plus_a @ B @ A ) )
= ( ord_less_a @ zero_zero_a @ B ) ) ).
% less_add_same_cancel2
thf(fact_762_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_763_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_764_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_765_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: a] :
( ( ord_less_a @ ( plus_plus_a @ A @ A ) @ zero_zero_a )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_766_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_767_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_768_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ A ) )
= ( ord_less_a @ zero_zero_a @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_769_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_770_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_771_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_772_linf__norm__vec__ge__0,axiom,
! [V2: vec_a] : ( ord_less_eq_a @ zero_zero_a @ ( linf_norm_vec_a @ V2 ) ) ).
% linf_norm_vec_ge_0
thf(fact_773_linf__norm__vec__ge__0,axiom,
! [V2: vec_int] : ( ord_less_eq_int @ zero_zero_int @ ( linf_norm_vec_int @ V2 ) ) ).
% linf_norm_vec_ge_0
thf(fact_774_linf__norm__vec__ge__0,axiom,
! [V2: vec_real] : ( ord_less_eq_real @ zero_zero_real @ ( linf_norm_vec_real @ V2 ) ) ).
% linf_norm_vec_ge_0
thf(fact_775_linf__norm__zero__vec,axiom,
! [N: nat] :
( ( linf_norm_vec_a @ ( zero_vec_a @ N ) )
= zero_zero_a ) ).
% linf_norm_zero_vec
thf(fact_776_linf__norm__zero__vec,axiom,
! [N: nat] :
( ( linf_norm_vec_int @ ( zero_vec_int @ N ) )
= zero_zero_int ) ).
% linf_norm_zero_vec
thf(fact_777_linf__norm__zero__vec,axiom,
! [N: nat] :
( ( linf_norm_vec_real @ ( zero_vec_real @ N ) )
= zero_zero_real ) ).
% linf_norm_zero_vec
thf(fact_778_linf__norm__vec__greater__0,axiom,
! [V2: vec_a,N: nat] :
( ( member_vec_a @ V2 @ ( carrier_vec_a @ N ) )
=> ( ( ord_less_a @ zero_zero_a @ ( linf_norm_vec_a @ V2 ) )
= ( V2
!= ( zero_vec_a @ N ) ) ) ) ).
% linf_norm_vec_greater_0
thf(fact_779_linf__norm__vec__greater__0,axiom,
! [V2: vec_int,N: nat] :
( ( member_vec_int @ V2 @ ( carrier_vec_int @ N ) )
=> ( ( ord_less_int @ zero_zero_int @ ( linf_norm_vec_int @ V2 ) )
= ( V2
!= ( zero_vec_int @ N ) ) ) ) ).
% linf_norm_vec_greater_0
thf(fact_780_linf__norm__vec__greater__0,axiom,
! [V2: vec_real,N: nat] :
( ( member_vec_real @ V2 @ ( carrier_vec_real @ N ) )
=> ( ( ord_less_real @ zero_zero_real @ ( linf_norm_vec_real @ V2 ) )
= ( V2
!= ( zero_vec_real @ N ) ) ) ) ).
% linf_norm_vec_greater_0
thf(fact_781_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_782_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_783_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_784_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_785_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_786_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_787_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_788_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_789_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_790_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_791_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_792_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_793_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_794_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_795_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_796_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_797_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_798_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_799_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_800_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_801_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_802_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_803_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_804_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_805_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_806_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_807_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_808_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_809_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_810_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_811_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_812_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_813_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_814_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_815_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_816_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_817_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_818_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_819_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_820_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_821_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_822_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_823_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_824_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_825_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_826_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_827_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_828_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_829_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_830_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_831_lt__ex,axiom,
! [X: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% lt_ex
thf(fact_832_lt__ex,axiom,
! [X: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).
% lt_ex
thf(fact_833_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_834_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_835_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_836_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_837_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_838_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_839_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_840_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_841_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_842_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_843_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_844_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_845_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_846_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_847_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_848_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_849_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_850_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_851_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_852_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_853_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_854_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_855_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_856_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_857_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_858_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_859_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_860_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_861_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_862_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_863_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_864_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_865_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_866_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_867_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_868_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_869_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_870_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_871_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_872_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_873_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_874_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_875_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_876_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_877_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_878_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_879_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_880_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_881_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_882_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_883_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_884_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_885_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_886_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_887_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_888_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_889_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_890_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_891_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_892_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_893_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_894_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_895_order__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_896_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_897_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_898_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_899_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_900_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_901_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_902_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_903_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_904_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_905_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_906_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_907_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_908_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_909_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_910_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_911_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_912_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_913_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_914_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_915_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_916_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_917_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,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_918_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,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_919_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,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_920_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,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_921_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,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_922_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,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_923_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,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_924_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,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_925_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,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_926_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,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_927_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,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_928_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,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_929_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,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_930_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,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_931_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,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_932_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,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_933_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,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_934_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,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_945_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_946_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_947_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_948_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_949_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_950_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_951_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_952_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_953_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_954_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_955_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_956_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_957_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_958_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_959_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_960_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_961_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_962_real__linear,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ( ord_less_nat @ A @ B )
| ( A = B )
| ( ord_less_nat @ B @ A ) ) ) ) ).
% real_linear
thf(fact_963_real__linear,axiom,
! [A: int,B: int] :
( ( ( ord_less_int @ A @ zero_zero_int )
| ( A = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ A ) )
=> ( ( ( ord_less_int @ B @ zero_zero_int )
| ( B = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ B ) )
=> ( ( ord_less_int @ A @ B )
| ( A = B )
| ( ord_less_int @ B @ A ) ) ) ) ).
% real_linear
thf(fact_964_real__linear,axiom,
! [A: real,B: real] :
( ( ( ord_less_real @ A @ zero_zero_real )
| ( A = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ A ) )
=> ( ( ( ord_less_real @ B @ zero_zero_real )
| ( B = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ B ) )
=> ( ( ord_less_real @ A @ B )
| ( A = B )
| ( ord_less_real @ B @ A ) ) ) ) ).
% real_linear
thf(fact_965_neg__neg__linear,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ( ord_less_nat @ A @ B )
| ( A = B )
| ( ord_less_nat @ B @ A ) ) ) ) ).
% neg_neg_linear
thf(fact_966_neg__neg__linear,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ A @ B )
| ( A = B )
| ( ord_less_int @ B @ A ) ) ) ) ).
% neg_neg_linear
thf(fact_967_neg__neg__linear,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_real @ A @ B )
| ( A = B )
| ( ord_less_real @ B @ A ) ) ) ) ).
% neg_neg_linear
thf(fact_968_pos__pos__linear,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ A @ B )
| ( A = B )
| ( ord_less_nat @ B @ A ) ) ) ) ).
% pos_pos_linear
thf(fact_969_pos__pos__linear,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ A @ B )
| ( A = B )
| ( ord_less_int @ B @ A ) ) ) ) ).
% pos_pos_linear
thf(fact_970_pos__pos__linear,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ A @ B )
| ( A = B )
| ( ord_less_real @ B @ A ) ) ) ) ).
% pos_pos_linear
thf(fact_971_real__linorder__cases,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_nat @ A @ zero_zero_nat )
| ( A = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ A ) )
=> ( ( ( ord_less_nat @ B @ zero_zero_nat )
| ( B = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ B ) )
=> ( ~ ( ord_less_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ B @ A ) ) ) ) ) ).
% real_linorder_cases
thf(fact_972_real__linorder__cases,axiom,
! [A: int,B: int] :
( ( ( ord_less_int @ A @ zero_zero_int )
| ( A = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ A ) )
=> ( ( ( ord_less_int @ B @ zero_zero_int )
| ( B = zero_zero_int )
| ( ord_less_int @ zero_zero_int @ B ) )
=> ( ~ ( ord_less_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ B @ A ) ) ) ) ) ).
% real_linorder_cases
thf(fact_973_real__linorder__cases,axiom,
! [A: real,B: real] :
( ( ( ord_less_real @ A @ zero_zero_real )
| ( A = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ A ) )
=> ( ( ( ord_less_real @ B @ zero_zero_real )
| ( B = zero_zero_real )
| ( ord_less_real @ zero_zero_real @ B ) )
=> ( ~ ( ord_less_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ B @ A ) ) ) ) ) ).
% real_linorder_cases
thf(fact_974_verit__comp__simplify1_I3_J,axiom,
! [B5: a,A5: a] :
( ( ~ ( ord_less_eq_a @ B5 @ A5 ) )
= ( ord_less_a @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_975_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_976_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_977_verit__comp__simplify1_I3_J,axiom,
! [B5: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B5 @ A5 ) )
= ( ord_less_real @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_978_leD,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_a @ X @ Y ) ) ).
% leD
thf(fact_979_leD,axiom,
! [Y: vec_a,X: vec_a] :
( ( ord_less_eq_vec_a @ Y @ X )
=> ~ ( ord_less_vec_a @ X @ Y ) ) ).
% leD
thf(fact_980_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_981_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_982_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_983_leI,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% leI
thf(fact_984_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_985_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_986_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_987_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_988_nless__le,axiom,
! [A: vec_a,B: vec_a] :
( ( ~ ( ord_less_vec_a @ A @ B ) )
= ( ~ ( ord_less_eq_vec_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_989_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_990_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_991_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_992_antisym__conv1,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_993_antisym__conv1,axiom,
! [X: vec_a,Y: vec_a] :
( ~ ( ord_less_vec_a @ X @ Y )
=> ( ( ord_less_eq_vec_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_994_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_995_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_996_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_997_antisym__conv2,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ~ ( ord_less_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_998_antisym__conv2,axiom,
! [X: vec_a,Y: vec_a] :
( ( ord_less_eq_vec_a @ X @ Y )
=> ( ( ~ ( ord_less_vec_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_999_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1000_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1001_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1002_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_1003_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_1004_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
& ~ ( ord_less_eq_a @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1005_less__le__not__le,axiom,
( ord_less_vec_a
= ( ^ [X2: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X2 @ Y3 )
& ~ ( ord_less_eq_vec_a @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1006_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1007_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1008_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1009_not__le__imp__less,axiom,
! [Y: a,X: a] :
( ~ ( ord_less_eq_a @ Y @ X )
=> ( ord_less_a @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1010_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1011_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1012_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1013_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1014_order_Oorder__iff__strict,axiom,
( ord_less_eq_vec_a
= ( ^ [A2: vec_a,B2: vec_a] :
( ( ord_less_vec_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1015_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1016_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1017_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1018_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1019_order_Ostrict__iff__order,axiom,
( ord_less_vec_a
= ( ^ [A2: vec_a,B2: vec_a] :
( ( ord_less_eq_vec_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1020_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1021_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1022_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1023_order_Ostrict__trans1,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1024_order_Ostrict__trans1,axiom,
! [A: vec_a,B: vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_vec_a @ B @ C )
=> ( ord_less_vec_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1025_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1026_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1027_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1028_order_Ostrict__trans2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1029_order_Ostrict__trans2,axiom,
! [A: vec_a,B: vec_a,C: vec_a] :
( ( ord_less_vec_a @ A @ B )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ord_less_vec_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1030_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1031_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1032_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1033_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ~ ( ord_less_eq_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1034_order_Ostrict__iff__not,axiom,
( ord_less_vec_a
= ( ^ [A2: vec_a,B2: vec_a] :
( ( ord_less_eq_vec_a @ A2 @ B2 )
& ~ ( ord_less_eq_vec_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1035_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1036_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1037_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1038_dense__ge__bounded,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W2: real] :
( ( ord_less_real @ Z @ W2 )
=> ( ( ord_less_real @ W2 @ X )
=> ( ord_less_eq_real @ Y @ W2 ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_1039_dense__le__bounded,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W2: real] :
( ( ord_less_real @ X @ W2 )
=> ( ( ord_less_real @ W2 @ Y )
=> ( ord_less_eq_real @ W2 @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_1040_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1041_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_vec_a
= ( ^ [B2: vec_a,A2: vec_a] :
( ( ord_less_vec_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1042_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1043_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1044_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1045_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1046_dual__order_Ostrict__iff__order,axiom,
( ord_less_vec_a
= ( ^ [B2: vec_a,A2: vec_a] :
( ( ord_less_eq_vec_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1047_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1048_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1049_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1050_dual__order_Ostrict__trans1,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1051_dual__order_Ostrict__trans1,axiom,
! [B: vec_a,A: vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ B @ A )
=> ( ( ord_less_vec_a @ C @ B )
=> ( ord_less_vec_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1052_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1053_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1054_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1055_dual__order_Ostrict__trans2,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1056_dual__order_Ostrict__trans2,axiom,
! [B: vec_a,A: vec_a,C: vec_a] :
( ( ord_less_vec_a @ B @ A )
=> ( ( ord_less_eq_vec_a @ C @ B )
=> ( ord_less_vec_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1057_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1058_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1059_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1060_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ~ ( ord_less_eq_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1061_dual__order_Ostrict__iff__not,axiom,
( ord_less_vec_a
= ( ^ [B2: vec_a,A2: vec_a] :
( ( ord_less_eq_vec_a @ B2 @ A2 )
& ~ ( ord_less_eq_vec_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1062_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1063_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1064_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ~ ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1065_order_Ostrict__implies__order,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_eq_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1066_order_Ostrict__implies__order,axiom,
! [A: vec_a,B: vec_a] :
( ( ord_less_vec_a @ A @ B )
=> ( ord_less_eq_vec_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1067_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1068_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1069_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1070_dual__order_Ostrict__implies__order,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_eq_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1071_dual__order_Ostrict__implies__order,axiom,
! [B: vec_a,A: vec_a] :
( ( ord_less_vec_a @ B @ A )
=> ( ord_less_eq_vec_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1072_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1073_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1074_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1075_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1076_order__le__less,axiom,
( ord_less_eq_vec_a
= ( ^ [X2: vec_a,Y3: vec_a] :
( ( ord_less_vec_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1077_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1078_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1079_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1080_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_1081_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_1082_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_1083_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_1084_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_1085_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1086_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1087_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_1088_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1089_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1090_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1091_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_1092_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_1093_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_1094_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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_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_1101_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_1102_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1103_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1104_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1105_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_1106_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1107_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1108_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 )
=> ( ! [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1109_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_1110_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1111_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_1112_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1113_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1114_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1115_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_1116_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W3: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1117_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1118_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1119_nat__int__comparison_I1_J,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A2 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1120_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1121_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1122_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1123_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1124_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1125_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1126_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1127_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1128_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1129_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1130_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1131_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1132_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1133_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1134_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1135_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1136_pos__mult__pos__ge,axiom,
! [X: int,N: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ ( times_times_int @ N @ one_one_int ) @ ( times_times_int @ N @ X ) ) ) ) ).
% pos_mult_pos_ge
thf(fact_1137_linear__plus__1__le__power,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_1138_Bernoulli__inequality,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_1139_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_1140_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_1141_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_1142_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_1143_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_1144_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_1145_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_1146_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1147_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_1148_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1149_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_1150_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_1151_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1152_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q: nat] :
( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q ) )
= ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q ) ) ) ).
% nat_mult_max_right
thf(fact_1153_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q: nat] :
( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q )
= ( ord_max_nat @ ( times_times_nat @ M @ Q ) @ ( times_times_nat @ N @ Q ) ) ) ).
% nat_mult_max_left
thf(fact_1154_nat__add__max__right,axiom,
! [M: nat,N: nat,Q: nat] :
( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q ) )
= ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q ) ) ) ).
% nat_add_max_right
thf(fact_1155_nat__add__max__left,axiom,
! [M: nat,N: nat,Q: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q )
= ( ord_max_nat @ ( plus_plus_nat @ M @ Q ) @ ( plus_plus_nat @ N @ Q ) ) ) ).
% nat_add_max_left
thf(fact_1156_int__cases2,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_1157_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_1158_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_1159_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1160_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1161_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_1162_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_1163_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1164_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1165_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1166_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1167_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_1168_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_1169_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_1170_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% max_Suc_Suc
thf(fact_1171_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1172_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1173_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_1174_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_1175_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_1176_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1177_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_1178_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_1179_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1180_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_1181_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1182_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_1183_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_1184_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1185_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1186_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_1187_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_1188_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1189_Nat_OAll__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 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_1190_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1191_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1192_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_1193_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_1194_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_1195_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_1196_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_1197_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1198_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1199_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_1200_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_1201_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1202_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1203_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_1204_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1205_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_1206_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1207_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1208_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_1209_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_1210_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_1211_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1212_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_1213_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1214_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_1215_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_1216_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1217_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1218_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1219_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M4 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1220_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_1221_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1222_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1223_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1224_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_1225_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_1226_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1227_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1228_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y4: nat,Z3: nat] :
( ( R2 @ X3 @ Y4 )
=> ( ( R2 @ Y4 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1229_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1230_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1231_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_1232_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1233_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_1234_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1235_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1236_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_1237_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1238_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_1239_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1240_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1241_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1242_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1243_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P @ X3 @ Y4 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1244_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1245_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1246_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1247_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1248_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1249_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1250_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_1251_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_1252_int__cases,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_1253_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_1254_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1255_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_1256_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_1257_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_1258_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_1259_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_1260_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1261_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1262_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W3: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1263_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_1264_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_1265_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1266_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_1267_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_1268_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1269_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1270_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
% 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 (1)
thf(conj_0,conjecture,
ord_less_eq_a @ ( scalar_prod_a @ c @ x ) @ ( scalar_prod_a @ b @ y ) ).
%------------------------------------------------------------------------------