TPTP Problem File: SLH0062^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : LP_Duality/0001_LP_Duality/prob_00273_012665__28861938_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1398 ( 583 unt; 121 typ; 0 def)
% Number of atoms : 3626 (1236 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11485 ( 268 ~; 197 |; 132 &;9426 @)
% ( 0 <=>;1462 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 6 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 254 ( 254 >; 0 *; 0 +; 0 <<)
% Number of symbols : 111 ( 108 usr; 29 con; 0-4 aty)
% Number of variables : 3230 ( 145 ^;3056 !; 29 ?;3230 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:59:50.636
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
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__Omat_It__Nat__Onat_J_J,type,
set_mat_nat: $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__Matrix__Ovec_It__Nat__Onat_J,type,
vec_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $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__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__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (108)
thf(sy_c_Field__as__Ring_Ofield__class_Oeuclidean__size__field_001tf__a,type,
field_8182332513790994628ield_a: a > nat ).
thf(sy_c_Field__as__Ring_Ofield__class_Omod__field_001tf__a,type,
field_4049668205554139175ield_a: a > a > a ).
thf(sy_c_Field__as__Ring_Ofield__class_Onormalize__field_001tf__a,type,
field_879120192484243794ield_a: a > a ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001tf__a,type,
inverse_inverse_a: a > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_Itf__a_J,type,
minus_minus_mat_a: mat_a > mat_a > mat_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Ovec_Itf__a_J,type,
minus_minus_vec_a: vec_a > vec_a > vec_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001tf__a,type,
minus_minus_a: a > a > a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001tf__a,type,
one_one_a: a ).
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__Nat__Onat_J,type,
plus_plus_vec_nat: vec_nat > vec_nat > vec_nat ).
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__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_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__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__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_Itf__a_J,type,
times_times_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a,type,
times_times_a: a > a > a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Omat_Itf__a_J,type,
uminus_uminus_mat_a: mat_a > mat_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Matrix__Ovec_Itf__a_J,type,
uminus_uminus_vec_a: vec_a > vec_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001tf__a,type,
uminus_uminus_a: a > a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Matrix_Oappend__rows_001tf__a,type,
append_rows_a: mat_a > mat_a > mat_a ).
thf(sy_c_Matrix_Oappend__vec_001t__Nat__Onat,type,
append_vec_nat: vec_nat > vec_nat > vec_nat ).
thf(sy_c_Matrix_Oappend__vec_001tf__a,type,
append_vec_a: vec_a > vec_a > vec_a ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Nat__Onat,type,
carrier_mat_nat: nat > nat > set_mat_nat ).
thf(sy_c_Matrix_Ocarrier__mat_001tf__a,type,
carrier_mat_a: nat > nat > set_mat_a ).
thf(sy_c_Matrix_Ocarrier__vec_001t__Nat__Onat,type,
carrier_vec_nat: nat > set_vec_nat ).
thf(sy_c_Matrix_Ocarrier__vec_001tf__a,type,
carrier_vec_a: nat > set_vec_a ).
thf(sy_c_Matrix_Odim__vec_001tf__a,type,
dim_vec_a: vec_a > nat ).
thf(sy_c_Matrix_Ofour__block__mat_001tf__a,type,
four_block_mat_a: mat_a > mat_a > mat_a > mat_a > mat_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__Nat__Onat,type,
mult_mat_vec_nat: mat_nat > vec_nat > vec_nat ).
thf(sy_c_Matrix_Omult__mat__vec_001tf__a,type,
mult_mat_vec_a: mat_a > vec_a > vec_a ).
thf(sy_c_Matrix_Oone__mat_001tf__a,type,
one_mat_a: nat > mat_a ).
thf(sy_c_Matrix_Oscalar__prod_001t__Nat__Onat,type,
scalar_prod_nat: vec_nat > vec_nat > nat ).
thf(sy_c_Matrix_Oscalar__prod_001tf__a,type,
scalar_prod_a: vec_a > vec_a > a ).
thf(sy_c_Matrix_Osmult__vec_001t__Nat__Onat,type,
smult_vec_nat: nat > vec_nat > vec_nat ).
thf(sy_c_Matrix_Osmult__vec_001tf__a,type,
smult_vec_a: a > vec_a > vec_a ).
thf(sy_c_Matrix_Otranspose__mat_001t__Nat__Onat,type,
transpose_mat_nat: mat_nat > mat_nat ).
thf(sy_c_Matrix_Otranspose__mat_001tf__a,type,
transpose_mat_a: mat_a > mat_a ).
thf(sy_c_Matrix_Ovec__first_001tf__a,type,
vec_first_a: vec_a > nat > vec_a ).
thf(sy_c_Matrix_Ovec__index_001tf__a,type,
vec_index_a: vec_a > nat > a ).
thf(sy_c_Matrix_Ovec__last_001tf__a,type,
vec_last_a: vec_a > nat > vec_a ).
thf(sy_c_Matrix_Ozero__mat_001tf__a,type,
zero_mat_a: nat > nat > mat_a ).
thf(sy_c_Matrix_Ozero__vec_001t__Nat__Onat,type,
zero_vec_nat: nat > vec_nat ).
thf(sy_c_Matrix_Ozero__vec_001tf__a,type,
zero_vec_a: nat > vec_a ).
thf(sy_c_Matrix__Kernel_Ovardim_Ounpadl_001tf__a,type,
matrix_unpadl_a: nat > vec_a > vec_a ).
thf(sy_c_Matrix__Kernel_Ovardim_Ounpadr_001tf__a,type,
matrix_unpadr_a: nat > vec_a > vec_a ).
thf(sy_c_Missing__Matrix_Oappend__cols_001tf__a,type,
missin386308114684349109cols_a: mat_a > mat_a > mat_a ).
thf(sy_c_Missing__Matrix_Omat__of__col_001tf__a,type,
missing_mat_of_col_a: vec_a > mat_a ).
thf(sy_c_Norms_Olinf__norm__vec_001tf__a,type,
linf_norm_vec_a: vec_a > a ).
thf(sy_c_Norms_Onorm1_001tf__a,type,
norm1_a: poly_a > a ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001tf__a,type,
neg_nu181380926503873385_dec_a: a > a ).
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__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_001tf__a,type,
ord_less_a: a > a > $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_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__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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Schur__Decomposition_Ovec__inv_001tf__a,type,
schur_vec_inv_a: vec_a > vec_a ).
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__Matrix__Omat_It__Nat__Onat_J,type,
member_mat_nat: mat_nat > set_mat_nat > $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__Nat__Onat_J,type,
member_vec_nat: vec_nat > set_vec_nat > $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_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: mat_a ).
thf(sy_v_L____,type,
l: vec_a ).
thf(sy_v_M____,type,
m: mat_a ).
thf(sy_v_M__last____,type,
m_last: mat_a ).
thf(sy_v_M__low____,type,
m_low: mat_a ).
thf(sy_v_M__up____,type,
m_up: mat_a ).
thf(sy_v_b,type,
b: vec_a ).
thf(sy_v_bc____,type,
bc: vec_a ).
thf(sy_v_c,type,
c: vec_a ).
thf(sy_v_lam____,type,
lam: a ).
thf(sy_v_nc,type,
nc: nat ).
thf(sy_v_nr,type,
nr: nat ).
thf(sy_v_t____,type,
t: vec_a ).
thf(sy_v_u1____,type,
u1: vec_a ).
thf(sy_v_u2____,type,
u2: vec_a ).
thf(sy_v_u3____,type,
u3: vec_a ).
thf(sy_v_u____,type,
u: vec_a ).
thf(sy_v_ulv____,type,
ulv: vec_a ).
thf(sy_v_v____,type,
v: vec_a ).
thf(sy_v_vec1____,type,
vec1: vec_a ).
thf(sy_v_vec2____,type,
vec2: vec_a ).
thf(sy_v_vec3____,type,
vec3: vec_a ).
thf(sy_v_w____,type,
w: vec_a ).
% Relevant facts (1271)
thf(fact_0_v,axiom,
member_vec_a @ v @ ( carrier_vec_a @ nc ) ).
% v
thf(fact_1_w,axiom,
member_vec_a @ w @ ( carrier_vec_a @ nc ) ).
% w
thf(fact_2_c,axiom,
member_vec_a @ c @ ( carrier_vec_a @ nc ) ).
% c
thf(fact_3_vec1,axiom,
member_vec_a @ vec1 @ ( carrier_vec_a @ nc ) ).
% vec1
thf(fact_4_minus__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 @ ( minus_minus_vec_a @ V_1 @ V_2 ) @ ( carrier_vec_a @ N ) ) ) ) ).
% minus_carrier_vec
thf(fact_5_w__def,axiom,
( w
= ( vec_last_a @ u3 @ nc ) ) ).
% w_def
thf(fact_6_v__def,axiom,
( v
= ( vec_first_a @ u3 @ nc ) ) ).
% v_def
thf(fact_7_diff__left__imp__eq,axiom,
! [A: a,B: a,C: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_8_diff__eq__diff__eq,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_9_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: a,C: a,B: a] :
( ( minus_minus_a @ ( minus_minus_a @ A @ C ) @ B )
= ( minus_minus_a @ ( minus_minus_a @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_10_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_11_u3id,axiom,
( u3
= ( append_vec_a @ v @ w ) ) ).
% u3id
thf(fact_12_u3,axiom,
member_vec_a @ u3 @ ( carrier_vec_a @ ( plus_plus_nat @ nc @ nc ) ) ).
% u3
thf(fact_13_preconds_I4_J,axiom,
ord_less_eq_vec_a @ ( zero_vec_a @ nc ) @ w ).
% preconds(4)
thf(fact_14_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_15_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_16_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_17_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_18__C01_C,axiom,
( vec1
= ( zero_vec_a @ nc ) ) ).
% "01"
thf(fact_19_preconds_I3_J,axiom,
ord_less_eq_vec_a @ ( zero_vec_a @ nc ) @ v ).
% preconds(3)
thf(fact_20_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_21_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_22_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_23_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_24_add__diff__cancel__right_H,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_25_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_26_add__diff__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( minus_minus_a @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_27_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_28_add__diff__cancel__left_H,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_29_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_30_add__diff__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( minus_minus_a @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_31_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_32_diff__add__cancel,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_33_add__diff__cancel,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_34_append__vec__eq,axiom,
! [V: vec_a,N: nat,V2: vec_a,W: vec_a,W2: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V2 @ ( carrier_vec_a @ N ) )
=> ( ( ( append_vec_a @ V @ W )
= ( append_vec_a @ V2 @ W2 ) )
= ( ( V = V2 )
& ( W = W2 ) ) ) ) ) ).
% append_vec_eq
thf(fact_35_append__carrier__vec,axiom,
! [V: vec_a,N1: nat,W: vec_a,N2: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N1 ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ N2 ) )
=> ( member_vec_a @ ( append_vec_a @ V @ W ) @ ( carrier_vec_a @ ( plus_plus_nat @ N1 @ N2 ) ) ) ) ) ).
% append_carrier_vec
thf(fact_36_minus__cancel__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( minus_minus_vec_a @ V @ V )
= ( zero_vec_a @ N ) ) ) ).
% minus_cancel_vec
thf(fact_37_minus__zero__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( minus_minus_vec_a @ V @ ( zero_vec_a @ N ) )
= V ) ) ).
% minus_zero_vec
thf(fact_38_vec__first__last__append,axiom,
! [V: vec_a,N: nat,M: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ ( plus_plus_nat @ N @ M ) ) )
=> ( ( append_vec_a @ ( vec_first_a @ V @ N ) @ ( vec_last_a @ V @ M ) )
= V ) ) ).
% vec_first_last_append
thf(fact_39_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_40_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_41_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_42_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_43_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_44_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_45_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_46_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_47_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_48_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_49_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_50_group__cancel_Oadd1,axiom,
! [A2: a,K: a,A: a,B: a] :
( ( A2
= ( plus_plus_a @ K @ A ) )
=> ( ( plus_plus_a @ A2 @ B )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_51_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_52_group__cancel_Oadd2,axiom,
! [B2: a,K: a,B: a,A: a] :
( ( B2
= ( plus_plus_a @ K @ B ) )
=> ( ( plus_plus_a @ A @ B2 )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_53_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_54_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_55_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_56_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_57_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_58_add_Ocommute,axiom,
( plus_plus_a
= ( ^ [A3: a,B3: a] : ( plus_plus_a @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_59_diff__le__eq,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ ( minus_minus_a @ A @ B ) @ C )
= ( ord_less_eq_a @ A @ ( plus_plus_a @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_60_le__diff__eq,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ A @ ( minus_minus_a @ C @ B ) )
= ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_61_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_62_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_63_Collect__mem__eq,axiom,
! [A2: set_vec_a] :
( ( collect_vec_a
@ ^ [X: vec_a] : ( member_vec_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A2: set_mat_a] :
( ( collect_mat_a
@ ^ [X: mat_a] : ( member_mat_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_65_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_66_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_67_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_68_add__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% add_mono
thf(fact_69_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_70_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_71_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_72_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_73_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_74_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_75_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_76_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_77_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_78_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_79_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_80_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_81_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_82_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_83_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_84_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_85_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_86_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_87_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_88_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_89_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_90_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_91_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_92_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_93_append__vec__le,axiom,
! [V: vec_a,N: nat,W: vec_a,V2: vec_a,W2: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ N ) )
=> ( ( ord_less_eq_vec_a @ ( append_vec_a @ V @ V2 ) @ ( append_vec_a @ W @ W2 ) )
= ( ( ord_less_eq_vec_a @ V @ W )
& ( ord_less_eq_vec_a @ V2 @ W2 ) ) ) ) ) ).
% append_vec_le
thf(fact_94_all__vec__append,axiom,
! [N: nat,M: nat,P: vec_a > $o] :
( ( ! [X: vec_a] :
( ( member_vec_a @ X @ ( carrier_vec_a @ ( plus_plus_nat @ N @ M ) ) )
=> ( P @ X ) ) )
= ( ! [X: vec_a] :
( ( member_vec_a @ X @ ( carrier_vec_a @ N ) )
=> ! [Y: vec_a] :
( ( member_vec_a @ Y @ ( carrier_vec_a @ M ) )
=> ( P @ ( append_vec_a @ X @ Y ) ) ) ) ) ) ).
% all_vec_append
thf(fact_95_zero__carrier__vec,axiom,
! [N: nat] : ( member_vec_a @ ( zero_vec_a @ N ) @ ( carrier_vec_a @ N ) ) ).
% zero_carrier_vec
thf(fact_96_diff__diff__eq,axiom,
! [A: a,B: a,C: a] :
( ( minus_minus_a @ ( minus_minus_a @ A @ B ) @ C )
= ( minus_minus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_97_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_98_add__implies__diff,axiom,
! [C: a,B: a,A: a] :
( ( ( plus_plus_a @ C @ B )
= A )
=> ( C
= ( minus_minus_a @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_99_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_100_diff__add__eq__diff__diff__swap,axiom,
! [A: a,B: a,C: a] :
( ( minus_minus_a @ A @ ( plus_plus_a @ B @ C ) )
= ( minus_minus_a @ ( minus_minus_a @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_101_diff__add__eq,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ C )
= ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_102_diff__diff__eq2,axiom,
! [A: a,B: a,C: a] :
( ( minus_minus_a @ A @ ( minus_minus_a @ B @ C ) )
= ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_103_add__diff__eq,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ A @ ( minus_minus_a @ B @ C ) )
= ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_104_eq__diff__eq,axiom,
! [A: a,C: a,B: a] :
( ( A
= ( minus_minus_a @ C @ B ) )
= ( ( plus_plus_a @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_105_diff__eq__eq,axiom,
! [A: a,B: a,C: a] :
( ( ( minus_minus_a @ A @ B )
= C )
= ( A
= ( plus_plus_a @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_106_group__cancel_Osub1,axiom,
! [A2: a,K: a,A: a,B: a] :
( ( A2
= ( plus_plus_a @ K @ A ) )
=> ( ( minus_minus_a @ A2 @ B )
= ( plus_plus_a @ K @ ( minus_minus_a @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_107_diff__eq__diff__less__eq,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D ) )
=> ( ( ord_less_eq_a @ A @ B )
= ( ord_less_eq_a @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_108_diff__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_109_diff__left__mono,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ord_less_eq_a @ ( minus_minus_a @ C @ A ) @ ( minus_minus_a @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_110_diff__mono,axiom,
! [A: a,B: a,D: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ D @ C )
=> ( ord_less_eq_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_111_vec__first__carrier,axiom,
! [V: vec_a,N: nat] : ( member_vec_a @ ( vec_first_a @ V @ N ) @ ( carrier_vec_a @ N ) ) ).
% vec_first_carrier
thf(fact_112_vec__last__carrier,axiom,
! [V: vec_a,N: nat] : ( member_vec_a @ ( vec_last_a @ V @ N ) @ ( carrier_vec_a @ N ) ) ).
% vec_last_carrier
thf(fact_113_u3__def,axiom,
( u3
= ( vec_last_a @ u1 @ ( plus_plus_nat @ nc @ nc ) ) ) ).
% u3_def
thf(fact_114_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_115_le__add__diff__inverse,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( plus_plus_a @ B @ ( minus_minus_a @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_116_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_117_le__add__diff__inverse2,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_118__092_060open_062c_A_092_060bullet_062_Av_A_N_Ac_A_092_060bullet_062_Aw_A_061_Ac_A_092_060bullet_062_A_Iv_A_N_Aw_J_092_060close_062,axiom,
( ( minus_minus_a @ ( scalar_prod_a @ c @ v ) @ ( scalar_prod_a @ c @ w ) )
= ( scalar_prod_a @ c @ ( minus_minus_vec_a @ v @ w ) ) ) ).
% \<open>c \<bullet> v - c \<bullet> w = c \<bullet> (v - w)\<close>
thf(fact_119_vec__first__append,axiom,
! [V: vec_a,N: nat,W: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( vec_first_a @ ( append_vec_a @ V @ W ) @ N )
= V ) ) ).
% vec_first_append
thf(fact_120_vardim_Opadr__padl__eq,axiom,
! [V: vec_a,N: nat,M: nat,U: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( ( append_vec_a @ V @ ( zero_vec_a @ M ) )
= ( append_vec_a @ ( zero_vec_a @ N ) @ U ) )
= ( ( V
= ( zero_vec_a @ N ) )
& ( U
= ( zero_vec_a @ M ) ) ) ) ) ).
% vardim.padr_padl_eq
thf(fact_121_exists__vec__append,axiom,
! [N: nat,M: nat,P: vec_a > $o] :
( ( ? [X: vec_a] :
( ( member_vec_a @ X @ ( carrier_vec_a @ ( plus_plus_nat @ N @ M ) ) )
& ( P @ X ) ) )
= ( ? [X: vec_a] :
( ( member_vec_a @ X @ ( carrier_vec_a @ N ) )
& ? [Y: vec_a] :
( ( member_vec_a @ Y @ ( carrier_vec_a @ M ) )
& ( P @ ( append_vec_a @ X @ Y ) ) ) ) ) ) ).
% exists_vec_append
thf(fact_122_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_123_add__le__imp__le__diff,axiom,
! [I: a,K: a,N: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ N )
=> ( ord_less_eq_a @ I @ ( minus_minus_a @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_124_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_125_add__le__add__imp__diff__le,axiom,
! [I: a,K: a,N: a,J: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ N )
=> ( ( ord_less_eq_a @ N @ ( plus_plus_a @ J @ K ) )
=> ( ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ N )
=> ( ( ord_less_eq_a @ N @ ( plus_plus_a @ J @ K ) )
=> ( ord_less_eq_a @ ( minus_minus_a @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_126_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_127_set__plus__intro,axiom,
! [A: mat_a,C4: set_mat_a,B: mat_a,D2: set_mat_a] :
( ( member_mat_a @ A @ C4 )
=> ( ( member_mat_a @ B @ D2 )
=> ( member_mat_a @ ( plus_plus_mat_a @ A @ B ) @ ( plus_plus_set_mat_a @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_128_set__plus__intro,axiom,
! [A: nat,C4: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A @ C4 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_129_set__plus__intro,axiom,
! [A: a,C4: set_a,B: a,D2: set_a] :
( ( member_a @ A @ C4 )
=> ( ( member_a @ B @ D2 )
=> ( member_a @ ( plus_plus_a @ A @ B ) @ ( plus_plus_set_a @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_130_set__plus__intro,axiom,
! [A: vec_a,C4: set_vec_a,B: vec_a,D2: set_vec_a] :
( ( member_vec_a @ A @ C4 )
=> ( ( member_vec_a @ B @ D2 )
=> ( member_vec_a @ ( plus_plus_vec_a @ A @ B ) @ ( plus_plus_set_vec_a @ C4 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_131_order__refl,axiom,
! [X2: vec_a] : ( ord_less_eq_vec_a @ X2 @ X2 ) ).
% order_refl
thf(fact_132_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_133_order__refl,axiom,
! [X2: a] : ( ord_less_eq_a @ X2 @ X2 ) ).
% order_refl
thf(fact_134_dual__order_Orefl,axiom,
! [A: vec_a] : ( ord_less_eq_vec_a @ A @ A ) ).
% dual_order.refl
thf(fact_135_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_136_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_137_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_138_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_139_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_140_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_141_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_142_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_143_right__zero__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ V @ ( zero_vec_a @ N ) )
= V ) ) ).
% right_zero_vec
thf(fact_144_left__zero__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ ( zero_vec_a @ N ) @ V )
= V ) ) ).
% left_zero_vec
thf(fact_145_vec__first__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( vec_first_a @ ( zero_vec_a @ N ) @ M )
= ( zero_vec_a @ M ) ) ) ).
% vec_first_zero
thf(fact_146_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_147_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_148_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_149_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_150_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_151_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_152_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_153_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_154_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_155_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_156_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_157_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_158_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_159_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_160_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_161_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_162_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_163_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_164_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_165_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_166_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_167_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_168_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_169_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_170_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_171_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_172_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_173_add__diff__eq__vec,axiom,
! [Y2: vec_a,N: nat,X2: vec_a,Z: vec_a] :
( ( member_vec_a @ Y2 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ X2 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ Z @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ Y2 @ ( minus_minus_vec_a @ X2 @ Z ) )
= ( minus_minus_vec_a @ ( plus_plus_vec_a @ Y2 @ X2 ) @ Z ) ) ) ) ) ).
% add_diff_eq_vec
thf(fact_174_add__diff__cancel__right__vec,axiom,
! [A: vec_a,N: nat,B: vec_a] :
( ( member_vec_a @ A @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ N ) )
=> ( ( minus_minus_vec_a @ ( plus_plus_vec_a @ A @ B ) @ B )
= A ) ) ) ).
% add_diff_cancel_right_vec
thf(fact_175_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_176_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_177_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_178_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_179_add__inv__exists__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ? [X3: vec_a] :
( ( member_vec_a @ X3 @ ( carrier_vec_a @ N ) )
& ( ( plus_plus_vec_a @ X3 @ V )
= ( zero_vec_a @ N ) )
& ( ( plus_plus_vec_a @ V @ X3 )
= ( zero_vec_a @ N ) ) ) ) ).
% add_inv_exists_vec
thf(fact_180_append__vec__add,axiom,
! [V: vec_a,N: nat,V2: vec_a,W: vec_a,M: nat,W2: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V2 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ M ) )
=> ( ( member_vec_a @ W2 @ ( carrier_vec_a @ M ) )
=> ( ( plus_plus_vec_a @ ( append_vec_a @ V @ W ) @ ( append_vec_a @ V2 @ W2 ) )
= ( append_vec_a @ ( plus_plus_vec_a @ V @ V2 ) @ ( plus_plus_vec_a @ W @ W2 ) ) ) ) ) ) ) ).
% append_vec_add
thf(fact_181_minus__add__minus__vec,axiom,
! [U: vec_a,N: nat,V: vec_a,W: vec_a] :
( ( member_vec_a @ U @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ N ) )
=> ( ( minus_minus_vec_a @ U @ ( plus_plus_vec_a @ V @ W ) )
= ( minus_minus_vec_a @ ( minus_minus_vec_a @ U @ V ) @ W ) ) ) ) ) ).
% minus_add_minus_vec
thf(fact_182_scalar__prod__append,axiom,
! [V1: vec_nat,N1: nat,V22: vec_nat,N2: nat,W1: vec_nat,W22: vec_nat] :
( ( member_vec_nat @ V1 @ ( carrier_vec_nat @ N1 ) )
=> ( ( member_vec_nat @ V22 @ ( carrier_vec_nat @ N2 ) )
=> ( ( member_vec_nat @ W1 @ ( carrier_vec_nat @ N1 ) )
=> ( ( member_vec_nat @ W22 @ ( carrier_vec_nat @ N2 ) )
=> ( ( scalar_prod_nat @ ( append_vec_nat @ V1 @ V22 ) @ ( append_vec_nat @ W1 @ W22 ) )
= ( plus_plus_nat @ ( scalar_prod_nat @ V1 @ W1 ) @ ( scalar_prod_nat @ V22 @ W22 ) ) ) ) ) ) ) ).
% scalar_prod_append
thf(fact_183_scalar__prod__append,axiom,
! [V1: vec_a,N1: nat,V22: vec_a,N2: nat,W1: vec_a,W22: vec_a] :
( ( member_vec_a @ V1 @ ( carrier_vec_a @ N1 ) )
=> ( ( member_vec_a @ V22 @ ( carrier_vec_a @ N2 ) )
=> ( ( member_vec_a @ W1 @ ( carrier_vec_a @ N1 ) )
=> ( ( member_vec_a @ W22 @ ( carrier_vec_a @ N2 ) )
=> ( ( scalar_prod_a @ ( append_vec_a @ V1 @ V22 ) @ ( append_vec_a @ W1 @ W22 ) )
= ( plus_plus_a @ ( scalar_prod_a @ V1 @ W1 ) @ ( scalar_prod_a @ V22 @ W22 ) ) ) ) ) ) ) ).
% scalar_prod_append
thf(fact_184_minus__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 @ ( minus_minus_vec_a @ V_1 @ V_2 ) @ V_3 )
= ( minus_minus_a @ ( scalar_prod_a @ V_1 @ V_3 ) @ ( scalar_prod_a @ V_2 @ V_3 ) ) ) ) ) ) ).
% minus_scalar_prod_distrib
thf(fact_185_scalar__prod__minus__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 @ ( minus_minus_vec_a @ V_2 @ V_3 ) )
= ( minus_minus_a @ ( scalar_prod_a @ V_1 @ V_2 ) @ ( scalar_prod_a @ V_1 @ V_3 ) ) ) ) ) ) ).
% scalar_prod_minus_distrib
thf(fact_186_order__antisym__conv,axiom,
! [Y2: vec_a,X2: vec_a] :
( ( ord_less_eq_vec_a @ Y2 @ X2 )
=> ( ( ord_less_eq_vec_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_187_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_188_order__antisym__conv,axiom,
! [Y2: a,X2: a] :
( ( ord_less_eq_a @ Y2 @ X2 )
=> ( ( ord_less_eq_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_189_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_190_linorder__le__cases,axiom,
! [X2: a,Y2: a] :
( ~ ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_191_ord__le__eq__subst,axiom,
! [A: vec_a,B: vec_a,F: vec_a > vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_192_ord__le__eq__subst,axiom,
! [A: vec_a,B: vec_a,F: vec_a > nat,C: nat] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_193_ord__le__eq__subst,axiom,
! [A: vec_a,B: vec_a,F: vec_a > a,C: a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_194_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > vec_a,C: vec_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_195_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_196_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_197_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > vec_a,C: vec_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_198_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,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_199_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,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_200_ord__eq__le__subst,axiom,
! [A: vec_a,F: vec_a > vec_a,B: vec_a,C: vec_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_201_ord__eq__le__subst,axiom,
! [A: nat,F: vec_a > nat,B: vec_a,C: vec_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_202_ord__eq__le__subst,axiom,
! [A: a,F: vec_a > a,B: vec_a,C: vec_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_203_ord__eq__le__subst,axiom,
! [A: vec_a,F: nat > vec_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_204_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_205_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_206_ord__eq__le__subst,axiom,
! [A: vec_a,F: a > vec_a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_207_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,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_208_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,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_209_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_210_linorder__linear,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
| ( ord_less_eq_a @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_211_order__eq__refl,axiom,
! [X2: vec_a,Y2: vec_a] :
( ( X2 = Y2 )
=> ( ord_less_eq_vec_a @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_212_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_213_order__eq__refl,axiom,
! [X2: a,Y2: a] :
( ( X2 = Y2 )
=> ( ord_less_eq_a @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_214_order__subst2,axiom,
! [A: vec_a,B: vec_a,F: vec_a > vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_eq_vec_a @ ( F @ B ) @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_215_order__subst2,axiom,
! [A: vec_a,B: vec_a,F: vec_a > nat,C: nat] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_216_order__subst2,axiom,
! [A: vec_a,B: vec_a,F: vec_a > a,C: a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_217_order__subst2,axiom,
! [A: nat,B: nat,F: nat > vec_a,C: vec_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_vec_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_218_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_219_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_220_order__subst2,axiom,
! [A: a,B: a,F: a > vec_a,C: vec_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_vec_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_221_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,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_222_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,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_223_order__subst1,axiom,
! [A: vec_a,F: vec_a > vec_a,B: vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_224_order__subst1,axiom,
! [A: vec_a,F: nat > vec_a,B: nat,C: nat] :
( ( ord_less_eq_vec_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_225_order__subst1,axiom,
! [A: vec_a,F: a > vec_a,B: a,C: a] :
( ( ord_less_eq_vec_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_226_order__subst1,axiom,
! [A: nat,F: vec_a > nat,B: vec_a,C: vec_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_227_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_228_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,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_229_order__subst1,axiom,
! [A: a,F: vec_a > a,B: vec_a,C: vec_a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_230_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_231_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,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_232_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: vec_a,Z2: vec_a] : ( Y4 = Z2 ) )
= ( ^ [A3: vec_a,B3: vec_a] :
( ( ord_less_eq_vec_a @ A3 @ B3 )
& ( ord_less_eq_vec_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_233_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_234_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
& ( ord_less_eq_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_235_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_236_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_237_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_238_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_239_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_240_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_241_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_242_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_243_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_244_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: vec_a,Z2: vec_a] : ( Y4 = Z2 ) )
= ( ^ [A3: vec_a,B3: vec_a] :
( ( ord_less_eq_vec_a @ B3 @ A3 )
& ( ord_less_eq_vec_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_245_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_246_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ B3 @ A3 )
& ( ord_less_eq_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_247_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_248_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: a,B4: a] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_249_order__trans,axiom,
! [X2: vec_a,Y2: vec_a,Z: vec_a] :
( ( ord_less_eq_vec_a @ X2 @ Y2 )
=> ( ( ord_less_eq_vec_a @ Y2 @ Z )
=> ( ord_less_eq_vec_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_250_order__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_251_order__trans,axiom,
! [X2: a,Y2: a,Z: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z )
=> ( ord_less_eq_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_252_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_253_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_254_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_255_order__antisym,axiom,
! [X2: vec_a,Y2: vec_a] :
( ( ord_less_eq_vec_a @ X2 @ Y2 )
=> ( ( ord_less_eq_vec_a @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_256_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_257_order__antisym,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_258_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_259_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_260_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_261_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_262_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_263_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_264_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: vec_a,Z2: vec_a] : ( Y4 = Z2 ) )
= ( ^ [X: vec_a,Y: vec_a] :
( ( ord_less_eq_vec_a @ X @ Y )
& ( ord_less_eq_vec_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_265_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_266_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ( ord_less_eq_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_267_le__cases3,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_268_le__cases3,axiom,
! [X2: a,Y2: a,Z: a] :
( ( ( ord_less_eq_a @ X2 @ Y2 )
=> ~ ( ord_less_eq_a @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_a @ Y2 @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Z ) )
=> ( ( ( ord_less_eq_a @ X2 @ Z )
=> ~ ( ord_less_eq_a @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_a @ Z @ Y2 )
=> ~ ( ord_less_eq_a @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_a @ Y2 @ Z )
=> ~ ( ord_less_eq_a @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_a @ Z @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_269_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_270_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_271_set__plus__elim,axiom,
! [X2: mat_a,A2: set_mat_a,B2: set_mat_a] :
( ( member_mat_a @ X2 @ ( plus_plus_set_mat_a @ A2 @ B2 ) )
=> ~ ! [A4: mat_a,B4: mat_a] :
( ( X2
= ( plus_plus_mat_a @ A4 @ B4 ) )
=> ( ( member_mat_a @ A4 @ A2 )
=> ~ ( member_mat_a @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_272_set__plus__elim,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X2 @ ( plus_plus_set_nat @ A2 @ B2 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X2
= ( plus_plus_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A2 )
=> ~ ( member_nat @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_273_set__plus__elim,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ( member_a @ X2 @ ( plus_plus_set_a @ A2 @ B2 ) )
=> ~ ! [A4: a,B4: a] :
( ( X2
= ( plus_plus_a @ A4 @ B4 ) )
=> ( ( member_a @ A4 @ A2 )
=> ~ ( member_a @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_274_set__plus__elim,axiom,
! [X2: vec_a,A2: set_vec_a,B2: set_vec_a] :
( ( member_vec_a @ X2 @ ( plus_plus_set_vec_a @ A2 @ B2 ) )
=> ~ ! [A4: vec_a,B4: vec_a] :
( ( X2
= ( plus_plus_vec_a @ A4 @ B4 ) )
=> ( ( member_vec_a @ A4 @ A2 )
=> ~ ( member_vec_a @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_275_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_276_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_277_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_278_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_279_u1id,axiom,
( u1
= ( append_vec_a @ u2 @ u3 ) ) ).
% u1id
thf(fact_280__092_060open_062c_A_092_060bullet_062_Av_A_L_A_N_Ac_A_092_060bullet_062_Aw_A_061_Ac_A_092_060bullet_062_Av_A_N_Ac_A_092_060bullet_062_Aw_092_060close_062,axiom,
( ( plus_plus_a @ ( scalar_prod_a @ c @ v ) @ ( scalar_prod_a @ ( uminus_uminus_vec_a @ c ) @ w ) )
= ( minus_minus_a @ ( scalar_prod_a @ c @ v ) @ ( scalar_prod_a @ c @ w ) ) ) ).
% \<open>c \<bullet> v + - c \<bullet> w = c \<bullet> v - c \<bullet> w\<close>
thf(fact_281__092_060open_062v_A_092_060bullet_062_Ac_A_L_Aw_A_092_060bullet_062_A_N_Ac_A_061_Ac_A_092_060bullet_062_Av_A_L_A_N_Ac_A_092_060bullet_062_Aw_092_060close_062,axiom,
( ( plus_plus_a @ ( scalar_prod_a @ v @ c ) @ ( scalar_prod_a @ w @ ( uminus_uminus_vec_a @ c ) ) )
= ( plus_plus_a @ ( scalar_prod_a @ c @ v ) @ ( scalar_prod_a @ ( uminus_uminus_vec_a @ c ) @ w ) ) ) ).
% \<open>v \<bullet> c + w \<bullet> - c = c \<bullet> v + - c \<bullet> w\<close>
thf(fact_282_sum__carrier__vec,axiom,
! [A2: set_vec_a,N: nat,B2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ B2 @ ( carrier_vec_a @ N ) )
=> ( ord_le4791951621262958845_vec_a @ ( plus_plus_set_vec_a @ A2 @ B2 ) @ ( carrier_vec_a @ N ) ) ) ) ).
% sum_carrier_vec
thf(fact_283_add__diff__add,axiom,
! [A: a,C: a,B: a,D: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) )
= ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ ( minus_minus_a @ C @ D ) ) ) ).
% add_diff_add
thf(fact_284__092_060open_0620_092_060_094sub_062v_A_Inc_A_L_Anr_J_A_061_A0_092_060_094sub_062v_Anc_A_064_092_060_094sub_062v_A0_092_060_094sub_062v_Anr_092_060close_062,axiom,
( ( zero_vec_a @ ( plus_plus_nat @ nc @ nr ) )
= ( append_vec_a @ ( zero_vec_a @ nc ) @ ( zero_vec_a @ nr ) ) ) ).
% \<open>0\<^sub>v (nc + nr) = 0\<^sub>v nc @\<^sub>v 0\<^sub>v nr\<close>
thf(fact_285_vardim_Ounpadr__padr,axiom,
! [M: nat,V: vec_a] :
( ( matrix_unpadr_a @ M @ ( append_vec_a @ V @ ( zero_vec_a @ M ) ) )
= V ) ).
% vardim.unpadr_padr
thf(fact_286_vardim_Ounpadl__padl,axiom,
! [M: nat,V: vec_a] :
( ( matrix_unpadl_a @ M @ ( append_vec_a @ ( zero_vec_a @ M ) @ V ) )
= V ) ).
% vardim.unpadl_padl
thf(fact_287_scalar__prod__right__zero,axiom,
! [V: vec_nat,N: nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( scalar_prod_nat @ V @ ( zero_vec_nat @ N ) )
= zero_zero_nat ) ) ).
% scalar_prod_right_zero
thf(fact_288_scalar__prod__right__zero,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ V @ ( zero_vec_a @ N ) )
= zero_zero_a ) ) ).
% scalar_prod_right_zero
thf(fact_289_scalar__prod__left__zero,axiom,
! [V: vec_nat,N: nat] :
( ( member_vec_nat @ V @ ( carrier_vec_nat @ N ) )
=> ( ( scalar_prod_nat @ ( zero_vec_nat @ N ) @ V )
= zero_zero_nat ) ) ).
% scalar_prod_left_zero
thf(fact_290_scalar__prod__left__zero,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ ( zero_vec_a @ N ) @ V )
= zero_zero_a ) ) ).
% scalar_prod_left_zero
thf(fact_291_neg__equal__iff__equal,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= ( uminus_uminus_a @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_292_add_Oinverse__inverse,axiom,
! [A: a] :
( ( uminus_uminus_a @ ( uminus_uminus_a @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_293_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_294_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_295_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_296_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_297_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_298_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_299_uminus__uminus__vec,axiom,
! [V: vec_a] :
( ( uminus_uminus_vec_a @ ( uminus_uminus_vec_a @ V ) )
= V ) ).
% uminus_uminus_vec
thf(fact_300_uminus__eq__vec,axiom,
! [V: vec_a,W: vec_a] :
( ( ( uminus_uminus_vec_a @ V )
= ( uminus_uminus_vec_a @ W ) )
= ( V = W ) ) ).
% uminus_eq_vec
thf(fact_301_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_302_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_303_add_Oright__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.right_neutral
thf(fact_304_double__zero__sym,axiom,
! [A: a] :
( ( zero_zero_a
= ( plus_plus_a @ A @ A ) )
= ( A = zero_zero_a ) ) ).
% double_zero_sym
thf(fact_305_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_306_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_307_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_308_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_309_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_310_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_311_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_312_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_313_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_314_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y2 ) )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_315_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_316_add__0,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% add_0
thf(fact_317_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: a] :
( ( minus_minus_a @ A @ A )
= zero_zero_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_318_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_319_diff__zero,axiom,
! [A: a] :
( ( minus_minus_a @ A @ zero_zero_a )
= A ) ).
% diff_zero
thf(fact_320_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_321_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_322_diff__0__right,axiom,
! [A: a] :
( ( minus_minus_a @ A @ zero_zero_a )
= A ) ).
% diff_0_right
thf(fact_323_diff__self,axiom,
! [A: a] :
( ( minus_minus_a @ A @ A )
= zero_zero_a ) ).
% diff_self
thf(fact_324_neg__equal__zero,axiom,
! [A: a] :
( ( ( uminus_uminus_a @ A )
= A )
= ( A = zero_zero_a ) ) ).
% neg_equal_zero
thf(fact_325_equal__neg__zero,axiom,
! [A: a] :
( ( A
= ( uminus_uminus_a @ A ) )
= ( A = zero_zero_a ) ) ).
% equal_neg_zero
thf(fact_326_neg__equal__0__iff__equal,axiom,
! [A: a] :
( ( ( uminus_uminus_a @ A )
= zero_zero_a )
= ( A = zero_zero_a ) ) ).
% neg_equal_0_iff_equal
thf(fact_327_neg__0__equal__iff__equal,axiom,
! [A: a] :
( ( zero_zero_a
= ( uminus_uminus_a @ A ) )
= ( zero_zero_a = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_328_add_Oinverse__neutral,axiom,
( ( uminus_uminus_a @ zero_zero_a )
= zero_zero_a ) ).
% add.inverse_neutral
thf(fact_329_neg__le__iff__le,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
= ( ord_less_eq_a @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_330_add__minus__cancel,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ A @ ( plus_plus_a @ ( uminus_uminus_a @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_331_minus__add__cancel,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ ( plus_plus_a @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_332_minus__add__distrib,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( plus_plus_a @ A @ B ) )
= ( plus_plus_a @ ( uminus_uminus_a @ A ) @ ( uminus_uminus_a @ B ) ) ) ).
% minus_add_distrib
thf(fact_333_minus__diff__eq,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( minus_minus_a @ A @ B ) )
= ( minus_minus_a @ B @ A ) ) ).
% minus_diff_eq
thf(fact_334_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_335_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_336_uminus__carrier__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ ( uminus_uminus_vec_a @ V ) @ ( carrier_vec_a @ N ) )
= ( member_vec_a @ V @ ( carrier_vec_a @ N ) ) ) ).
% uminus_carrier_vec
thf(fact_337_uminus__zero__vec,axiom,
! [N: nat] :
( ( uminus_uminus_vec_a @ ( zero_vec_a @ N ) )
= ( zero_vec_a @ N ) ) ).
% uminus_zero_vec
thf(fact_338_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_339_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_340_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_341_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_342_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_343_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_344_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_345_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_346_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_347_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_348_diff__ge__0__iff__ge,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( minus_minus_a @ A @ B ) )
= ( ord_less_eq_a @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_349_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_350_neg__0__le__iff__le,axiom,
! [A: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% neg_0_le_iff_le
thf(fact_351_neg__le__0__iff__le,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% neg_le_0_iff_le
thf(fact_352_less__eq__neg__nonpos,axiom,
! [A: a] :
( ( ord_less_eq_a @ A @ ( uminus_uminus_a @ A ) )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% less_eq_neg_nonpos
thf(fact_353_neg__less__eq__nonneg,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ A )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_354_add_Oright__inverse,axiom,
! [A: a] :
( ( plus_plus_a @ A @ ( uminus_uminus_a @ A ) )
= zero_zero_a ) ).
% add.right_inverse
thf(fact_355_ab__left__minus,axiom,
! [A: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ A )
= zero_zero_a ) ).
% ab_left_minus
thf(fact_356_diff__0,axiom,
! [A: a] :
( ( minus_minus_a @ zero_zero_a @ A )
= ( uminus_uminus_a @ A ) ) ).
% diff_0
thf(fact_357_diff__minus__eq__add,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ A @ ( uminus_uminus_a @ B ) )
= ( plus_plus_a @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_358_uminus__add__conv__diff,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ B )
= ( minus_minus_a @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_359_b,axiom,
member_vec_a @ b @ ( carrier_vec_a @ nr ) ).
% b
thf(fact_360_uminus__l__inv__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ ( uminus_uminus_vec_a @ V ) @ V )
= ( zero_vec_a @ N ) ) ) ).
% uminus_l_inv_vec
thf(fact_361_uminus__r__inv__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_vec_a @ V @ ( uminus_uminus_vec_a @ V ) )
= ( zero_vec_a @ N ) ) ) ).
% uminus_r_inv_vec
thf(fact_362_zero__minus__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( minus_minus_vec_a @ ( zero_vec_a @ N ) @ V )
= ( uminus_uminus_vec_a @ V ) ) ) ).
% zero_minus_vec
thf(fact_363_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_364_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_365_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_366_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_367_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_368_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_369_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_370_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_371_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_372_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_373_neg__eq__iff__add__eq__0,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= B )
= ( ( plus_plus_a @ A @ B )
= zero_zero_a ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_374_eq__neg__iff__add__eq__0,axiom,
! [A: a,B: a] :
( ( A
= ( uminus_uminus_a @ B ) )
= ( ( plus_plus_a @ A @ B )
= zero_zero_a ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_375_add_Oinverse__unique,axiom,
! [A: a,B: a] :
( ( ( plus_plus_a @ A @ B )
= zero_zero_a )
=> ( ( uminus_uminus_a @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_376_ab__group__add__class_Oab__left__minus,axiom,
! [A: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ A )
= zero_zero_a ) ).
% ab_group_add_class.ab_left_minus
thf(fact_377_add__eq__0__iff,axiom,
! [A: a,B: a] :
( ( ( plus_plus_a @ A @ B )
= zero_zero_a )
= ( B
= ( uminus_uminus_a @ A ) ) ) ).
% add_eq_0_iff
thf(fact_378_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_379_zero__reorient,axiom,
! [X2: a] :
( ( zero_zero_a = X2 )
= ( X2 = zero_zero_a ) ) ).
% zero_reorient
thf(fact_380_minus__equation__iff,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= B )
= ( ( uminus_uminus_a @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_381_equation__minus__iff,axiom,
! [A: a,B: a] :
( ( A
= ( uminus_uminus_a @ B ) )
= ( B
= ( uminus_uminus_a @ A ) ) ) ).
% equation_minus_iff
thf(fact_382_minus__diff__minus,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( uminus_uminus_a @ A ) @ ( uminus_uminus_a @ B ) )
= ( uminus_uminus_a @ ( minus_minus_a @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_383_uminus__scalar__prod,axiom,
! [V: vec_a,N: nat,W: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ N ) )
=> ( ( uminus_uminus_a @ ( scalar_prod_a @ V @ W ) )
= ( scalar_prod_a @ ( uminus_uminus_vec_a @ V ) @ W ) ) ) ) ).
% uminus_scalar_prod
thf(fact_384_le__minus__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ ( uminus_uminus_a @ B ) )
= ( ord_less_eq_a @ B @ ( uminus_uminus_a @ A ) ) ) ).
% le_minus_iff
thf(fact_385_minus__le__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ B )
= ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_386_le__imp__neg__le,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% le_imp_neg_le
thf(fact_387_group__cancel_Oneg1,axiom,
! [A2: a,K: a,A: a] :
( ( A2
= ( plus_plus_a @ K @ A ) )
=> ( ( uminus_uminus_a @ A2 )
= ( plus_plus_a @ ( uminus_uminus_a @ K ) @ ( uminus_uminus_a @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_388_add_Oinverse__distrib__swap,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( plus_plus_a @ A @ B ) )
= ( plus_plus_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_389_minus__diff__commute,axiom,
! [B: a,A: a] :
( ( minus_minus_a @ ( uminus_uminus_a @ B ) @ A )
= ( minus_minus_a @ ( uminus_uminus_a @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_390_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_391_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_392_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_393_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_394_add_Ocomm__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.comm_neutral
thf(fact_395_add_Ogroup__left__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% add.group_left_neutral
thf(fact_396_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
= ( ^ [A3: a,B3: a] :
( ( minus_minus_a @ A3 @ B3 )
= zero_zero_a ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_397_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_398_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_399_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_400_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_401_set__zero__plus2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( member_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ ( plus_plus_set_nat @ A2 @ B2 ) ) ) ).
% set_zero_plus2
thf(fact_402_set__zero__plus2,axiom,
! [A2: set_a,B2: set_a] :
( ( member_a @ zero_zero_a @ A2 )
=> ( ord_less_eq_set_a @ B2 @ ( plus_plus_set_a @ A2 @ B2 ) ) ) ).
% set_zero_plus2
thf(fact_403_group__cancel_Osub2,axiom,
! [B2: a,K: a,B: a,A: a] :
( ( B2
= ( plus_plus_a @ K @ B ) )
=> ( ( minus_minus_a @ A @ B2 )
= ( plus_plus_a @ ( uminus_uminus_a @ K ) @ ( minus_minus_a @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_404_diff__conv__add__uminus,axiom,
( minus_minus_a
= ( ^ [A3: a,B3: a] : ( plus_plus_a @ A3 @ ( uminus_uminus_a @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_405_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_a
= ( ^ [A3: a,B3: a] : ( plus_plus_a @ A3 @ ( uminus_uminus_a @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_406_uminus__zero__vec__eq,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( ( uminus_uminus_vec_a @ V )
= ( zero_vec_a @ N ) )
= ( V
= ( zero_vec_a @ N ) ) ) ) ).
% uminus_zero_vec_eq
thf(fact_407_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_408_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_409_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_410_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_411_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_412_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_413_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_414_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_415_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_416_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_417_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_418_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_419_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_420_add__nonneg__eq__0__iff,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ zero_zero_a @ X2 )
=> ( ( ord_less_eq_a @ zero_zero_a @ Y2 )
=> ( ( ( plus_plus_a @ X2 @ Y2 )
= zero_zero_a )
= ( ( X2 = zero_zero_a )
& ( Y2 = zero_zero_a ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_421_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_422_add__nonpos__eq__0__iff,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ zero_zero_a )
=> ( ( ord_less_eq_a @ Y2 @ zero_zero_a )
=> ( ( ( plus_plus_a @ X2 @ Y2 )
= zero_zero_a )
= ( ( X2 = zero_zero_a )
& ( Y2 = zero_zero_a ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_423_le__iff__diff__le__0,axiom,
( ord_less_eq_a
= ( ^ [A3: a,B3: a] : ( ord_less_eq_a @ ( minus_minus_a @ A3 @ B3 ) @ zero_zero_a ) ) ) ).
% le_iff_diff_le_0
thf(fact_424_uminus__add__minus__vec,axiom,
! [L: vec_a,N: nat,R: vec_a] :
( ( member_vec_a @ L @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ R @ ( carrier_vec_a @ N ) )
=> ( ( uminus_uminus_vec_a @ ( plus_plus_vec_a @ L @ R ) )
= ( minus_minus_vec_a @ ( uminus_uminus_vec_a @ L ) @ R ) ) ) ) ).
% uminus_add_minus_vec
thf(fact_425_minus__add__uminus__vec,axiom,
! [V: vec_a,N: nat,W: vec_a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ N ) )
=> ( ( minus_minus_vec_a @ V @ W )
= ( plus_plus_vec_a @ V @ ( uminus_uminus_vec_a @ W ) ) ) ) ) ).
% minus_add_uminus_vec
thf(fact_426_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_427_assoc__add__vecset,axiom,
! [A2: set_vec_a,N: nat,B2: set_vec_a,C4: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ B2 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ C4 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_set_vec_a @ A2 @ ( plus_plus_set_vec_a @ B2 @ C4 ) )
= ( plus_plus_set_vec_a @ ( plus_plus_set_vec_a @ A2 @ B2 ) @ C4 ) ) ) ) ) ).
% assoc_add_vecset
thf(fact_428_comm__add__vecset,axiom,
! [A2: set_vec_a,N: nat,B2: set_vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ ( carrier_vec_a @ N ) )
=> ( ( ord_le4791951621262958845_vec_a @ B2 @ ( carrier_vec_a @ N ) )
=> ( ( plus_plus_set_vec_a @ A2 @ B2 )
= ( plus_plus_set_vec_a @ B2 @ A2 ) ) ) ) ).
% comm_add_vecset
thf(fact_429_u2__def,axiom,
( u2
= ( vec_first_a @ u1 @ ( plus_plus_nat @ nr @ one_one_nat ) ) ) ).
% u2_def
thf(fact_430_u1,axiom,
member_vec_a @ u1 @ ( carrier_vec_a @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) ) ).
% u1
thf(fact_431_verit__minus__simplify_I3_J,axiom,
! [B: a] :
( ( minus_minus_a @ zero_zero_a @ B )
= ( uminus_uminus_a @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_432_id0,axiom,
( ( zero_vec_a @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) @ nr ) )
= ( append_vec_a @ ( append_vec_a @ ( append_vec_a @ ( zero_vec_a @ nr ) @ ( zero_vec_a @ one_one_nat ) ) @ ( append_vec_a @ ( zero_vec_a @ nc ) @ ( zero_vec_a @ nc ) ) ) @ ( zero_vec_a @ nr ) ) ) ).
% id0
thf(fact_433_u2,axiom,
member_vec_a @ u2 @ ( carrier_vec_a @ ( plus_plus_nat @ nr @ one_one_nat ) ) ).
% u2
thf(fact_434_class__ring_Ominus__zero,axiom,
( ( uminus_uminus_a @ zero_zero_a )
= zero_zero_a ) ).
% class_ring.minus_zero
thf(fact_435_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_436_verit__minus__simplify_I4_J,axiom,
! [B: a] :
( ( uminus_uminus_a @ ( uminus_uminus_a @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_437_ulv0,axiom,
ord_less_eq_vec_a @ ( zero_vec_a @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) @ nr ) ) @ ulv ).
% ulv0
thf(fact_438_bc,axiom,
member_vec_a @ bc @ ( carrier_vec_a @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) @ nr ) ) ).
% bc
thf(fact_439_bc__def,axiom,
( bc
= ( append_vec_a @ ( append_vec_a @ ( append_vec_a @ b @ ( zero_vec_a @ one_one_nat ) ) @ ( append_vec_a @ c @ ( uminus_uminus_vec_a @ c ) ) ) @ ( zero_vec_a @ nr ) ) ) ).
% bc_def
thf(fact_440_class__field_Ozero__not__one,axiom,
zero_zero_a != one_one_a ).
% class_field.zero_not_one
thf(fact_441_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_442_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_443_zero__neq__one,axiom,
zero_zero_a != one_one_a ).
% zero_neq_one
thf(fact_444_class__field_Oneg__1__not__0,axiom,
( ( uminus_uminus_a @ one_one_a )
!= zero_zero_a ) ).
% class_field.neg_1_not_0
thf(fact_445_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_446_not__one__le__zero,axiom,
~ ( ord_less_eq_a @ one_one_a @ zero_zero_a ) ).
% not_one_le_zero
thf(fact_447_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_448_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_449_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_450_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_451_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_452_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_453_verit__comp__simplify1_I2_J,axiom,
! [A: vec_a] : ( ord_less_eq_vec_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_454_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_455_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_456_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( A = B )
=> ( ( C = D )
=> ( ( plus_plus_nat @ A @ C )
= ( plus_plus_nat @ B @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_457_class__semiring_Oadd_Ofactors__equal,axiom,
! [A: a,B: a,C: a,D: a] :
( ( A = B )
=> ( ( C = D )
=> ( ( plus_plus_a @ A @ C )
= ( plus_plus_a @ B @ D ) ) ) ) ).
% class_semiring.add.factors_equal
thf(fact_458_verit__negate__coefficient_I3_J,axiom,
! [A: a,B: a] :
( ( A = B )
=> ( ( uminus_uminus_a @ A )
= ( uminus_uminus_a @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_459_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_460_verit__sum__simplify,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% verit_sum_simplify
thf(fact_461_class__cring_Ocring__simprules_I22_J,axiom,
( ( uminus_uminus_a @ zero_zero_a )
= zero_zero_a ) ).
% class_cring.cring_simprules(22)
thf(fact_462_class__ring_Ominus__eq,axiom,
( minus_minus_a
= ( ^ [X: a,Y: a] : ( plus_plus_a @ X @ ( uminus_uminus_a @ Y ) ) ) ) ).
% class_ring.minus_eq
thf(fact_463_u1__def,axiom,
( u1
= ( vec_first_a @ ulv @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) ) ) ).
% u1_def
thf(fact_464_diff__numeral__special_I12_J,axiom,
( ( minus_minus_a @ ( uminus_uminus_a @ one_one_a ) @ ( uminus_uminus_a @ one_one_a ) )
= zero_zero_a ) ).
% diff_numeral_special(12)
thf(fact_465_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_a @ one_one_a @ ( uminus_uminus_a @ one_one_a ) )
= zero_zero_a ) ).
% add_neg_numeral_special(7)
thf(fact_466_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_a @ ( uminus_uminus_a @ one_one_a ) @ one_one_a )
= zero_zero_a ) ).
% add_neg_numeral_special(8)
thf(fact_467_ulv,axiom,
member_vec_a @ ulv @ ( carrier_vec_a @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) @ nr ) ) ).
% ulv
thf(fact_468_diff__numeral__special_I9_J,axiom,
( ( minus_minus_a @ one_one_a @ one_one_a )
= zero_zero_a ) ).
% diff_numeral_special(9)
thf(fact_469_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_470_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_471_le__numeral__extra_I3_J,axiom,
ord_less_eq_a @ zero_zero_a @ zero_zero_a ).
% le_numeral_extra(3)
thf(fact_472_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_473_le__numeral__extra_I4_J,axiom,
ord_less_eq_a @ one_one_a @ one_one_a ).
% le_numeral_extra(4)
thf(fact_474_one__neq__neg__one,axiom,
( one_one_a
!= ( uminus_uminus_a @ one_one_a ) ) ).
% one_neq_neg_one
thf(fact_475_is__num__normalize_I8_J,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( plus_plus_a @ A @ B ) )
= ( plus_plus_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_476_zero__neq__neg__one,axiom,
( zero_zero_a
!= ( uminus_uminus_a @ one_one_a ) ) ).
% zero_neq_neg_one
thf(fact_477_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_a @ one_one_a @ ( uminus_uminus_a @ one_one_a ) ) ).
% le_minus_one_simps(4)
thf(fact_478_le__minus__one__simps_I2_J,axiom,
ord_less_eq_a @ ( uminus_uminus_a @ one_one_a ) @ one_one_a ).
% le_minus_one_simps(2)
thf(fact_479_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_a @ zero_zero_a @ ( uminus_uminus_a @ one_one_a ) ) ).
% le_minus_one_simps(3)
thf(fact_480_le__minus__one__simps_I1_J,axiom,
ord_less_eq_a @ ( uminus_uminus_a @ one_one_a ) @ zero_zero_a ).
% le_minus_one_simps(1)
thf(fact_481__092_060open_062ulv_A_092_060bullet_062_Abc_A_061_Au_A_092_060bullet_062_Ab_A_L_A_Iv_A_092_060bullet_062_Ac_A_L_Aw_A_092_060bullet_062_A_N_Ac_J_092_060close_062,axiom,
( ( scalar_prod_a @ ulv @ bc )
= ( plus_plus_a @ ( scalar_prod_a @ u @ b ) @ ( plus_plus_a @ ( scalar_prod_a @ v @ c ) @ ( scalar_prod_a @ w @ ( uminus_uminus_vec_a @ c ) ) ) ) ) ).
% \<open>ulv \<bullet> bc = u \<bullet> b + (v \<bullet> c + w \<bullet> - c)\<close>
thf(fact_482_ulvbc,axiom,
( ( scalar_prod_a @ ulv @ bc )
= ( plus_plus_a @ ( scalar_prod_a @ u @ b ) @ ( scalar_prod_a @ c @ ( minus_minus_vec_a @ v @ w ) ) ) ) ).
% ulvbc
thf(fact_483_L__def,axiom,
( l
= ( vec_last_a @ u2 @ one_one_nat ) ) ).
% L_def
thf(fact_484_ineqs_I2_J,axiom,
ord_less_eq_vec_a @ ( zero_vec_a @ one_one_nat ) @ l ).
% ineqs(2)
thf(fact_485_ulvid,axiom,
( ulv
= ( append_vec_a @ u1 @ t ) ) ).
% ulvid
thf(fact_486_t__def,axiom,
( t
= ( vec_last_a @ ulv @ nr ) ) ).
% t_def
thf(fact_487_L,axiom,
member_vec_a @ l @ ( carrier_vec_a @ one_one_nat ) ).
% L
thf(fact_488_u,axiom,
member_vec_a @ u @ ( carrier_vec_a @ nr ) ).
% u
thf(fact_489_t,axiom,
member_vec_a @ t @ ( carrier_vec_a @ nr ) ).
% t
thf(fact_490_preconds_I2_J,axiom,
ord_less_eq_vec_a @ ( zero_vec_a @ nr ) @ u ).
% preconds(2)
thf(fact_491_ineqs_I5_J,axiom,
ord_less_eq_vec_a @ ( zero_vec_a @ nr ) @ t ).
% ineqs(5)
thf(fact_492_u2id,axiom,
( u2
= ( append_vec_a @ u @ l ) ) ).
% u2id
thf(fact_493_u__def,axiom,
( u
= ( vec_first_a @ u2 @ nr ) ) ).
% u_def
thf(fact_494__092_060open_0620_092_060_094sub_062v_Anr_A_092_060le_062_Au_A_092_060and_062_A0_092_060_094sub_062v_A1_A_092_060le_062_AL_A_092_060and_062_A0_092_060_094sub_062v_Anc_A_092_060le_062_Av_A_092_060and_062_A0_092_060_094sub_062v_Anc_A_092_060le_062_Aw_A_092_060and_062_A0_092_060_094sub_062v_Anr_A_092_060le_062_At_092_060close_062,axiom,
( ( ord_less_eq_vec_a @ ( zero_vec_a @ nr ) @ u )
& ( ord_less_eq_vec_a @ ( zero_vec_a @ one_one_nat ) @ l )
& ( ord_less_eq_vec_a @ ( zero_vec_a @ nc ) @ v )
& ( ord_less_eq_vec_a @ ( zero_vec_a @ nc ) @ w )
& ( ord_less_eq_vec_a @ ( zero_vec_a @ nr ) @ t ) ) ).
% \<open>0\<^sub>v nr \<le> u \<and> 0\<^sub>v 1 \<le> L \<and> 0\<^sub>v nc \<le> v \<and> 0\<^sub>v nc \<le> w \<and> 0\<^sub>v nr \<le> t\<close>
thf(fact_495__092_060open_062_I_I0_092_060_094sub_062v_Anr_A_064_092_060_094sub_062v_A0_092_060_094sub_062v_A1_J_A_064_092_060_094sub_062v_A0_092_060_094sub_062v_Anc_A_064_092_060_094sub_062v_A0_092_060_094sub_062v_Anc_J_A_064_092_060_094sub_062v_A0_092_060_094sub_062v_Anr_A_092_060le_062_A_I_Iu_A_064_092_060_094sub_062v_AL_J_A_064_092_060_094sub_062v_Av_A_064_092_060_094sub_062v_Aw_J_A_064_092_060_094sub_062v_At_092_060close_062,axiom,
ord_less_eq_vec_a @ ( append_vec_a @ ( append_vec_a @ ( append_vec_a @ ( zero_vec_a @ nr ) @ ( zero_vec_a @ one_one_nat ) ) @ ( append_vec_a @ ( zero_vec_a @ nc ) @ ( zero_vec_a @ nc ) ) ) @ ( zero_vec_a @ nr ) ) @ ( append_vec_a @ ( append_vec_a @ ( append_vec_a @ u @ l ) @ ( append_vec_a @ v @ w ) ) @ t ) ).
% \<open>((0\<^sub>v nr @\<^sub>v 0\<^sub>v 1) @\<^sub>v 0\<^sub>v nc @\<^sub>v 0\<^sub>v nc) @\<^sub>v 0\<^sub>v nr \<le> ((u @\<^sub>v L) @\<^sub>v v @\<^sub>v w) @\<^sub>v t\<close>
thf(fact_496_vec2,axiom,
member_vec_a @ vec2 @ ( carrier_vec_a @ nr ) ).
% vec2
thf(fact_497__092_060open_062vec1_A_064_092_060_094sub_062v_Avec3_A_L_Avec2_A_N_At_A_061_A0_092_060_094sub_062v_A_Inc_A_L_Anr_J_092_060close_062,axiom,
( ( append_vec_a @ vec1 @ ( minus_minus_vec_a @ ( plus_plus_vec_a @ vec3 @ vec2 ) @ t ) )
= ( zero_vec_a @ ( plus_plus_nat @ nc @ nr ) ) ) ).
% \<open>vec1 @\<^sub>v vec3 + vec2 - t = 0\<^sub>v (nc + nr)\<close>
thf(fact_498_scalar__prod__ge__0,axiom,
! [X2: vec_a] : ( ord_less_eq_a @ zero_zero_a @ ( scalar_prod_a @ X2 @ X2 ) ) ).
% scalar_prod_ge_0
thf(fact_499__092_060open_062vec1_A_064_092_060_094sub_062v_Avec3_A_L_A_I0_092_060_094sub_062v_Anc_A_064_092_060_094sub_062v_Avec2_J_A_L_A_I0_092_060_094sub_062v_Anc_A_064_092_060_094sub_062v_A_N_At_J_A_061_Avec1_A_064_092_060_094sub_062v_Avec3_A_L_Avec2_A_N_At_092_060close_062,axiom,
( ( plus_plus_vec_a @ ( plus_plus_vec_a @ ( append_vec_a @ vec1 @ vec3 ) @ ( append_vec_a @ ( zero_vec_a @ nc ) @ vec2 ) ) @ ( append_vec_a @ ( zero_vec_a @ nc ) @ ( uminus_uminus_vec_a @ t ) ) )
= ( append_vec_a @ vec1 @ ( minus_minus_vec_a @ ( plus_plus_vec_a @ vec3 @ vec2 ) @ t ) ) ) ).
% \<open>vec1 @\<^sub>v vec3 + (0\<^sub>v nc @\<^sub>v vec2) + (0\<^sub>v nc @\<^sub>v - t) = vec1 @\<^sub>v vec3 + vec2 - t\<close>
thf(fact_500__092_060open_062vec1_A_061_A0_092_060_094sub_062v_Anc_A_092_060and_062_Avec3_A_L_Avec2_A_N_At_A_061_A0_092_060_094sub_062v_Anr_092_060close_062,axiom,
( ( vec1
= ( zero_vec_a @ nc ) )
& ( ( minus_minus_vec_a @ ( plus_plus_vec_a @ vec3 @ vec2 ) @ t )
= ( zero_vec_a @ nr ) ) ) ).
% \<open>vec1 = 0\<^sub>v nc \<and> vec3 + vec2 - t = 0\<^sub>v nr\<close>
thf(fact_501_dbl__dec__simps_I2_J,axiom,
( ( neg_nu181380926503873385_dec_a @ zero_zero_a )
= ( uminus_uminus_a @ one_one_a ) ) ).
% dbl_dec_simps(2)
thf(fact_502_t23,axiom,
( t
= ( plus_plus_vec_a @ vec2 @ vec3 ) ) ).
% t23
thf(fact_503__092_060open_062vec3_A_L_Avec2_A_N_At_A_L_At_A_061_Avec2_A_L_Avec3_092_060close_062,axiom,
( ( plus_plus_vec_a @ ( minus_minus_vec_a @ ( plus_plus_vec_a @ vec3 @ vec2 ) @ t ) @ t )
= ( plus_plus_vec_a @ vec2 @ vec3 ) ) ).
% \<open>vec3 + vec2 - t + t = vec2 + vec3\<close>
thf(fact_504__092_060open_062vec3_A_L_Avec2_A_N_At_A_L_At_A_061_A0_092_060_094sub_062v_Anr_A_L_At_092_060close_062,axiom,
( ( plus_plus_vec_a @ ( minus_minus_vec_a @ ( plus_plus_vec_a @ vec3 @ vec2 ) @ t ) @ t )
= ( plus_plus_vec_a @ ( zero_vec_a @ nr ) @ t ) ) ).
% \<open>vec3 + vec2 - t + t = 0\<^sub>v nr + t\<close>
thf(fact_505__C02_C,axiom,
( ( minus_minus_vec_a @ ( plus_plus_vec_a @ vec3 @ vec2 ) @ t )
= ( zero_vec_a @ nr ) ) ).
% "02"
thf(fact_506_dbl__dec__def,axiom,
( neg_nu181380926503873385_dec_a
= ( ^ [X: a] : ( minus_minus_a @ ( plus_plus_a @ X @ X ) @ one_one_a ) ) ) ).
% dbl_dec_def
thf(fact_507_subsetI,axiom,
! [A2: set_vec_a,B2: set_vec_a] :
( ! [X3: vec_a] :
( ( member_vec_a @ X3 @ A2 )
=> ( member_vec_a @ X3 @ B2 ) )
=> ( ord_le4791951621262958845_vec_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_508_subsetI,axiom,
! [A2: set_mat_a,B2: set_mat_a] :
( ! [X3: mat_a] :
( ( member_mat_a @ X3 @ A2 )
=> ( member_mat_a @ X3 @ B2 ) )
=> ( ord_le3318621148231462513_mat_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_509_norm1__ge__0,axiom,
! [F: poly_a] : ( ord_less_eq_a @ zero_zero_a @ ( norm1_a @ F ) ) ).
% norm1_ge_0
thf(fact_510_subset__iff,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [A5: set_vec_a,B5: set_vec_a] :
! [T: vec_a] :
( ( member_vec_a @ T @ A5 )
=> ( member_vec_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_511_subset__iff,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A5: set_mat_a,B5: set_mat_a] :
! [T: mat_a] :
( ( member_mat_a @ T @ A5 )
=> ( member_mat_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_512_subset__eq,axiom,
( ord_le4791951621262958845_vec_a
= ( ^ [A5: set_vec_a,B5: set_vec_a] :
! [X: vec_a] :
( ( member_vec_a @ X @ A5 )
=> ( member_vec_a @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_513_subset__eq,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A5: set_mat_a,B5: set_mat_a] :
! [X: mat_a] :
( ( member_mat_a @ X @ A5 )
=> ( member_mat_a @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_514_subsetD,axiom,
! [A2: set_vec_a,B2: set_vec_a,C: vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ B2 )
=> ( ( member_vec_a @ C @ A2 )
=> ( member_vec_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_515_subsetD,axiom,
! [A2: set_mat_a,B2: set_mat_a,C: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A2 @ B2 )
=> ( ( member_mat_a @ C @ A2 )
=> ( member_mat_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_516_in__mono,axiom,
! [A2: set_vec_a,B2: set_vec_a,X2: vec_a] :
( ( ord_le4791951621262958845_vec_a @ A2 @ B2 )
=> ( ( member_vec_a @ X2 @ A2 )
=> ( member_vec_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_517_in__mono,axiom,
! [A2: set_mat_a,B2: set_mat_a,X2: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A2 @ B2 )
=> ( ( member_mat_a @ X2 @ A2 )
=> ( member_mat_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_518_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_519_vec__inv,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( V
!= ( zero_vec_a @ N ) )
=> ( ( scalar_prod_a @ ( schur_vec_inv_a @ V ) @ V )
= one_one_a ) ) ) ).
% vec_inv
thf(fact_520_linf__norm__vec__eq__0,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( ( linf_norm_vec_a @ V )
= zero_zero_a )
= ( V
= ( zero_vec_a @ N ) ) ) ) ).
% linf_norm_vec_eq_0
thf(fact_521_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_522_add__0__iff,axiom,
! [B: a,A: a] :
( ( B
= ( plus_plus_a @ B @ A ) )
= ( A = zero_zero_a ) ) ).
% add_0_iff
thf(fact_523_euclidean__size__field__def,axiom,
( field_8182332513790994628ield_a
= ( ^ [X: a] : ( if_nat @ ( X = zero_zero_a ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% euclidean_size_field_def
thf(fact_524_linf__norm__vec__ge__0,axiom,
! [V: vec_a] : ( ord_less_eq_a @ zero_zero_a @ ( linf_norm_vec_a @ V ) ) ).
% linf_norm_vec_ge_0
thf(fact_525_linf__norm__zero__vec,axiom,
! [N: nat] :
( ( linf_norm_vec_a @ ( zero_vec_a @ N ) )
= zero_zero_a ) ).
% linf_norm_zero_vec
thf(fact_526_vec__inv__closed,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( member_vec_a @ ( schur_vec_inv_a @ V ) @ ( carrier_vec_a @ N ) ) ) ).
% vec_inv_closed
thf(fact_527_normalize__field__def,axiom,
( field_879120192484243794ield_a
= ( ^ [X: a] : ( if_a @ ( X = zero_zero_a ) @ zero_zero_a @ one_one_a ) ) ) ).
% normalize_field_def
thf(fact_528_mod__field__def,axiom,
( field_4049668205554139175ield_a
= ( ^ [X: a,Y: a] : ( if_a @ ( Y = zero_zero_a ) @ X @ zero_zero_a ) ) ) ).
% mod_field_def
thf(fact_529_primal,axiom,
? [X3: vec_a] :
( ( member_vec_a @ X3 @ ( carrier_vec_a @ nc ) )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ a2 @ X3 ) @ b ) ) ).
% primal
thf(fact_530__092_060open_062vec1_A_L_Amat__of__col_Ac_A_K_092_060_094sub_062v_AL_A_061_Amat__of__col_Ac_A_K_092_060_094sub_062v_AL_092_060close_062,axiom,
( ( plus_plus_vec_a @ vec1 @ ( mult_mat_vec_a @ ( missing_mat_of_col_a @ c ) @ l ) )
= ( mult_mat_vec_a @ ( missing_mat_of_col_a @ c ) @ l ) ) ).
% \<open>vec1 + mat_of_col c *\<^sub>v L = mat_of_col c *\<^sub>v L\<close>
thf(fact_531_dbl__inc__simps_I4_J,axiom,
( ( neg_nu6917059380386235053_inc_a @ ( uminus_uminus_a @ one_one_a ) )
= ( uminus_uminus_a @ one_one_a ) ) ).
% dbl_inc_simps(4)
thf(fact_532_vec3__def,axiom,
( vec3
= ( mult_mat_vec_a @ ( missing_mat_of_col_a @ b ) @ l ) ) ).
% vec3_def
thf(fact_533__092_060open_062A_A_K_092_060_094sub_062v_Av_A_L_A_N_AA_A_K_092_060_094sub_062v_Aw_A_061_Avec2_092_060close_062,axiom,
( ( plus_plus_vec_a @ ( mult_mat_vec_a @ a2 @ v ) @ ( mult_mat_vec_a @ ( uminus_uminus_mat_a @ a2 ) @ w ) )
= vec2 ) ).
% \<open>A *\<^sub>v v + - A *\<^sub>v w = vec2\<close>
thf(fact_534_vec2__def,axiom,
( vec2
= ( mult_mat_vec_a @ a2 @ ( minus_minus_vec_a @ v @ w ) ) ) ).
% vec2_def
thf(fact_535_dbl__inc__simps_I2_J,axiom,
( ( neg_nu6917059380386235053_inc_a @ zero_zero_a )
= one_one_a ) ).
% dbl_inc_simps(2)
thf(fact_536_A,axiom,
member_mat_a @ a2 @ ( carrier_mat_a @ nr @ nc ) ).
% A
thf(fact_537_dual,axiom,
? [Y3: vec_a] :
( ( ord_less_eq_vec_a @ ( zero_vec_a @ nr ) @ Y3 )
& ( ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ Y3 )
= c ) ) ).
% dual
thf(fact_538_dbl__inc__def,axiom,
( neg_nu6917059380386235053_inc_a
= ( ^ [X: a] : ( plus_plus_a @ ( plus_plus_a @ X @ X ) @ one_one_a ) ) ) ).
% dbl_inc_def
thf(fact_539_vec1__def,axiom,
( vec1
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ u ) @ ( mult_mat_vec_a @ ( missing_mat_of_col_a @ ( uminus_uminus_vec_a @ c ) ) @ l ) ) ) ).
% vec1_def
thf(fact_540__092_060open_062vec1_A_L_Amat__of__col_Ac_A_K_092_060_094sub_062v_AL_A_061_AA_092_060_094sup_062T_A_K_092_060_094sub_062v_Au_092_060close_062,axiom,
( ( plus_plus_vec_a @ vec1 @ ( mult_mat_vec_a @ ( missing_mat_of_col_a @ c ) @ l ) )
= ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ u ) ) ).
% \<open>vec1 + mat_of_col c *\<^sub>v L = A\<^sup>T *\<^sub>v u\<close>
thf(fact_541__092_060open_0620_092_060_094sub_062m_Anc_Anc_A_K_092_060_094sub_062v_Av_A_L_A0_092_060_094sub_062m_Anc_Anc_A_K_092_060_094sub_062v_Aw_A_061_A0_092_060_094sub_062v_Anc_092_060close_062,axiom,
( ( plus_plus_vec_a @ ( mult_mat_vec_a @ ( zero_mat_a @ nc @ nc ) @ v ) @ ( mult_mat_vec_a @ ( zero_mat_a @ nc @ nc ) @ w ) )
= ( zero_vec_a @ nc ) ) ).
% \<open>0\<^sub>m nc nc *\<^sub>v v + 0\<^sub>m nc nc *\<^sub>v w = 0\<^sub>v nc\<close>
thf(fact_542_Mulv,axiom,
( ( mult_mat_vec_a @ ( transpose_mat_a @ m ) @ ulv )
= ( zero_vec_a @ ( plus_plus_nat @ nc @ nr ) ) ) ).
% Mulv
thf(fact_543_preconds_I5_J,axiom,
ord_less_eq_vec_a @ ( zero_vec_a @ nr ) @ ( plus_plus_vec_a @ ( mult_mat_vec_a @ a2 @ ( minus_minus_vec_a @ v @ w ) ) @ ( smult_vec_a @ lam @ b ) ) ).
% preconds(5)
thf(fact_544_lam0,axiom,
ord_less_eq_a @ zero_zero_a @ lam ).
% lam0
thf(fact_545_assoc__add__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a,C4: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ C4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( plus_plus_mat_a @ A2 @ B2 ) @ C4 )
= ( plus_plus_mat_a @ A2 @ ( plus_plus_mat_a @ B2 @ C4 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_546_Matrix_Otranspose__transpose,axiom,
! [A2: mat_a] :
( ( transpose_mat_a @ ( transpose_mat_a @ A2 ) )
= A2 ) ).
% Matrix.transpose_transpose
thf(fact_547_transpose__mat__eq,axiom,
! [A2: mat_a,B2: mat_a] :
( ( ( transpose_mat_a @ A2 )
= ( transpose_mat_a @ B2 ) )
= ( A2 = B2 ) ) ).
% transpose_mat_eq
thf(fact_548_True,axiom,
ord_less_a @ zero_zero_a @ lam ).
% True
thf(fact_549_vec3,axiom,
( vec3
= ( smult_vec_a @ lam @ b ) ) ).
% vec3
thf(fact_550_scalar__vec__one,axiom,
! [V: vec_a] :
( ( smult_vec_a @ one_one_a @ V )
= V ) ).
% scalar_vec_one
thf(fact_551_scalar__vec__one,axiom,
! [V: vec_nat] :
( ( smult_vec_nat @ one_one_nat @ V )
= V ) ).
% scalar_vec_one
thf(fact_552_one__smult__vec,axiom,
! [V: vec_a] :
( ( smult_vec_a @ one_one_a @ V )
= V ) ).
% one_smult_vec
thf(fact_553_left__add__zero__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( zero_mat_a @ Nr @ Nc ) @ A2 )
= A2 ) ) ).
% left_add_zero_mat
thf(fact_554_right__add__zero__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ A2 @ ( zero_mat_a @ Nr @ Nc ) )
= A2 ) ) ).
% right_add_zero_mat
thf(fact_555_minus__r__inv__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( minus_minus_mat_a @ A2 @ A2 )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% minus_r_inv_mat
thf(fact_556_smult__carrier__vec,axiom,
! [A: a,V: vec_a,N: nat] :
( ( member_vec_a @ ( smult_vec_a @ A @ V ) @ ( carrier_vec_a @ N ) )
= ( member_vec_a @ V @ ( carrier_vec_a @ N ) ) ) ).
% smult_carrier_vec
thf(fact_557_transpose__carrier__mat,axiom,
! [A2: mat_a,Nc: nat,Nr: nat] :
( ( member_mat_a @ ( transpose_mat_a @ A2 ) @ ( carrier_mat_a @ Nc @ Nr ) )
= ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% transpose_carrier_mat
thf(fact_558_zero__transpose__mat,axiom,
! [N: nat,M: nat] :
( ( transpose_mat_a @ ( zero_mat_a @ N @ M ) )
= ( zero_mat_a @ M @ N ) ) ).
% zero_transpose_mat
thf(fact_559_uminus__carrier__iff__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ ( uminus_uminus_mat_a @ A2 ) @ ( carrier_mat_a @ Nr @ Nc ) )
= ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% uminus_carrier_iff_mat
thf(fact_560_M,axiom,
member_mat_a @ m @ ( carrier_mat_a @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) @ nr ) @ ( plus_plus_nat @ nc @ nr ) ) ).
% M
thf(fact_561_As,axiom,
( ( mult_mat_vec_a @ ( transpose_mat_a @ a2 ) @ u )
= ( smult_vec_a @ lam @ c ) ) ).
% As
thf(fact_562_uminus__l__inv__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( uminus_uminus_mat_a @ A2 ) @ A2 )
= ( zero_mat_a @ Nr @ Nc ) ) ) ).
% uminus_l_inv_mat
thf(fact_563_zero__mat__mult__vector,axiom,
! [X2: vec_a,Nc: nat,Nr: nat] :
( ( member_vec_a @ X2 @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ ( zero_mat_a @ Nr @ Nc ) @ X2 )
= ( zero_vec_a @ Nr ) ) ) ).
% zero_mat_mult_vector
thf(fact_564_mult__mat__vec,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,V: vec_a,K: a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ A2 @ ( smult_vec_a @ K @ V ) )
= ( smult_vec_a @ K @ ( mult_mat_vec_a @ A2 @ V ) ) ) ) ) ).
% mult_mat_vec
thf(fact_565_uminus__add__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( uminus_uminus_mat_a @ ( plus_plus_mat_a @ A2 @ B2 ) )
= ( plus_plus_mat_a @ ( uminus_uminus_mat_a @ B2 ) @ ( uminus_uminus_mat_a @ A2 ) ) ) ) ) ).
% uminus_add_mat
thf(fact_566_transpose__uminus,axiom,
! [A2: mat_a] :
( ( transpose_mat_a @ ( uminus_uminus_mat_a @ A2 ) )
= ( uminus_uminus_mat_a @ ( transpose_mat_a @ A2 ) ) ) ).
% transpose_uminus
thf(fact_567_uminus__carrier__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( uminus_uminus_mat_a @ A2 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% uminus_carrier_mat
thf(fact_568_zero__carrier__mat,axiom,
! [Nr: nat,Nc: nat] : ( member_mat_a @ ( zero_mat_a @ Nr @ Nc ) @ ( carrier_mat_a @ Nr @ Nc ) ) ).
% zero_carrier_mat
thf(fact_569_smult__vec__nonneg__eq,axiom,
! [C: a,X2: vec_a,Y2: vec_a] :
( ( C != zero_zero_a )
=> ( ( ( smult_vec_a @ C @ X2 )
= ( smult_vec_a @ C @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% smult_vec_nonneg_eq
thf(fact_570_comm__add__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ A2 @ B2 )
= ( plus_plus_mat_a @ B2 @ A2 ) ) ) ) ).
% comm_add_mat
thf(fact_571_transpose__add,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( transpose_mat_a @ ( plus_plus_mat_a @ A2 @ B2 ) )
= ( plus_plus_mat_a @ ( transpose_mat_a @ A2 ) @ ( transpose_mat_a @ B2 ) ) ) ) ) ).
% transpose_add
thf(fact_572_add__carrier__mat,axiom,
! [B2: mat_a,Nr: nat,Nc: nat,A2: mat_a] :
( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( plus_plus_mat_a @ A2 @ B2 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% add_carrier_mat
thf(fact_573_add__inv__exists__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ? [X3: mat_a] :
( ( member_mat_a @ X3 @ ( carrier_mat_a @ Nr @ Nc ) )
& ( ( plus_plus_mat_a @ X3 @ A2 )
= ( zero_mat_a @ Nr @ Nc ) )
& ( ( plus_plus_mat_a @ A2 @ X3 )
= ( zero_mat_a @ Nr @ Nc ) ) ) ) ).
% add_inv_exists_mat
thf(fact_574_minus__add__minus__mat,axiom,
! [U: mat_a,Nr: nat,Nc: nat,V: mat_a,W: mat_a] :
( ( member_mat_a @ U @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ V @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ W @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( minus_minus_mat_a @ U @ ( plus_plus_mat_a @ V @ W ) )
= ( minus_minus_mat_a @ ( minus_minus_mat_a @ U @ V ) @ W ) ) ) ) ) ).
% minus_add_minus_mat
thf(fact_575_minus__carrier__mat,axiom,
! [B2: mat_a,Nr: nat,Nc: nat,A2: mat_a] :
( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( minus_minus_mat_a @ A2 @ B2 ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_576_transpose__minus,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( transpose_mat_a @ ( minus_minus_mat_a @ A2 @ B2 ) )
= ( minus_minus_mat_a @ ( transpose_mat_a @ A2 ) @ ( transpose_mat_a @ B2 ) ) ) ) ) ).
% transpose_minus
thf(fact_577_uminus__add__minus__mat,axiom,
! [L: mat_a,Nr: nat,Nc: nat,R: mat_a] :
( ( member_mat_a @ L @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ R @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( uminus_uminus_mat_a @ ( plus_plus_mat_a @ L @ R ) )
= ( minus_minus_mat_a @ ( uminus_uminus_mat_a @ L ) @ R ) ) ) ) ).
% uminus_add_minus_mat
thf(fact_578_minus__add__uminus__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( minus_minus_mat_a @ A2 @ B2 )
= ( plus_plus_mat_a @ A2 @ ( uminus_uminus_mat_a @ B2 ) ) ) ) ) ).
% minus_add_uminus_mat
thf(fact_579_add__uminus__minus__mat,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ A2 @ ( uminus_uminus_mat_a @ B2 ) )
= ( minus_minus_mat_a @ A2 @ B2 ) ) ) ) ).
% add_uminus_minus_mat
thf(fact_580_transpose__vec__mult__scalar,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,X2: vec_a,Y2: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ X2 @ ( carrier_vec_a @ Nc ) )
=> ( ( member_vec_a @ Y2 @ ( carrier_vec_a @ Nr ) )
=> ( ( scalar_prod_a @ ( mult_mat_vec_a @ ( transpose_mat_a @ A2 ) @ Y2 ) @ X2 )
= ( scalar_prod_a @ Y2 @ ( mult_mat_vec_a @ A2 @ X2 ) ) ) ) ) ) ).
% transpose_vec_mult_scalar
thf(fact_581_add__smult__distrib__vec,axiom,
! [A: a,B: a,V: vec_a] :
( ( smult_vec_a @ ( plus_plus_a @ A @ B ) @ V )
= ( plus_plus_vec_a @ ( smult_vec_a @ A @ V ) @ ( smult_vec_a @ B @ V ) ) ) ).
% add_smult_distrib_vec
thf(fact_582_smult__add__distrib__vec,axiom,
! [V: vec_a,N: nat,W: vec_a,A: a] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ N ) )
=> ( ( smult_vec_a @ A @ ( plus_plus_vec_a @ V @ W ) )
= ( plus_plus_vec_a @ ( smult_vec_a @ A @ V ) @ ( smult_vec_a @ A @ W ) ) ) ) ) ).
% smult_add_distrib_vec
thf(fact_583_diff__smult__distrib__vec,axiom,
! [X2: a,Y2: a,V: vec_a] :
( ( smult_vec_a @ ( minus_minus_a @ X2 @ Y2 ) @ V )
= ( minus_minus_vec_a @ ( smult_vec_a @ X2 @ V ) @ ( smult_vec_a @ Y2 @ V ) ) ) ).
% diff_smult_distrib_vec
thf(fact_584_mult__mat__vec__carrier,axiom,
! [A2: mat_a,Nr: nat,N: nat,V: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( member_vec_a @ ( mult_mat_vec_a @ A2 @ V ) @ ( carrier_vec_a @ Nr ) ) ) ) ).
% mult_mat_vec_carrier
thf(fact_585_weak__duality__theorem,axiom,
! [A2: mat_nat,Nr: nat,Nc: nat,B: vec_nat,C: vec_nat,X2: vec_nat,Y2: vec_nat] :
( ( member_mat_nat @ A2 @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_vec_nat @ B @ ( carrier_vec_nat @ Nr ) )
=> ( ( member_vec_nat @ C @ ( carrier_vec_nat @ Nc ) )
=> ( ( member_vec_nat @ X2 @ ( carrier_vec_nat @ Nc ) )
=> ( ( ord_less_eq_vec_nat @ ( mult_mat_vec_nat @ A2 @ X2 ) @ B )
=> ( ( ord_less_eq_vec_nat @ ( zero_vec_nat @ Nr ) @ Y2 )
=> ( ( ( mult_mat_vec_nat @ ( transpose_mat_nat @ A2 ) @ Y2 )
= C )
=> ( ord_less_eq_nat @ ( scalar_prod_nat @ C @ X2 ) @ ( scalar_prod_nat @ B @ Y2 ) ) ) ) ) ) ) ) ) ).
% weak_duality_theorem
thf(fact_586_weak__duality__theorem,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B: vec_a,C: vec_a,X2: vec_a,Y2: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ Nr ) )
=> ( ( member_vec_a @ C @ ( carrier_vec_a @ Nc ) )
=> ( ( member_vec_a @ X2 @ ( carrier_vec_a @ Nc ) )
=> ( ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A2 @ X2 ) @ B )
=> ( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ Y2 )
=> ( ( ( mult_mat_vec_a @ ( transpose_mat_a @ A2 ) @ Y2 )
= C )
=> ( ord_less_eq_a @ ( scalar_prod_a @ C @ X2 ) @ ( scalar_prod_a @ B @ Y2 ) ) ) ) ) ) ) ) ) ).
% weak_duality_theorem
thf(fact_587_unbounded__dual__solutions,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B: vec_a,C: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ Nr ) )
=> ( ( member_vec_a @ C @ ( carrier_vec_a @ Nc ) )
=> ( ! [V3: a] :
? [Y5: vec_a] :
( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ Y5 )
& ( ( mult_mat_vec_a @ ( transpose_mat_a @ A2 ) @ Y5 )
= C )
& ( ord_less_eq_a @ ( scalar_prod_a @ B @ Y5 ) @ V3 ) )
=> ~ ? [X4: vec_a] :
( ( member_vec_a @ X4 @ ( carrier_vec_a @ Nc ) )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A2 @ X4 ) @ B ) ) ) ) ) ) ).
% unbounded_dual_solutions
thf(fact_588_unbounded__primal__solutions,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B: vec_a,C: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ Nr ) )
=> ( ( member_vec_a @ C @ ( carrier_vec_a @ Nc ) )
=> ( ! [V3: a] :
? [X4: vec_a] :
( ( member_vec_a @ X4 @ ( carrier_vec_a @ Nc ) )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A2 @ X4 ) @ B )
& ( ord_less_eq_a @ V3 @ ( scalar_prod_a @ C @ X4 ) ) )
=> ~ ? [Y5: vec_a] :
( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ Y5 )
& ( ( mult_mat_vec_a @ ( transpose_mat_a @ A2 ) @ Y5 )
= C ) ) ) ) ) ) ).
% unbounded_primal_solutions
thf(fact_589_vec__first__smult,axiom,
! [M: nat,N: nat,X2: vec_a,C: a] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( member_vec_a @ X2 @ ( carrier_vec_a @ N ) )
=> ( ( vec_first_a @ ( smult_vec_a @ C @ X2 ) @ M )
= ( smult_vec_a @ C @ ( vec_first_a @ X2 @ M ) ) ) ) ) ).
% vec_first_smult
thf(fact_590_add__mult__distrib__mat__vec,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a,V: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ ( plus_plus_mat_a @ A2 @ B2 ) @ V )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ A2 @ V ) @ ( mult_mat_vec_a @ B2 @ V ) ) ) ) ) ) ).
% add_mult_distrib_mat_vec
thf(fact_591_mult__add__distrib__mat__vec,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,V_1: vec_a,V_2: vec_a] :
( ( member_mat_a @ A2 @ ( 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 @ A2 @ ( plus_plus_vec_a @ V_1 @ V_2 ) )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ A2 @ V_1 ) @ ( mult_mat_vec_a @ A2 @ V_2 ) ) ) ) ) ) ).
% mult_add_distrib_mat_vec
thf(fact_592_minus__mult__distrib__mat__vec,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B2: mat_a,V: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ ( minus_minus_mat_a @ A2 @ B2 ) @ V )
= ( minus_minus_vec_a @ ( mult_mat_vec_a @ A2 @ V ) @ ( mult_mat_vec_a @ B2 @ V ) ) ) ) ) ) ).
% minus_mult_distrib_mat_vec
thf(fact_593_mult__minus__distrib__mat__vec,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,V: vec_a,W: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( member_vec_a @ W @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ A2 @ ( minus_minus_vec_a @ V @ W ) )
= ( minus_minus_vec_a @ ( mult_mat_vec_a @ A2 @ V ) @ ( mult_mat_vec_a @ A2 @ W ) ) ) ) ) ) ).
% mult_minus_distrib_mat_vec
thf(fact_594_smult__nneg__npos__vec,axiom,
! [L: nat,V: vec_nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ L )
=> ( ( ord_less_eq_vec_nat @ V @ ( zero_vec_nat @ N ) )
=> ( ord_less_eq_vec_nat @ ( smult_vec_nat @ L @ V ) @ ( zero_vec_nat @ N ) ) ) ) ).
% smult_nneg_npos_vec
thf(fact_595_smult__nneg__npos__vec,axiom,
! [L: a,V: vec_a,N: nat] :
( ( ord_less_eq_a @ zero_zero_a @ L )
=> ( ( ord_less_eq_vec_a @ V @ ( zero_vec_a @ N ) )
=> ( ord_less_eq_vec_a @ ( smult_vec_a @ L @ V ) @ ( zero_vec_a @ N ) ) ) ) ).
% smult_nneg_npos_vec
thf(fact_596_M__last,axiom,
member_mat_a @ m_last @ ( carrier_mat_a @ nr @ ( plus_plus_nat @ nc @ nr ) ) ).
% M_last
thf(fact_597_M__up,axiom,
member_mat_a @ m_up @ ( carrier_mat_a @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nr ) ) ).
% M_up
thf(fact_598__092_060open_062four__block__mat_AA_092_060_094sup_062T_A_Imat__of__col_A_I_N_Ac_J_J_A_I0_092_060_094sub_062m_Anr_Anr_J_A_Imat__of__col_Ab_J_A_K_092_060_094sub_062v_Au2_A_061_Avec1_A_064_092_060_094sub_062v_Avec3_092_060close_062,axiom,
( ( mult_mat_vec_a @ ( four_block_mat_a @ ( transpose_mat_a @ a2 ) @ ( missing_mat_of_col_a @ ( uminus_uminus_vec_a @ c ) ) @ ( zero_mat_a @ nr @ nr ) @ ( missing_mat_of_col_a @ b ) ) @ u2 )
= ( append_vec_a @ vec1 @ vec3 ) ) ).
% \<open>four_block_mat A\<^sup>T (mat_of_col (- c)) (0\<^sub>m nr nr) (mat_of_col b) *\<^sub>v u2 = vec1 @\<^sub>v vec3\<close>
thf(fact_599_M__low,axiom,
member_mat_a @ m_low @ ( carrier_mat_a @ ( plus_plus_nat @ nc @ nc ) @ ( plus_plus_nat @ nc @ nr ) ) ).
% M_low
thf(fact_600__092_060open_062four__block__mat_A_I0_092_060_094sub_062m_Anc_Anc_J_A_I0_092_060_094sub_062m_Anc_Anc_J_AA_A_I_N_AA_J_A_K_092_060_094sub_062v_Au3_A_061_A_I0_092_060_094sub_062m_Anc_Anc_A_K_092_060_094sub_062v_Av_A_L_A0_092_060_094sub_062m_Anc_Anc_A_K_092_060_094sub_062v_Aw_J_A_064_092_060_094sub_062v_AA_A_K_092_060_094sub_062v_Av_A_L_A_N_AA_A_K_092_060_094sub_062v_Aw_092_060close_062,axiom,
( ( mult_mat_vec_a @ ( four_block_mat_a @ ( zero_mat_a @ nc @ nc ) @ ( zero_mat_a @ nc @ nc ) @ a2 @ ( uminus_uminus_mat_a @ a2 ) ) @ u3 )
= ( append_vec_a @ ( plus_plus_vec_a @ ( mult_mat_vec_a @ ( zero_mat_a @ nc @ nc ) @ v ) @ ( mult_mat_vec_a @ ( zero_mat_a @ nc @ nc ) @ w ) ) @ ( plus_plus_vec_a @ ( mult_mat_vec_a @ a2 @ v ) @ ( mult_mat_vec_a @ ( uminus_uminus_mat_a @ a2 ) @ w ) ) ) ) ).
% \<open>four_block_mat (0\<^sub>m nc nc) (0\<^sub>m nc nc) A (- A) *\<^sub>v u3 = (0\<^sub>m nc nc *\<^sub>v v + 0\<^sub>m nc nc *\<^sub>v w) @\<^sub>v A *\<^sub>v v + - A *\<^sub>v w\<close>
thf(fact_601_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_602_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_603_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_604_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_605_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_606_neg__less__iff__less,axiom,
! [B: a,A: a] :
( ( ord_less_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
= ( ord_less_a @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_607_M__low__def,axiom,
( m_low
= ( four_block_mat_a @ ( zero_mat_a @ nc @ nc ) @ ( transpose_mat_a @ a2 ) @ ( zero_mat_a @ nc @ nc ) @ ( uminus_uminus_mat_a @ ( transpose_mat_a @ a2 ) ) ) ) ).
% M_low_def
thf(fact_608__092_060open_062M__low_092_060_094sup_062T_A_061_Afour__block__mat_A_I0_092_060_094sub_062m_Anc_Anc_J_A_I0_092_060_094sub_062m_Anc_Anc_J_AA_A_I_N_AA_J_092_060close_062,axiom,
( ( transpose_mat_a @ m_low )
= ( four_block_mat_a @ ( zero_mat_a @ nc @ nc ) @ ( zero_mat_a @ nc @ nc ) @ a2 @ ( uminus_uminus_mat_a @ a2 ) ) ) ).
% \<open>M_low\<^sup>T = four_block_mat (0\<^sub>m nc nc) (0\<^sub>m nc nc) A (- A)\<close>
thf(fact_609_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_610_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_611_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_612_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_613_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_614_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_615_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_616_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_617_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_618_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_619_diff__gt__0__iff__gt,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ ( minus_minus_a @ A @ B ) )
= ( ord_less_a @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_620_less__neg__neg,axiom,
! [A: a] :
( ( ord_less_a @ A @ ( uminus_uminus_a @ A ) )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% less_neg_neg
thf(fact_621_neg__less__pos,axiom,
! [A: a] :
( ( ord_less_a @ ( uminus_uminus_a @ A ) @ A )
= ( ord_less_a @ zero_zero_a @ A ) ) ).
% neg_less_pos
thf(fact_622_neg__0__less__iff__less,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% neg_0_less_iff_less
thf(fact_623_neg__less__0__iff__less,axiom,
! [A: a] :
( ( ord_less_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
= ( ord_less_a @ zero_zero_a @ A ) ) ).
% neg_less_0_iff_less
thf(fact_624_four__block__zero__mat,axiom,
! [Nr1: nat,Nc1: nat,Nc2: nat,Nr2: nat] :
( ( four_block_mat_a @ ( zero_mat_a @ Nr1 @ Nc1 ) @ ( zero_mat_a @ Nr1 @ Nc2 ) @ ( zero_mat_a @ Nr2 @ Nc1 ) @ ( zero_mat_a @ Nr2 @ Nc2 ) )
= ( zero_mat_a @ ( plus_plus_nat @ Nr1 @ Nr2 ) @ ( plus_plus_nat @ Nc1 @ Nc2 ) ) ) ).
% four_block_zero_mat
thf(fact_625__092_060open_062M__up_092_060_094sup_062T_A_061_Afour__block__mat_AA_092_060_094sup_062T_A_Imat__of__col_A_I_N_Ac_J_J_A_I0_092_060_094sub_062m_Anr_Anr_J_A_Imat__of__col_Ab_J_092_060close_062,axiom,
( ( transpose_mat_a @ m_up )
= ( four_block_mat_a @ ( transpose_mat_a @ a2 ) @ ( missing_mat_of_col_a @ ( uminus_uminus_vec_a @ c ) ) @ ( zero_mat_a @ nr @ nr ) @ ( missing_mat_of_col_a @ b ) ) ) ).
% \<open>M_up\<^sup>T = four_block_mat A\<^sup>T (mat_of_col (- c)) (0\<^sub>m nr nr) (mat_of_col b)\<close>
thf(fact_626_smult__pos__vec,axiom,
! [L: a,V: vec_a,N: nat] :
( ( ord_less_a @ zero_zero_a @ L )
=> ( ( ord_less_eq_vec_a @ ( smult_vec_a @ L @ V ) @ ( zero_vec_a @ N ) )
= ( ord_less_eq_vec_a @ V @ ( zero_vec_a @ N ) ) ) ) ).
% smult_pos_vec
thf(fact_627_linf__norm__vec__greater__0,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( ord_less_a @ zero_zero_a @ ( linf_norm_vec_a @ V ) )
= ( V
!= ( zero_vec_a @ N ) ) ) ) ).
% linf_norm_vec_greater_0
thf(fact_628_add__four__block__mat,axiom,
! [A1: mat_a,Nr1: nat,Nc1: nat,B1: mat_a,Nc2: nat,C1: mat_a,Nr2: nat,D1: mat_a,A22: mat_a,B22: mat_a,C22: mat_a,D22: mat_a] :
( ( member_mat_a @ A1 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B1 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C1 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D1 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( member_mat_a @ A22 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B22 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C22 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D22 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( plus_plus_mat_a @ ( four_block_mat_a @ A1 @ B1 @ C1 @ D1 ) @ ( four_block_mat_a @ A22 @ B22 @ C22 @ D22 ) )
= ( four_block_mat_a @ ( plus_plus_mat_a @ A1 @ A22 ) @ ( plus_plus_mat_a @ B1 @ B22 ) @ ( plus_plus_mat_a @ C1 @ C22 ) @ ( plus_plus_mat_a @ D1 @ D22 ) ) ) ) ) ) ) ) ) ) ) ).
% add_four_block_mat
thf(fact_629_verit__negate__coefficient_I2_J,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_630_less__minus__iff,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ ( uminus_uminus_a @ B ) )
= ( ord_less_a @ B @ ( uminus_uminus_a @ A ) ) ) ).
% less_minus_iff
thf(fact_631_minus__less__iff,axiom,
! [A: a,B: a] :
( ( ord_less_a @ ( uminus_uminus_a @ A ) @ B )
= ( ord_less_a @ ( uminus_uminus_a @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_632_real__linear,axiom,
! [A: a,B: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ A @ B )
| ( A = B )
| ( ord_less_a @ B @ A ) ) ) ) ).
% real_linear
thf(fact_633_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_634_neg__neg__linear,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ( ord_less_a @ A @ B )
| ( A = B )
| ( ord_less_a @ B @ A ) ) ) ) ).
% neg_neg_linear
thf(fact_635_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_636_pos__pos__linear,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ( ord_less_a @ A @ B )
| ( A = B )
| ( ord_less_a @ B @ A ) ) ) ) ).
% pos_pos_linear
thf(fact_637_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_638_real__linorder__cases,axiom,
! [A: a,B: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ~ ( ord_less_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ B @ A ) ) ) ) ) ).
% real_linorder_cases
thf(fact_639_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_640_less__numeral__extra_I3_J,axiom,
~ ( ord_less_a @ zero_zero_a @ zero_zero_a ) ).
% less_numeral_extra(3)
thf(fact_641_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_642_field__lbound__gt__zero,axiom,
! [D12: a,D23: a] :
( ( ord_less_a @ zero_zero_a @ D12 )
=> ( ( ord_less_a @ zero_zero_a @ D23 )
=> ? [E: a] :
( ( ord_less_a @ zero_zero_a @ E )
& ( ord_less_a @ E @ D12 )
& ( ord_less_a @ E @ D23 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_643_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_644_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_645_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_646_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_647_leD,axiom,
! [Y2: vec_a,X2: vec_a] :
( ( ord_less_eq_vec_a @ Y2 @ X2 )
=> ~ ( ord_less_vec_a @ X2 @ Y2 ) ) ).
% leD
thf(fact_648_leD,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_649_leD,axiom,
! [Y2: a,X2: a] :
( ( ord_less_eq_a @ Y2 @ X2 )
=> ~ ( ord_less_a @ X2 @ Y2 ) ) ).
% leD
thf(fact_650_leI,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% leI
thf(fact_651_leI,axiom,
! [X2: a,Y2: a] :
( ~ ( ord_less_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) ) ).
% leI
thf(fact_652_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_653_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_654_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_655_antisym__conv1,axiom,
! [X2: vec_a,Y2: vec_a] :
( ~ ( ord_less_vec_a @ X2 @ Y2 )
=> ( ( ord_less_eq_vec_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_656_antisym__conv1,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_657_antisym__conv1,axiom,
! [X2: a,Y2: a] :
( ~ ( ord_less_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_658_antisym__conv2,axiom,
! [X2: vec_a,Y2: vec_a] :
( ( ord_less_eq_vec_a @ X2 @ Y2 )
=> ( ( ~ ( ord_less_vec_a @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_659_antisym__conv2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_660_antisym__conv2,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ~ ( ord_less_a @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_661_dense__ge,axiom,
! [Z: a,Y2: a] :
( ! [X3: a] :
( ( ord_less_a @ Z @ X3 )
=> ( ord_less_eq_a @ Y2 @ X3 ) )
=> ( ord_less_eq_a @ Y2 @ Z ) ) ).
% dense_ge
thf(fact_662_dense__le,axiom,
! [Y2: a,Z: a] :
( ! [X3: a] :
( ( ord_less_a @ X3 @ Y2 )
=> ( ord_less_eq_a @ X3 @ Z ) )
=> ( ord_less_eq_a @ Y2 @ Z ) ) ).
% dense_le
thf(fact_663_less__le__not__le,axiom,
( ord_less_vec_a
= ( ^ [X: vec_a,Y: vec_a] :
( ( ord_less_eq_vec_a @ X @ Y )
& ~ ( ord_less_eq_vec_a @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_664_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_665_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ~ ( ord_less_eq_a @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_666_not__le__imp__less,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_667_not__le__imp__less,axiom,
! [Y2: a,X2: a] :
( ~ ( ord_less_eq_a @ Y2 @ X2 )
=> ( ord_less_a @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_668_order_Oorder__iff__strict,axiom,
( ord_less_eq_vec_a
= ( ^ [A3: vec_a,B3: vec_a] :
( ( ord_less_vec_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_669_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_670_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A3: a,B3: a] :
( ( ord_less_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_671_order_Ostrict__iff__order,axiom,
( ord_less_vec_a
= ( ^ [A3: vec_a,B3: vec_a] :
( ( ord_less_eq_vec_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_672_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_673_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_674_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_675_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_676_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_677_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_678_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_679_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_680_order_Ostrict__iff__not,axiom,
( ord_less_vec_a
= ( ^ [A3: vec_a,B3: vec_a] :
( ( ord_less_eq_vec_a @ A3 @ B3 )
& ~ ( ord_less_eq_vec_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_681_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_682_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
& ~ ( ord_less_eq_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_683_dense__ge__bounded,axiom,
! [Z: a,X2: a,Y2: a] :
( ( ord_less_a @ Z @ X2 )
=> ( ! [W3: a] :
( ( ord_less_a @ Z @ W3 )
=> ( ( ord_less_a @ W3 @ X2 )
=> ( ord_less_eq_a @ Y2 @ W3 ) ) )
=> ( ord_less_eq_a @ Y2 @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_684_dense__le__bounded,axiom,
! [X2: a,Y2: a,Z: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ( ! [W3: a] :
( ( ord_less_a @ X2 @ W3 )
=> ( ( ord_less_a @ W3 @ Y2 )
=> ( ord_less_eq_a @ W3 @ Z ) ) )
=> ( ord_less_eq_a @ Y2 @ Z ) ) ) ).
% dense_le_bounded
thf(fact_685_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_vec_a
= ( ^ [B3: vec_a,A3: vec_a] :
( ( ord_less_vec_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_686_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_687_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B3: a,A3: a] :
( ( ord_less_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_688_dual__order_Ostrict__iff__order,axiom,
( ord_less_vec_a
= ( ^ [B3: vec_a,A3: vec_a] :
( ( ord_less_eq_vec_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_689_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_690_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B3: a,A3: a] :
( ( ord_less_eq_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_691_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_692_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_693_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_694_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_695_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_696_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_697_dual__order_Ostrict__iff__not,axiom,
( ord_less_vec_a
= ( ^ [B3: vec_a,A3: vec_a] :
( ( ord_less_eq_vec_a @ B3 @ A3 )
& ~ ( ord_less_eq_vec_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_698_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_699_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B3: a,A3: a] :
( ( ord_less_eq_a @ B3 @ A3 )
& ~ ( ord_less_eq_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_700_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_701_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_702_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_703_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_704_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_705_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_706_order__le__less,axiom,
( ord_less_eq_vec_a
= ( ^ [X: vec_a,Y: vec_a] :
( ( ord_less_vec_a @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_707_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_708_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_709_order__less__le,axiom,
( ord_less_vec_a
= ( ^ [X: vec_a,Y: vec_a] :
( ( ord_less_eq_vec_a @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_710_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_711_order__less__le,axiom,
( ord_less_a
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_712_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_713_linorder__not__le,axiom,
! [X2: a,Y2: a] :
( ( ~ ( ord_less_eq_a @ X2 @ Y2 ) )
= ( ord_less_a @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_714_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_715_linorder__not__less,axiom,
! [X2: a,Y2: a] :
( ( ~ ( ord_less_a @ X2 @ Y2 ) )
= ( ord_less_eq_a @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_716_order__less__imp__le,axiom,
! [X2: vec_a,Y2: vec_a] :
( ( ord_less_vec_a @ X2 @ Y2 )
=> ( ord_less_eq_vec_a @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_717_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_718_order__less__imp__le,axiom,
! [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_719_order__le__neq__trans,axiom,
! [A: vec_a,B: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_vec_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_720_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_721_order__le__neq__trans,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_722_order__neq__le__trans,axiom,
! [A: vec_a,B: vec_a] :
( ( A != B )
=> ( ( ord_less_eq_vec_a @ A @ B )
=> ( ord_less_vec_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_723_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_724_order__neq__le__trans,axiom,
! [A: a,B: a] :
( ( A != B )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_725_order__le__less__trans,axiom,
! [X2: vec_a,Y2: vec_a,Z: vec_a] :
( ( ord_less_eq_vec_a @ X2 @ Y2 )
=> ( ( ord_less_vec_a @ Y2 @ Z )
=> ( ord_less_vec_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_726_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_727_order__le__less__trans,axiom,
! [X2: a,Y2: a,Z: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_a @ Y2 @ Z )
=> ( ord_less_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_728_order__less__le__trans,axiom,
! [X2: vec_a,Y2: vec_a,Z: vec_a] :
( ( ord_less_vec_a @ X2 @ Y2 )
=> ( ( ord_less_eq_vec_a @ Y2 @ Z )
=> ( ord_less_vec_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_729_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_730_order__less__le__trans,axiom,
! [X2: a,Y2: a,Z: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z )
=> ( ord_less_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_731_order__le__less__subst1,axiom,
! [A: vec_a,F: a > vec_a,B: a,C: a] :
( ( ord_less_eq_vec_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_732_order__le__less__subst1,axiom,
! [A: vec_a,F: nat > vec_a,B: nat,C: nat] :
( ( ord_less_eq_vec_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_733_order__le__less__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_734_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_735_order__le__less__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_736_order__le__less__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_737_order__le__less__subst2,axiom,
! [A: vec_a,B: vec_a,F: vec_a > vec_a,C: vec_a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_vec_a @ ( F @ B ) @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_738_order__le__less__subst2,axiom,
! [A: vec_a,B: vec_a,F: vec_a > nat,C: nat] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_739_order__le__less__subst2,axiom,
! [A: vec_a,B: vec_a,F: vec_a > a,C: a] :
( ( ord_less_eq_vec_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_740_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > vec_a,C: vec_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_vec_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_741_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_742_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_743_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > vec_a,C: vec_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_vec_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_744_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_745_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_746_order__less__le__subst1,axiom,
! [A: vec_a,F: vec_a > vec_a,B: vec_a,C: vec_a] :
( ( ord_less_vec_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_747_order__less__le__subst1,axiom,
! [A: nat,F: vec_a > nat,B: vec_a,C: vec_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_748_order__less__le__subst1,axiom,
! [A: a,F: vec_a > a,B: vec_a,C: vec_a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_vec_a @ B @ C )
=> ( ! [X3: vec_a,Y3: vec_a] :
( ( ord_less_eq_vec_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_749_order__less__le__subst1,axiom,
! [A: vec_a,F: nat > vec_a,B: nat,C: nat] :
( ( ord_less_vec_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_750_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_751_order__less__le__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_752_order__less__le__subst1,axiom,
! [A: vec_a,F: a > vec_a,B: a,C: a] :
( ( ord_less_vec_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_vec_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_753_order__less__le__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_754_order__less__le__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_755_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > vec_a,C: vec_a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_vec_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_756_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > vec_a,C: vec_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_vec_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_vec_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_vec_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_757_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_758_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_759_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_760_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_761_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_762_linorder__le__less__linear,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
| ( ord_less_a @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_763_order__le__imp__less__or__eq,axiom,
! [X2: vec_a,Y2: vec_a] :
( ( ord_less_eq_vec_a @ X2 @ Y2 )
=> ( ( ord_less_vec_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_764_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_765_order__le__imp__less__or__eq,axiom,
! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ( ord_less_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_766_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_767_verit__comp__simplify1_I3_J,axiom,
! [B6: a,A6: a] :
( ( ~ ( ord_less_eq_a @ B6 @ A6 ) )
= ( ord_less_a @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_768_less__numeral__extra_I4_J,axiom,
~ ( ord_less_a @ one_one_a @ one_one_a ) ).
% less_numeral_extra(4)
thf(fact_769_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_770_add__mono__thms__linordered__field_I5_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_a @ I @ J )
& ( ord_less_a @ K @ L ) )
=> ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_771_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_772_add__mono__thms__linordered__field_I2_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( I = J )
& ( ord_less_a @ K @ L ) )
=> ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_773_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_774_add__mono__thms__linordered__field_I1_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_a @ I @ J )
& ( K = L ) )
=> ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_775_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_776_add__strict__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_777_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_778_add__strict__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_779_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_780_add__strict__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_781_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_782_add__less__imp__less__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_imp_less_left
thf(fact_783_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_784_add__less__imp__less__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_imp_less_right
thf(fact_785_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_786_diff__strict__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_787_diff__strict__left__mono,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_a @ ( minus_minus_a @ C @ A ) @ ( minus_minus_a @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_788_diff__eq__diff__less,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D ) )
=> ( ( ord_less_a @ A @ B )
= ( ord_less_a @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_789_diff__strict__mono,axiom,
! [A: a,B: a,D: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ D @ C )
=> ( ord_less_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_790_cong__four__block__mat,axiom,
! [A1: mat_a,B1: mat_a,A22: mat_a,B22: mat_a,A32: mat_a,B32: mat_a,A42: mat_a,B42: mat_a] :
( ( A1 = B1 )
=> ( ( A22 = B22 )
=> ( ( A32 = B32 )
=> ( ( A42 = B42 )
=> ( ( four_block_mat_a @ A1 @ A22 @ A32 @ A42 )
= ( four_block_mat_a @ B1 @ B22 @ B32 @ B42 ) ) ) ) ) ) ).
% cong_four_block_mat
thf(fact_791_lt__ex,axiom,
! [X2: a] :
? [Y3: a] : ( ord_less_a @ Y3 @ X2 ) ).
% lt_ex
thf(fact_792_gt__ex,axiom,
! [X2: a] :
? [X_1: a] : ( ord_less_a @ X2 @ X_1 ) ).
% gt_ex
thf(fact_793_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_794_dense,axiom,
! [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ? [Z3: a] :
( ( ord_less_a @ X2 @ Z3 )
& ( ord_less_a @ Z3 @ Y2 ) ) ) ).
% dense
thf(fact_795_less__imp__neq,axiom,
! [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_796_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_797_order_Oasym,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ~ ( ord_less_a @ B @ A ) ) ).
% order.asym
thf(fact_798_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_799_ord__eq__less__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_800_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_801_ord__less__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_802_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_803_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_804_antisym__conv3,axiom,
! [Y2: a,X2: a] :
( ~ ( ord_less_a @ Y2 @ X2 )
=> ( ( ~ ( ord_less_a @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_805_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_806_linorder__cases,axiom,
! [X2: a,Y2: a] :
( ~ ( ord_less_a @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_a @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_807_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_808_dual__order_Oasym,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ~ ( ord_less_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_809_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_810_dual__order_Oirrefl,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_811_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_812_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_813_linorder__less__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A4: a,B4: a] :
( ( ord_less_a @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: a] : ( P @ A4 @ A4 )
=> ( ! [A4: a,B4: a] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_814_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_815_order_Ostrict__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_816_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_817_not__less__iff__gr__or__eq,axiom,
! [X2: a,Y2: a] :
( ( ~ ( ord_less_a @ X2 @ Y2 ) )
= ( ( ord_less_a @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_818_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_819_dual__order_Ostrict__trans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_820_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_821_order_Ostrict__implies__not__eq,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_822_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_823_dual__order_Ostrict__implies__not__eq,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_824_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_825_linorder__neqE__linordered__idom,axiom,
! [X2: a,Y2: a] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_a @ X2 @ Y2 )
=> ( ord_less_a @ Y2 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_826_linorder__neqE,axiom,
! [X2: a,Y2: a] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_a @ X2 @ Y2 )
=> ( ord_less_a @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_827_linorder__neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_828_order__less__asym,axiom,
! [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ~ ( ord_less_a @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_829_order__less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_830_linorder__neq__iff,axiom,
! [X2: a,Y2: a] :
( ( X2 != Y2 )
= ( ( ord_less_a @ X2 @ Y2 )
| ( ord_less_a @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_831_linorder__neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_832_order__less__asym_H,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ~ ( ord_less_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_833_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_834_order__less__trans,axiom,
! [X2: a,Y2: a,Z: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ( ( ord_less_a @ Y2 @ Z )
=> ( ord_less_a @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_835_order__less__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_836_ord__eq__less__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_837_ord__eq__less__subst,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_838_ord__eq__less__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_839_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_840_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_841_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_842_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_843_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_844_order__less__irrefl,axiom,
! [X2: a] :
~ ( ord_less_a @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_845_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_846_order__less__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_847_order__less__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_848_order__less__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_849_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_850_order__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_851_order__less__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_852_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_853_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_854_order__less__not__sym,axiom,
! [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ~ ( ord_less_a @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_855_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_856_order__less__imp__triv,axiom,
! [X2: a,Y2: a,P: $o] :
( ( ord_less_a @ X2 @ Y2 )
=> ( ( ord_less_a @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_857_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_858_linorder__less__linear,axiom,
! [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_a @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_859_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_860_order__less__imp__not__eq,axiom,
! [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_861_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_862_order__less__imp__not__eq2,axiom,
! [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_863_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_864_order__less__imp__not__less,axiom,
! [X2: a,Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ~ ( ord_less_a @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_865_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_866_verit__comp__simplify1_I1_J,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_867_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_868_four__block__carrier__mat,axiom,
! [A2: mat_a,Nr1: nat,Nc1: nat,D2: mat_a,Nr2: nat,Nc2: nat,B2: mat_a,C4: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ D2 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( member_mat_a @ ( four_block_mat_a @ A2 @ B2 @ C4 @ D2 ) @ ( carrier_mat_a @ ( plus_plus_nat @ Nr1 @ Nr2 ) @ ( plus_plus_nat @ Nc1 @ Nc2 ) ) ) ) ) ).
% four_block_carrier_mat
thf(fact_869_nonpos__linorder__cases,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ~ ( ord_less_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ B @ A ) ) ) ) ) ).
% nonpos_linorder_cases
thf(fact_870_nonpos__linorder__cases,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ~ ( ord_less_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ B @ A ) ) ) ) ) ).
% nonpos_linorder_cases
thf(fact_871_nonneg__linorder__cases,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ~ ( ord_less_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ B @ A ) ) ) ) ) ).
% nonneg_linorder_cases
thf(fact_872_nonneg__linorder__cases,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ~ ( ord_less_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ B @ A ) ) ) ) ) ).
% nonneg_linorder_cases
thf(fact_873_not__less__real,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 @ B @ A ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% not_less_real
thf(fact_874_not__less__real,axiom,
! [A: a,B: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ~ ( ord_less_a @ B @ A ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ).
% not_less_real
thf(fact_875_not__le__real,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_eq_nat @ B @ A ) )
= ( ord_less_nat @ A @ B ) ) ) ) ).
% not_le_real
thf(fact_876_not__le__real,axiom,
! [A: a,B: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ~ ( ord_less_eq_a @ B @ A ) )
= ( ord_less_a @ A @ B ) ) ) ) ).
% not_le_real
thf(fact_877_not__one__less__zero,axiom,
~ ( ord_less_a @ one_one_a @ zero_zero_a ) ).
% not_one_less_zero
thf(fact_878_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_879_zero__less__one__class_Ozero__less__one,axiom,
ord_less_a @ zero_zero_a @ one_one_a ).
% zero_less_one_class.zero_less_one
thf(fact_880_zero__less__one__class_Ozero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_less_one
thf(fact_881_less__numeral__extra_I1_J,axiom,
ord_less_a @ zero_zero_a @ one_one_a ).
% less_numeral_extra(1)
thf(fact_882_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_883_add__neg__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_neg_neg
thf(fact_884_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_885_add__pos__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_886_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_887_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_888_pos__add__strict,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_889_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_890_add__less__zeroD,axiom,
! [X2: a,Y2: a] :
( ( ord_less_a @ ( plus_plus_a @ X2 @ Y2 ) @ zero_zero_a )
=> ( ( ord_less_a @ X2 @ zero_zero_a )
| ( ord_less_a @ Y2 @ zero_zero_a ) ) ) ).
% add_less_zeroD
thf(fact_891_real__add__less__cancel__right__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( ord_less_a @ A @ B ) ) ) ) ).
% real_add_less_cancel_right_pos
thf(fact_892_real__add__less__cancel__right__pos,axiom,
! [A: nat,B: nat,C: 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 @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ) ) ).
% real_add_less_cancel_right_pos
thf(fact_893_real__add__less__cancel__left__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( ord_less_a @ A @ B ) ) ) ) ).
% real_add_less_cancel_left_pos
thf(fact_894_real__add__less__cancel__left__pos,axiom,
! [A: nat,B: nat,C: 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 @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ) ) ).
% real_add_less_cancel_left_pos
thf(fact_895_add__pos__neg__is__real,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a )
| ( ( plus_plus_a @ A @ B )
= zero_zero_a )
| ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ) ).
% add_pos_neg_is_real
thf(fact_896_add__pos__neg__is__real,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat )
| ( ( plus_plus_nat @ A @ B )
= zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ) ).
% add_pos_neg_is_real
thf(fact_897_add__neg__pos__is__real,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a )
| ( ( plus_plus_a @ A @ B )
= zero_zero_a )
| ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ) ).
% add_neg_pos_is_real
thf(fact_898_add__neg__pos__is__real,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat )
| ( ( plus_plus_nat @ A @ B )
= zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ) ).
% add_neg_pos_is_real
thf(fact_899_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_900_add__less__le__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_901_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_902_add__le__less__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_903_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_904_add__mono__thms__linordered__field_I3_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_a @ I @ J )
& ( ord_less_eq_a @ K @ L ) )
=> ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_905_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_906_add__mono__thms__linordered__field_I4_J,axiom,
! [I: a,J: a,K: a,L: a] :
( ( ( ord_less_eq_a @ I @ J )
& ( ord_less_a @ K @ L ) )
=> ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_907_less__iff__diff__less__0,axiom,
( ord_less_a
= ( ^ [A3: a,B3: a] : ( ord_less_a @ ( minus_minus_a @ A3 @ B3 ) @ zero_zero_a ) ) ) ).
% less_iff_diff_less_0
thf(fact_908_transpose__four__block__mat,axiom,
! [A2: mat_a,Nr1: nat,Nc1: nat,B2: mat_a,Nc2: nat,C4: mat_a,Nr2: nat,D2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C4 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D2 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( transpose_mat_a @ ( four_block_mat_a @ A2 @ B2 @ C4 @ D2 ) )
= ( four_block_mat_a @ ( transpose_mat_a @ A2 ) @ ( transpose_mat_a @ C4 ) @ ( transpose_mat_a @ B2 ) @ ( transpose_mat_a @ D2 ) ) ) ) ) ) ) ).
% transpose_four_block_mat
thf(fact_909_less__add__one,axiom,
! [A: a] : ( ord_less_a @ A @ ( plus_plus_a @ A @ one_one_a ) ) ).
% less_add_one
thf(fact_910_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_911_add__mono1,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( plus_plus_a @ A @ one_one_a ) @ ( plus_plus_a @ B @ one_one_a ) ) ) ).
% add_mono1
thf(fact_912_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_913_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: a,B: a] :
( ~ ( ord_less_a @ A @ B )
=> ( ( plus_plus_a @ B @ ( minus_minus_a @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_914_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_915_less__diff__eq,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ A @ ( minus_minus_a @ C @ B ) )
= ( ord_less_a @ ( plus_plus_a @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_916_diff__less__eq,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ ( minus_minus_a @ A @ B ) @ C )
= ( ord_less_a @ A @ ( plus_plus_a @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_917_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_a @ one_one_a @ ( uminus_uminus_a @ one_one_a ) ) ).
% less_minus_one_simps(4)
thf(fact_918_less__minus__one__simps_I2_J,axiom,
ord_less_a @ ( uminus_uminus_a @ one_one_a ) @ one_one_a ).
% less_minus_one_simps(2)
thf(fact_919_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_920_add__strict__increasing2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_921_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_922_add__strict__increasing,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_923_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_924_add__pos__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_925_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_926_add__nonpos__neg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_nonpos_neg
thf(fact_927_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_928_add__nonneg__pos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_929_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_930_add__neg__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_neg_nonpos
thf(fact_931_real__add__le__cancel__right__pos,axiom,
! [A: nat,B: nat,C: 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_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% real_add_le_cancel_right_pos
thf(fact_932_real__add__le__cancel__right__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ).
% real_add_le_cancel_right_pos
thf(fact_933_real__add__le__cancel__left__pos,axiom,
! [A: nat,B: nat,C: 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_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% real_add_le_cancel_left_pos
thf(fact_934_real__add__le__cancel__left__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ).
% real_add_le_cancel_left_pos
thf(fact_935_zero__less__two,axiom,
ord_less_a @ zero_zero_a @ ( plus_plus_a @ one_one_a @ one_one_a ) ).
% zero_less_two
thf(fact_936_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_937_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_a @ zero_zero_a @ ( uminus_uminus_a @ one_one_a ) ) ).
% less_minus_one_simps(3)
thf(fact_938_less__minus__one__simps_I1_J,axiom,
ord_less_a @ ( uminus_uminus_a @ one_one_a ) @ zero_zero_a ).
% less_minus_one_simps(1)
thf(fact_939_four__block__mat__mult__vec,axiom,
! [A2: mat_a,Nr1: nat,Nc1: nat,B2: mat_a,Nc2: nat,C4: mat_a,Nr2: nat,D2: mat_a,A: vec_a,D: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc1 ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr1 @ Nc2 ) )
=> ( ( member_mat_a @ C4 @ ( carrier_mat_a @ Nr2 @ Nc1 ) )
=> ( ( member_mat_a @ D2 @ ( carrier_mat_a @ Nr2 @ Nc2 ) )
=> ( ( member_vec_a @ A @ ( carrier_vec_a @ Nc1 ) )
=> ( ( member_vec_a @ D @ ( carrier_vec_a @ Nc2 ) )
=> ( ( mult_mat_vec_a @ ( four_block_mat_a @ A2 @ B2 @ C4 @ D2 ) @ ( append_vec_a @ A @ D ) )
= ( append_vec_a @ ( plus_plus_vec_a @ ( mult_mat_vec_a @ A2 @ A ) @ ( mult_mat_vec_a @ B2 @ D ) ) @ ( plus_plus_vec_a @ ( mult_mat_vec_a @ C4 @ A ) @ ( mult_mat_vec_a @ D2 @ D ) ) ) ) ) ) ) ) ) ) ).
% four_block_mat_mult_vec
thf(fact_940_mult__mat__vec__split,axiom,
! [A2: mat_a,N: nat,D2: mat_a,M: nat,A: vec_a,D: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ D2 @ ( carrier_mat_a @ M @ M ) )
=> ( ( member_vec_a @ A @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ D @ ( carrier_vec_a @ M ) )
=> ( ( mult_mat_vec_a @ ( four_block_mat_a @ A2 @ ( zero_mat_a @ N @ M ) @ ( zero_mat_a @ M @ N ) @ D2 ) @ ( append_vec_a @ A @ D ) )
= ( append_vec_a @ ( mult_mat_vec_a @ A2 @ A ) @ ( mult_mat_vec_a @ D2 @ D ) ) ) ) ) ) ) ).
% mult_mat_vec_split
thf(fact_941__092_060open_062M_092_060_094sup_062T_A_K_092_060_094sub_062v_Aulv_A_061_A_IM__up_092_060_094sup_062T_A_064_092_060_094sub_062c_AM__low_092_060_094sup_062T_J_A_K_092_060_094sub_062v_Au1_A_L_AM__last_092_060_094sup_062T_A_K_092_060_094sub_062v_At_092_060close_062,axiom,
( ( mult_mat_vec_a @ ( transpose_mat_a @ m ) @ ulv )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ ( missin386308114684349109cols_a @ ( transpose_mat_a @ m_up ) @ ( transpose_mat_a @ m_low ) ) @ u1 ) @ ( mult_mat_vec_a @ ( transpose_mat_a @ m_last ) @ t ) ) ) ).
% \<open>M\<^sup>T *\<^sub>v ulv = (M_up\<^sup>T @\<^sub>c M_low\<^sup>T) *\<^sub>v u1 + M_last\<^sup>T *\<^sub>v t\<close>
thf(fact_942__092_060open_062_IM__up_092_060_094sup_062T_A_064_092_060_094sub_062c_AM__low_092_060_094sup_062T_J_A_K_092_060_094sub_062v_Au1_A_061_AM__up_092_060_094sup_062T_A_K_092_060_094sub_062v_Au2_A_L_AM__low_092_060_094sup_062T_A_K_092_060_094sub_062v_Au3_092_060close_062,axiom,
( ( mult_mat_vec_a @ ( missin386308114684349109cols_a @ ( transpose_mat_a @ m_up ) @ ( transpose_mat_a @ m_low ) ) @ u1 )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ ( transpose_mat_a @ m_up ) @ u2 ) @ ( mult_mat_vec_a @ ( transpose_mat_a @ m_low ) @ u3 ) ) ) ).
% \<open>(M_up\<^sup>T @\<^sub>c M_low\<^sup>T) *\<^sub>v u1 = M_up\<^sup>T *\<^sub>v u2 + M_low\<^sup>T *\<^sub>v u3\<close>
thf(fact_943_Mt,axiom,
( ( transpose_mat_a @ m )
= ( missin386308114684349109cols_a @ ( missin386308114684349109cols_a @ ( transpose_mat_a @ m_up ) @ ( transpose_mat_a @ m_low ) ) @ ( transpose_mat_a @ m_last ) ) ) ).
% Mt
thf(fact_944_M__def,axiom,
( m
= ( append_rows_a @ ( append_rows_a @ m_up @ m_low ) @ m_last ) ) ).
% M_def
thf(fact_945_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_946_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_947_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_948_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_949_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_950_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_951_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_952_carrier__append__rows,axiom,
! [A2: mat_a,Nr1: nat,Nc: nat,B2: mat_a,Nr2: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr2 @ Nc ) )
=> ( member_mat_a @ ( append_rows_a @ A2 @ B2 ) @ ( carrier_mat_a @ ( plus_plus_nat @ Nr1 @ Nr2 ) @ Nc ) ) ) ) ).
% carrier_append_rows
thf(fact_953_carrier__append__cols,axiom,
! [A2: mat_a,Nr: nat,Nc1: nat,B2: mat_a,Nc2: nat] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc1 ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc2 ) )
=> ( member_mat_a @ ( missin386308114684349109cols_a @ A2 @ B2 ) @ ( carrier_mat_a @ Nr @ ( plus_plus_nat @ Nc1 @ Nc2 ) ) ) ) ) ).
% carrier_append_cols
thf(fact_954_append__cols__def,axiom,
( missin386308114684349109cols_a
= ( ^ [A5: mat_a,B5: mat_a] : ( transpose_mat_a @ ( append_rows_a @ ( transpose_mat_a @ A5 ) @ ( transpose_mat_a @ B5 ) ) ) ) ) ).
% append_cols_def
thf(fact_955_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_956_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_957_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_958_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_959_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_960_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_961_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_962_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_963_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_964_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_965_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_966_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_967_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_968_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_969_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_970_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_971_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_972_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_973_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_974_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_975_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_976_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_977_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_978_less__vec__def,axiom,
( ord_less_vec_a
= ( ^ [V4: vec_a,W4: vec_a] :
( ( ord_less_eq_vec_a @ V4 @ W4 )
& ~ ( ord_less_eq_vec_a @ W4 @ V4 ) ) ) ) ).
% less_vec_def
thf(fact_979_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_980_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_981_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_982_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_983_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_984_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_985_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_986_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_987_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_988_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_989_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_990_mat__mult__append,axiom,
! [A2: mat_a,Nr1: nat,Nc: nat,B2: mat_a,Nr2: nat,V: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr2 @ Nc ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( mult_mat_vec_a @ ( append_rows_a @ A2 @ B2 ) @ V )
= ( append_vec_a @ ( mult_mat_vec_a @ A2 @ V ) @ ( mult_mat_vec_a @ B2 @ V ) ) ) ) ) ) ).
% mat_mult_append
thf(fact_991_append__rows__le,axiom,
! [A2: mat_a,Nr1: nat,Nc: nat,B2: mat_a,Nr2: nat,A: vec_a,V: vec_a,B: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr1 @ Nc ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr2 @ Nc ) )
=> ( ( member_vec_a @ A @ ( carrier_vec_a @ Nr1 ) )
=> ( ( member_vec_a @ V @ ( carrier_vec_a @ Nc ) )
=> ( ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ ( append_rows_a @ A2 @ B2 ) @ V ) @ ( append_vec_a @ A @ B ) )
= ( ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A2 @ V ) @ A )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ B2 @ V ) @ B ) ) ) ) ) ) ) ).
% append_rows_le
thf(fact_992_mat__mult__append__cols,axiom,
! [A2: mat_a,Nr: nat,Nc1: nat,B2: mat_a,Nc2: nat,V1: vec_a,V22: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc1 ) )
=> ( ( member_mat_a @ B2 @ ( carrier_mat_a @ Nr @ Nc2 ) )
=> ( ( member_vec_a @ V1 @ ( carrier_vec_a @ Nc1 ) )
=> ( ( member_vec_a @ V22 @ ( carrier_vec_a @ Nc2 ) )
=> ( ( mult_mat_vec_a @ ( missin386308114684349109cols_a @ A2 @ B2 ) @ ( append_vec_a @ V1 @ V22 ) )
= ( plus_plus_vec_a @ ( mult_mat_vec_a @ A2 @ V1 ) @ ( mult_mat_vec_a @ B2 @ V22 ) ) ) ) ) ) ) ).
% mat_mult_append_cols
thf(fact_993_M__up__def,axiom,
( m_up
= ( four_block_mat_a @ a2 @ ( zero_mat_a @ nr @ nr ) @ ( mat_of_row_a @ ( uminus_uminus_vec_a @ c ) ) @ ( mat_of_row_a @ b ) ) ) ).
% M_up_def
thf(fact_994_gram__schmidt_OFarkas__Lemma,axiom,
! [A2: mat_a,N: nat,Nr: nat,B: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ N @ Nr ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ N ) )
=> ( ( ? [X: vec_a] :
( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ X )
& ( ( mult_mat_vec_a @ A2 @ X )
= B ) ) )
= ( ! [Y: vec_a] :
( ( member_vec_a @ Y @ ( carrier_vec_a @ N ) )
=> ( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ ( mult_mat_vec_a @ ( transpose_mat_a @ A2 ) @ Y ) )
=> ( ord_less_eq_a @ zero_zero_a @ ( scalar_prod_a @ Y @ B ) ) ) ) ) ) ) ) ).
% gram_schmidt.Farkas_Lemma
thf(fact_995_gram__schmidt_OFarkas__Lemma_H,axiom,
! [A2: mat_a,Nr: nat,Nc: nat,B: vec_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_vec_a @ B @ ( carrier_vec_a @ Nr ) )
=> ( ( ? [X: vec_a] :
( ( member_vec_a @ X @ ( carrier_vec_a @ Nc ) )
& ( ord_less_eq_vec_a @ ( mult_mat_vec_a @ A2 @ X ) @ B ) ) )
= ( ! [Y: vec_a] :
( ( ( ord_less_eq_vec_a @ ( zero_vec_a @ Nr ) @ Y )
& ( ( mult_mat_vec_a @ ( transpose_mat_a @ A2 ) @ Y )
= ( zero_vec_a @ Nc ) ) )
=> ( ord_less_eq_a @ zero_zero_a @ ( scalar_prod_a @ Y @ B ) ) ) ) ) ) ) ).
% gram_schmidt.Farkas_Lemma'
thf(fact_996_mat__of__row__carrier_I1_J,axiom,
! [Y2: vec_a,N: nat] :
( ( member_vec_a @ Y2 @ ( carrier_vec_a @ N ) )
=> ( member_mat_a @ ( mat_of_row_a @ Y2 ) @ ( carrier_mat_a @ one_one_nat @ N ) ) ) ).
% mat_of_row_carrier(1)
thf(fact_997_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_998_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_999_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1000_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_1001_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1002_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( P @ M3 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1003_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1004_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_1005_mat__of__row__uminus,axiom,
! [V: vec_a] :
( ( mat_of_row_a @ ( uminus_uminus_vec_a @ V ) )
= ( uminus_uminus_mat_a @ ( mat_of_row_a @ V ) ) ) ).
% mat_of_row_uminus
thf(fact_1006_mat__of__col__def,axiom,
( missing_mat_of_col_a
= ( ^ [V4: vec_a] : ( transpose_mat_a @ ( mat_of_row_a @ V4 ) ) ) ) ).
% mat_of_col_def
thf(fact_1007__092_060open_062_I0_092_060_094sub_062m_Anc_Anr_A_064_092_060_094sub_062r_A_N_A1_092_060_094sub_062m_Anr_J_A_K_092_060_094sub_062v_At_A_061_A0_092_060_094sub_062v_Anc_A_064_092_060_094sub_062v_A_N_At_092_060close_062,axiom,
( ( mult_mat_vec_a @ ( append_rows_a @ ( zero_mat_a @ nc @ nr ) @ ( uminus_uminus_mat_a @ ( one_mat_a @ nr ) ) ) @ t )
= ( append_vec_a @ ( zero_vec_a @ nc ) @ ( uminus_uminus_vec_a @ t ) ) ) ).
% \<open>(0\<^sub>m nc nr @\<^sub>r - 1\<^sub>m nr) *\<^sub>v t = 0\<^sub>v nc @\<^sub>v - t\<close>
thf(fact_1008__092_060open_062M__last_092_060_094sup_062T_A_061_A0_092_060_094sub_062m_Anc_Anr_A_064_092_060_094sub_062r_A_N_A1_092_060_094sub_062m_Anr_092_060close_062,axiom,
( ( transpose_mat_a @ m_last )
= ( append_rows_a @ ( zero_mat_a @ nc @ nr ) @ ( uminus_uminus_mat_a @ ( one_mat_a @ nr ) ) ) ) ).
% \<open>M_last\<^sup>T = 0\<^sub>m nc nr @\<^sub>r - 1\<^sub>m nr\<close>
thf(fact_1009_M__last__def,axiom,
( m_last
= ( missin386308114684349109cols_a @ ( zero_mat_a @ nr @ nc ) @ ( uminus_uminus_mat_a @ ( one_mat_a @ nr ) ) ) ) ).
% M_last_def
thf(fact_1010_transpose__one,axiom,
! [N: nat] :
( ( transpose_mat_a @ ( one_mat_a @ N ) )
= ( one_mat_a @ N ) ) ).
% transpose_one
thf(fact_1011_one__mult__mat__vec,axiom,
! [V: vec_a,N: nat] :
( ( member_vec_a @ V @ ( carrier_vec_a @ N ) )
=> ( ( mult_mat_vec_a @ ( one_mat_a @ N ) @ V )
= V ) ) ).
% one_mult_mat_vec
thf(fact_1012_four__block__one__mat,axiom,
! [N1: nat,N2: nat] :
( ( four_block_mat_a @ ( one_mat_a @ N1 ) @ ( zero_mat_a @ N1 @ N2 ) @ ( zero_mat_a @ N2 @ N1 ) @ ( one_mat_a @ N2 ) )
= ( one_mat_a @ ( plus_plus_nat @ N1 @ N2 ) ) ) ).
% four_block_one_mat
thf(fact_1013_one__carrier__mat,axiom,
! [N: nat] : ( member_mat_a @ ( one_mat_a @ N ) @ ( carrier_mat_a @ N @ N ) ) ).
% one_carrier_mat
thf(fact_1014__092_060open_062inverse_Alam_A_K_A_I_N_A_IA_A_K_092_060_094sub_062v_A_Iv_A_N_Aw_J_J_A_092_060bullet_062_Au_J_A_092_060le_062_Ainverse_Alam_A_K_A_I_Ilam_A_092_060cdot_062_092_060_094sub_062v_Ab_J_A_092_060bullet_062_Au_J_092_060close_062,axiom,
ord_less_eq_a @ ( times_times_a @ ( inverse_inverse_a @ lam ) @ ( scalar_prod_a @ ( uminus_uminus_vec_a @ ( mult_mat_vec_a @ a2 @ ( minus_minus_vec_a @ v @ w ) ) ) @ u ) ) @ ( times_times_a @ ( inverse_inverse_a @ lam ) @ ( scalar_prod_a @ ( smult_vec_a @ lam @ b ) @ u ) ) ).
% \<open>inverse lam * (- (A *\<^sub>v (v - w)) \<bullet> u) \<le> inverse lam * ((lam \<cdot>\<^sub>v b) \<bullet> u)\<close>
thf(fact_1015_field__le__epsilon,axiom,
! [X2: a,Y2: a] :
( ! [E: a] :
( ( ord_less_a @ zero_zero_a @ E )
=> ( ord_less_eq_a @ X2 @ ( plus_plus_a @ Y2 @ E ) ) )
=> ( ord_less_eq_a @ X2 @ Y2 ) ) ).
% field_le_epsilon
thf(fact_1016_set__times__intro,axiom,
! [A: mat_a,C4: set_mat_a,B: mat_a,D2: set_mat_a] :
( ( member_mat_a @ A @ C4 )
=> ( ( member_mat_a @ B @ D2 )
=> ( member_mat_a @ ( times_times_mat_a @ A @ B ) @ ( times_1230744552615602198_mat_a @ C4 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_1017_set__times__intro,axiom,
! [A: a,C4: set_a,B: a,D2: set_a] :
( ( member_a @ A @ C4 )
=> ( ( member_a @ B @ D2 )
=> ( member_a @ ( times_times_a @ A @ B ) @ ( times_times_set_a @ C4 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_1018_set__times__intro,axiom,
! [A: nat,C4: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A @ C4 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C4 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_1019_mult__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ( times_times_a @ A @ C )
= ( times_times_a @ B @ C ) )
= ( ( C = zero_zero_a )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1020_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_1021_mult__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ( times_times_a @ C @ A )
= ( times_times_a @ C @ B ) )
= ( ( C = zero_zero_a )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1022_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_1023_mult__eq__0__iff,axiom,
! [A: a,B: a] :
( ( ( times_times_a @ A @ B )
= zero_zero_a )
= ( ( A = zero_zero_a )
| ( B = zero_zero_a ) ) ) ).
% mult_eq_0_iff
thf(fact_1024_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_1025_mult__zero__right,axiom,
! [A: a] :
( ( times_times_a @ A @ zero_zero_a )
= zero_zero_a ) ).
% mult_zero_right
thf(fact_1026_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1027_mult__zero__left,axiom,
! [A: a] :
( ( times_times_a @ zero_zero_a @ A )
= zero_zero_a ) ).
% mult_zero_left
thf(fact_1028_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1029_mult__1,axiom,
! [A: a] :
( ( times_times_a @ one_one_a @ A )
= A ) ).
% mult_1
thf(fact_1030_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1031_mult_Oright__neutral,axiom,
! [A: a] :
( ( times_times_a @ A @ one_one_a )
= A ) ).
% mult.right_neutral
thf(fact_1032_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1033_mult__minus__right,axiom,
! [A: a,B: a] :
( ( times_times_a @ A @ ( uminus_uminus_a @ B ) )
= ( uminus_uminus_a @ ( times_times_a @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_1034_minus__mult__minus,axiom,
! [A: a,B: a] :
( ( times_times_a @ ( uminus_uminus_a @ A ) @ ( uminus_uminus_a @ B ) )
= ( times_times_a @ A @ B ) ) ).
% minus_mult_minus
thf(fact_1035_mult__minus__left,axiom,
! [A: a,B: a] :
( ( times_times_a @ ( uminus_uminus_a @ A ) @ B )
= ( uminus_uminus_a @ ( times_times_a @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_1036_inverse__zero,axiom,
( ( inverse_inverse_a @ zero_zero_a )
= zero_zero_a ) ).
% inverse_zero
thf(fact_1037_inverse__nonzero__iff__nonzero,axiom,
! [A: a] :
( ( ( inverse_inverse_a @ A )
= zero_zero_a )
= ( A = zero_zero_a ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_1038_inverse__1,axiom,
( ( inverse_inverse_a @ one_one_a )
= one_one_a ) ).
% inverse_1
thf(fact_1039_inverse__eq__1__iff,axiom,
! [X2: a] :
( ( ( inverse_inverse_a @ X2 )
= one_one_a )
= ( X2 = one_one_a ) ) ).
% inverse_eq_1_iff
thf(fact_1040_inverse__minus__eq,axiom,
! [A: a] :
( ( inverse_inverse_a @ ( uminus_uminus_a @ A ) )
= ( uminus_uminus_a @ ( inverse_inverse_a @ A ) ) ) ).
% inverse_minus_eq
thf(fact_1041__092_060open_062u_A_092_060bullet_062_Ab_A_061_Ainverse_Alam_A_K_A_Ilam_A_K_A_Ib_A_092_060bullet_062_Au_J_J_092_060close_062,axiom,
( ( scalar_prod_a @ u @ b )
= ( times_times_a @ ( inverse_inverse_a @ lam ) @ ( times_times_a @ lam @ ( scalar_prod_a @ b @ u ) ) ) ) ).
% \<open>u \<bullet> b = inverse lam * (lam * (b \<bullet> u))\<close>
thf(fact_1042_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_1043_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_1044_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_1045_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_1046_mult__minus1,axiom,
! [Z: a] :
( ( times_times_a @ ( uminus_uminus_a @ one_one_a ) @ Z )
= ( uminus_uminus_a @ Z ) ) ).
% mult_minus1
thf(fact_1047_mult__minus1__right,axiom,
! [Z: a] :
( ( times_times_a @ Z @ ( uminus_uminus_a @ one_one_a ) )
= ( uminus_uminus_a @ Z ) ) ).
% mult_minus1_right
thf(fact_1048__092_060open_062inverse_Alam_A_K_A_Ilam_A_K_A_Ib_A_092_060bullet_062_Au_J_J_A_061_Ainverse_Alam_A_K_A_I_Ilam_A_092_060cdot_062_092_060_094sub_062v_Ab_J_A_092_060bullet_062_Au_J_092_060close_062,axiom,
( ( times_times_a @ ( inverse_inverse_a @ lam ) @ ( times_times_a @ lam @ ( scalar_prod_a @ b @ u ) ) )
= ( times_times_a @ ( inverse_inverse_a @ lam ) @ ( scalar_prod_a @ ( smult_vec_a @ lam @ b ) @ u ) ) ) ).
% \<open>inverse lam * (lam * (b \<bullet> u)) = inverse lam * ((lam \<cdot>\<^sub>v b) \<bullet> u)\<close>
thf(fact_1049_inverse__nonnegative__iff__nonnegative,axiom,
! [A: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( inverse_inverse_a @ A ) )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_1050_inverse__nonpositive__iff__nonpositive,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( inverse_inverse_a @ A ) @ zero_zero_a )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_1051_inverse__positive__iff__positive,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ ( inverse_inverse_a @ A ) )
= ( ord_less_a @ zero_zero_a @ A ) ) ).
% inverse_positive_iff_positive
thf(fact_1052_inverse__negative__iff__negative,axiom,
! [A: a] :
( ( ord_less_a @ ( inverse_inverse_a @ A ) @ zero_zero_a )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% inverse_negative_iff_negative
thf(fact_1053_inverse__less__iff__less__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ( ord_less_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
= ( ord_less_a @ B @ A ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_1054_inverse__less__iff__less,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ( ord_less_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
= ( ord_less_a @ B @ A ) ) ) ) ).
% inverse_less_iff_less
thf(fact_1055_inverse__le__iff__le,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
= ( ord_less_eq_a @ B @ A ) ) ) ) ).
% inverse_le_iff_le
thf(fact_1056_inverse__le__iff__le__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ( ord_less_eq_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
= ( ord_less_eq_a @ B @ A ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_1057_right__inverse,axiom,
! [A: a] :
( ( A != zero_zero_a )
=> ( ( times_times_a @ A @ ( inverse_inverse_a @ A ) )
= one_one_a ) ) ).
% right_inverse
thf(fact_1058_left__inverse,axiom,
! [A: a] :
( ( A != zero_zero_a )
=> ( ( times_times_a @ ( inverse_inverse_a @ A ) @ A )
= one_one_a ) ) ).
% left_inverse
thf(fact_1059_smult__scalar__prod__distrib,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a,A: a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ ( smult_vec_a @ A @ V_1 ) @ V_2 )
= ( times_times_a @ A @ ( scalar_prod_a @ V_1 @ V_2 ) ) ) ) ) ).
% smult_scalar_prod_distrib
thf(fact_1060_smult__scalar__prod__distrib,axiom,
! [V_1: vec_nat,N: nat,V_2: vec_nat,A: nat] :
( ( member_vec_nat @ V_1 @ ( carrier_vec_nat @ N ) )
=> ( ( member_vec_nat @ V_2 @ ( carrier_vec_nat @ N ) )
=> ( ( scalar_prod_nat @ ( smult_vec_nat @ A @ V_1 ) @ V_2 )
= ( times_times_nat @ A @ ( scalar_prod_nat @ V_1 @ V_2 ) ) ) ) ) ).
% smult_scalar_prod_distrib
thf(fact_1061_scalar__prod__smult__distrib,axiom,
! [V_1: vec_a,N: nat,V_2: vec_a,A: a] :
( ( member_vec_a @ V_1 @ ( carrier_vec_a @ N ) )
=> ( ( member_vec_a @ V_2 @ ( carrier_vec_a @ N ) )
=> ( ( scalar_prod_a @ V_1 @ ( smult_vec_a @ A @ V_2 ) )
= ( times_times_a @ A @ ( scalar_prod_a @ V_1 @ V_2 ) ) ) ) ) ).
% scalar_prod_smult_distrib
thf(fact_1062__092_060open_062inverse_Alam_A_K_A_I_N_A_IA_A_K_092_060_094sub_062v_A_Iv_A_N_Aw_J_J_A_092_060bullet_062_Au_J_A_061_A_N_A_Iinverse_Alam_A_K_A_I_IA_A_K_092_060_094sub_062v_A_Iv_A_N_Aw_J_J_A_092_060bullet_062_Au_J_J_092_060close_062,axiom,
( ( times_times_a @ ( inverse_inverse_a @ lam ) @ ( scalar_prod_a @ ( uminus_uminus_vec_a @ ( mult_mat_vec_a @ a2 @ ( minus_minus_vec_a @ v @ w ) ) ) @ u ) )
= ( uminus_uminus_a @ ( times_times_a @ ( inverse_inverse_a @ lam ) @ ( scalar_prod_a @ ( mult_mat_vec_a @ a2 @ ( minus_minus_vec_a @ v @ w ) ) @ u ) ) ) ) ).
% \<open>inverse lam * (- (A *\<^sub>v (v - w)) \<bullet> u) = - (inverse lam * ((A *\<^sub>v (v - w)) \<bullet> u))\<close>
thf(fact_1063_calculation,axiom,
ord_less_eq_a @ ( uminus_uminus_a @ ( times_times_a @ ( inverse_inverse_a @ lam ) @ ( scalar_prod_a @ ( mult_mat_vec_a @ a2 @ ( minus_minus_vec_a @ v @ w ) ) @ u ) ) ) @ ( scalar_prod_a @ u @ b ) ).
% calculation
thf(fact_1064_square__eq__1__iff,axiom,
! [X2: a] :
( ( ( times_times_a @ X2 @ X2 )
= one_one_a )
= ( ( X2 = one_one_a )
| ( X2
= ( uminus_uminus_a @ one_one_a ) ) ) ) ).
% square_eq_1_iff
thf(fact_1065_inverse__diff__inverse,axiom,
! [A: a,B: a] :
( ( A != zero_zero_a )
=> ( ( B != zero_zero_a )
=> ( ( minus_minus_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
= ( uminus_uminus_a @ ( times_times_a @ ( times_times_a @ ( inverse_inverse_a @ A ) @ ( minus_minus_a @ A @ B ) ) @ ( inverse_inverse_a @ B ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_1066_inverse__le__1__iff,axiom,
! [X2: a] :
( ( ord_less_eq_a @ ( inverse_inverse_a @ X2 ) @ one_one_a )
= ( ( ord_less_eq_a @ X2 @ zero_zero_a )
| ( ord_less_eq_a @ one_one_a @ X2 ) ) ) ).
% inverse_le_1_iff
thf(fact_1067_mult_Ocomm__neutral,axiom,
! [A: a] :
( ( times_times_a @ A @ one_one_a )
= A ) ).
% mult.comm_neutral
thf(fact_1068_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1069_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_1070_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_1071_inverse__unique,axiom,
! [A: a,B: a] :
( ( ( times_times_a @ A @ B )
= one_one_a )
=> ( ( inverse_inverse_a @ A )
= B ) ) ).
% inverse_unique
thf(fact_1072_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ ( times_times_a @ A @ B ) @ C )
= ( times_times_a @ A @ ( times_times_a @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1073_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_1074_mult_Oassoc,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ ( times_times_a @ A @ B ) @ C )
= ( times_times_a @ A @ ( times_times_a @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1075_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_1076_mult_Ocommute,axiom,
( times_times_a
= ( ^ [A3: a,B3: a] : ( times_times_a @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_1077_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_1078_mult_Oleft__commute,axiom,
! [B: a,A: a,C: a] :
( ( times_times_a @ B @ ( times_times_a @ A @ C ) )
= ( times_times_a @ A @ ( times_times_a @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1079_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_1080_set__times__elim,axiom,
! [X2: mat_a,A2: set_mat_a,B2: set_mat_a] :
( ( member_mat_a @ X2 @ ( times_1230744552615602198_mat_a @ A2 @ B2 ) )
=> ~ ! [A4: mat_a,B4: mat_a] :
( ( X2
= ( times_times_mat_a @ A4 @ B4 ) )
=> ( ( member_mat_a @ A4 @ A2 )
=> ~ ( member_mat_a @ B4 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_1081_set__times__elim,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ( member_a @ X2 @ ( times_times_set_a @ A2 @ B2 ) )
=> ~ ! [A4: a,B4: a] :
( ( X2
= ( times_times_a @ A4 @ B4 ) )
=> ( ( member_a @ A4 @ A2 )
=> ~ ( member_a @ B4 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_1082_set__times__elim,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X2 @ ( times_times_set_nat @ A2 @ B2 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X2
= ( times_times_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A2 )
=> ~ ( member_nat @ B4 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_1083_class__cring_Ofactors__equal,axiom,
! [A: a,B: a,C: a,D: a] :
( ( A = B )
=> ( ( C = D )
=> ( ( times_times_a @ A @ C )
= ( times_times_a @ B @ D ) ) ) ) ).
% class_cring.factors_equal
thf(fact_1084_smult__smult__assoc,axiom,
! [A: a,B: a,V: vec_a] :
( ( smult_vec_a @ A @ ( smult_vec_a @ B @ V ) )
= ( smult_vec_a @ ( times_times_a @ A @ B ) @ V ) ) ).
% smult_smult_assoc
thf(fact_1085_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_a @ zero_zero_a )
= zero_zero_a ) ).
% field_class.field_inverse_zero
thf(fact_1086_inverse__zero__imp__zero,axiom,
! [A: a] :
( ( ( inverse_inverse_a @ A )
= zero_zero_a )
=> ( A = zero_zero_a ) ) ).
% inverse_zero_imp_zero
thf(fact_1087_nonzero__inverse__eq__imp__eq,axiom,
! [A: a,B: a] :
( ( ( inverse_inverse_a @ A )
= ( inverse_inverse_a @ B ) )
=> ( ( A != zero_zero_a )
=> ( ( B != zero_zero_a )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_1088_nonzero__inverse__inverse__eq,axiom,
! [A: a] :
( ( A != zero_zero_a )
=> ( ( inverse_inverse_a @ ( inverse_inverse_a @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_1089_nonzero__imp__inverse__nonzero,axiom,
! [A: a] :
( ( A != zero_zero_a )
=> ( ( inverse_inverse_a @ A )
!= zero_zero_a ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_1090_nonzero__inverse__mult__distrib,axiom,
! [A: a,B: a] :
( ( A != zero_zero_a )
=> ( ( B != zero_zero_a )
=> ( ( inverse_inverse_a @ ( times_times_a @ A @ B ) )
= ( times_times_a @ ( inverse_inverse_a @ B ) @ ( inverse_inverse_a @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_1091_field__class_Ofield__inverse,axiom,
! [A: a] :
( ( A != zero_zero_a )
=> ( ( times_times_a @ ( inverse_inverse_a @ A ) @ A )
= one_one_a ) ) ).
% field_class.field_inverse
thf(fact_1092_division__ring__inverse__add,axiom,
! [A: a,B: a] :
( ( A != zero_zero_a )
=> ( ( B != zero_zero_a )
=> ( ( plus_plus_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
= ( times_times_a @ ( times_times_a @ ( inverse_inverse_a @ A ) @ ( plus_plus_a @ A @ B ) ) @ ( inverse_inverse_a @ B ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_1093_inverse__add,axiom,
! [A: a,B: a] :
( ( A != zero_zero_a )
=> ( ( B != zero_zero_a )
=> ( ( plus_plus_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
= ( times_times_a @ ( times_times_a @ ( plus_plus_a @ A @ B ) @ ( inverse_inverse_a @ A ) ) @ ( inverse_inverse_a @ B ) ) ) ) ) ).
% inverse_add
thf(fact_1094_left__diff__distrib,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ ( minus_minus_a @ A @ B ) @ C )
= ( minus_minus_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1095_right__diff__distrib,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ A @ ( minus_minus_a @ B @ C ) )
= ( minus_minus_a @ ( times_times_a @ A @ B ) @ ( times_times_a @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1096_left__diff__distrib_H,axiom,
! [B: a,C: a,A: a] :
( ( times_times_a @ ( minus_minus_a @ B @ C ) @ A )
= ( minus_minus_a @ ( times_times_a @ B @ A ) @ ( times_times_a @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1097_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1098_right__diff__distrib_H,axiom,
! [A: a,B: a,C: a] :
( ( times_times_a @ A @ ( minus_minus_a @ B @ C ) )
= ( minus_minus_a @ ( times_times_a @ A @ B ) @ ( times_times_a @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1099_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1100_combine__common__factor,axiom,
! [A: a,E2: a,B: a,C: a] :
( ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ C ) )
= ( plus_plus_a @ ( times_times_a @ ( plus_plus_a @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_1101_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_mult__not__zero,axiom,
! [A: a,B: a] :
( ( ( times_times_a @ A @ B )
!= zero_zero_a )
=> ( ( A != zero_zero_a )
& ( B != zero_zero_a ) ) ) ).
% mult_not_zero
thf(fact_1111_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_1112_divisors__zero,axiom,
! [A: a,B: a] :
( ( ( times_times_a @ A @ B )
= zero_zero_a )
=> ( ( A = zero_zero_a )
| ( B = zero_zero_a ) ) ) ).
% divisors_zero
thf(fact_1113_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_1114_no__zero__divisors,axiom,
! [A: a,B: a] :
( ( A != zero_zero_a )
=> ( ( B != zero_zero_a )
=> ( ( times_times_a @ A @ B )
!= zero_zero_a ) ) ) ).
% no_zero_divisors
thf(fact_1115_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_1116_mult__left__cancel,axiom,
! [C: a,A: a,B: a] :
( ( C != zero_zero_a )
=> ( ( ( times_times_a @ C @ A )
= ( times_times_a @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1117_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_1118_mult__right__cancel,axiom,
! [C: a,A: a,B: a] :
( ( C != zero_zero_a )
=> ( ( ( times_times_a @ A @ C )
= ( times_times_a @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1119_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_1120_crossproduct__noteq,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D ) )
!= ( plus_plus_a @ ( times_times_a @ A @ D ) @ ( times_times_a @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_1121_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_1122_crossproduct__eq,axiom,
! [W: a,Y2: a,X2: a,Z: a] :
( ( ( plus_plus_a @ ( times_times_a @ W @ Y2 ) @ ( times_times_a @ X2 @ Z ) )
= ( plus_plus_a @ ( times_times_a @ W @ Z ) @ ( times_times_a @ X2 @ Y2 ) ) )
= ( ( W = X2 )
| ( Y2 = Z ) ) ) ).
% crossproduct_eq
thf(fact_1123_crossproduct__eq,axiom,
! [W: nat,Y2: nat,X2: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y2 ) @ ( times_times_nat @ X2 @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X2 @ Y2 ) ) )
= ( ( W = X2 )
| ( Y2 = Z ) ) ) ).
% crossproduct_eq
thf(fact_1124_division__ring__inverse__diff,axiom,
! [A: a,B: a] :
( ( A != zero_zero_a )
=> ( ( B != zero_zero_a )
=> ( ( minus_minus_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
= ( times_times_a @ ( times_times_a @ ( inverse_inverse_a @ A ) @ ( minus_minus_a @ B @ A ) ) @ ( inverse_inverse_a @ B ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_1125_nonzero__inverse__minus__eq,axiom,
! [A: a] :
( ( A != zero_zero_a )
=> ( ( inverse_inverse_a @ ( uminus_uminus_a @ A ) )
= ( uminus_uminus_a @ ( inverse_inverse_a @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_1126_minus__mult__commute,axiom,
! [A: a,B: a] :
( ( times_times_a @ ( uminus_uminus_a @ A ) @ B )
= ( times_times_a @ A @ ( uminus_uminus_a @ B ) ) ) ).
% minus_mult_commute
thf(fact_1127_square__eq__iff,axiom,
! [A: a,B: a] :
( ( ( times_times_a @ A @ A )
= ( times_times_a @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_a @ B ) ) ) ) ).
% square_eq_iff
thf(fact_1128_mult__diff__mult,axiom,
! [X2: a,Y2: a,A: a,B: a] :
( ( minus_minus_a @ ( times_times_a @ X2 @ Y2 ) @ ( times_times_a @ A @ B ) )
= ( plus_plus_a @ ( times_times_a @ X2 @ ( minus_minus_a @ Y2 @ B ) ) @ ( times_times_a @ ( minus_minus_a @ X2 @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_1129_eq__add__iff1,axiom,
! [A: a,E2: a,C: a,B: a,D: a] :
( ( ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ C )
= ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ D ) )
= ( ( plus_plus_a @ ( times_times_a @ ( minus_minus_a @ A @ B ) @ E2 ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1130_eq__add__iff2,axiom,
! [A: a,E2: a,C: a,B: a,D: a] :
( ( ( plus_plus_a @ ( times_times_a @ A @ E2 ) @ C )
= ( plus_plus_a @ ( times_times_a @ B @ E2 ) @ D ) )
= ( C
= ( plus_plus_a @ ( times_times_a @ ( minus_minus_a @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1131_square__diff__square__factored,axiom,
! [X2: a,Y2: a] :
( ( minus_minus_a @ ( times_times_a @ X2 @ X2 ) @ ( times_times_a @ Y2 @ Y2 ) )
= ( times_times_a @ ( plus_plus_a @ X2 @ Y2 ) @ ( minus_minus_a @ X2 @ Y2 ) ) ) ).
% square_diff_square_factored
thf(fact_1132_less__1__mult,axiom,
! [M: a,N: a] :
( ( ord_less_a @ one_one_a @ M )
=> ( ( ord_less_a @ one_one_a @ N )
=> ( ord_less_a @ one_one_a @ ( times_times_a @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1133_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1134_add__scale__eq__noteq,axiom,
! [R: a,A: a,B: a,C: a,D: a] :
( ( R != zero_zero_a )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_a @ A @ ( times_times_a @ R @ C ) )
!= ( plus_plus_a @ B @ ( times_times_a @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1135_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1136_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1137_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1138_mult__less__cancel__right__disj,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
& ( ord_less_a @ A @ B ) )
| ( ( ord_less_a @ C @ zero_zero_a )
& ( ord_less_a @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1139_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1140_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1141_mult__strict__right__mono__neg,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1142_mult__less__cancel__left__disj,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
& ( ord_less_a @ A @ B ) )
| ( ( ord_less_a @ C @ zero_zero_a )
& ( ord_less_a @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1143_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1144_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1145_mult__strict__left__mono__neg,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1146_mult__less__cancel__left__pos,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ord_less_a @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1147_mult__less__cancel__left__neg,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ C @ zero_zero_a )
=> ( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ord_less_a @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1148_zero__less__mult__pos2,axiom,
! [B: a,A: a] :
( ( ord_less_a @ zero_zero_a @ ( times_times_a @ B @ A ) )
=> ( ( ord_less_a @ zero_zero_a @ A )
=> ( ord_less_a @ zero_zero_a @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1149_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1150_zero__less__mult__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ A )
=> ( ord_less_a @ zero_zero_a @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1151_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1152_zero__less__mult__iff,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) )
= ( ( ( ord_less_a @ zero_zero_a @ A )
& ( ord_less_a @ zero_zero_a @ B ) )
| ( ( ord_less_a @ A @ zero_zero_a )
& ( ord_less_a @ B @ zero_zero_a ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1153_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( times_times_a @ B @ A ) @ zero_zero_a ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_1154_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_1155_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_1156_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_1157_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_1158_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_1159_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_1160_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_1161_mult__less__0__iff,axiom,
! [A: a,B: a] :
( ( ord_less_a @ ( times_times_a @ A @ B ) @ zero_zero_a )
= ( ( ( ord_less_a @ zero_zero_a @ A )
& ( ord_less_a @ B @ zero_zero_a ) )
| ( ( ord_less_a @ A @ zero_zero_a )
& ( ord_less_a @ zero_zero_a @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_1162_not__square__less__zero,axiom,
! [A: a] :
~ ( ord_less_a @ ( times_times_a @ A @ A ) @ zero_zero_a ) ).
% not_square_less_zero
thf(fact_1163_linordered__ring__strict__class_Omult__neg__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).
% linordered_ring_strict_class.mult_neg_neg
thf(fact_1164_real__mult__less__cancel__right__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
= ( ord_less_a @ A @ B ) ) ) ) ) ).
% real_mult_less_cancel_right_pos
thf(fact_1165_real__mult__less__cancel__right__pos,axiom,
! [A: nat,B: nat,C: 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 @ zero_zero_nat @ C )
=> ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ) ) ) ).
% real_mult_less_cancel_right_pos
thf(fact_1166_real__mult__less__cancel__left__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ord_less_a @ A @ B ) ) ) ) ) ).
% real_mult_less_cancel_left_pos
thf(fact_1167_real__mult__less__cancel__left__pos,axiom,
! [A: nat,B: nat,C: 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 @ zero_zero_nat @ C )
=> ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ) ) ) ).
% real_mult_less_cancel_left_pos
thf(fact_1168_real__mult__eq__0__iff,axiom,
! [A: a,B: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ( times_times_a @ A @ B )
= zero_zero_a )
= ( ( A = zero_zero_a )
| ( B = zero_zero_a ) ) ) ) ) ).
% real_mult_eq_0_iff
thf(fact_1169_real__mult__eq__0__iff,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 ) )
=> ( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ) ) ).
% real_mult_eq_0_iff
thf(fact_1170_semiring__real__line__class_Omult__neg__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).
% semiring_real_line_class.mult_neg_neg
thf(fact_1171_semiring__real__line__class_Omult__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% semiring_real_line_class.mult_neg_neg
thf(fact_1172_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1173_mult__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1174_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1175_mult__mono_H,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1176_zero__le__square,axiom,
! [A: a] : ( ord_less_eq_a @ zero_zero_a @ ( times_times_a @ A @ A ) ) ).
% zero_le_square
thf(fact_1177_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_1178_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_1179_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_1180_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_1181_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_1182_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_1183_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_1184_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_1185_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_1186_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_1187_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_1188_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_1189_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_1190_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_1191_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_1192_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_1193_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_1194_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_1195_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_1196_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_1197_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_1198_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_1199_one__less__inverse,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ A @ one_one_a )
=> ( ord_less_a @ one_one_a @ ( inverse_inverse_a @ A ) ) ) ) ).
% one_less_inverse
thf(fact_1200_one__less__inverse__iff,axiom,
! [X2: a] :
( ( ord_less_a @ one_one_a @ ( inverse_inverse_a @ X2 ) )
= ( ( ord_less_a @ zero_zero_a @ X2 )
& ( ord_less_a @ X2 @ one_one_a ) ) ) ).
% one_less_inverse_iff
thf(fact_1201_le__imp__inverse__le__neg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ ( inverse_inverse_a @ B ) @ ( inverse_inverse_a @ A ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_1202_inverse__le__imp__le__neg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ B @ A ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_1203_le__imp__inverse__le,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ zero_zero_a @ A )
=> ( ord_less_eq_a @ ( inverse_inverse_a @ B ) @ ( inverse_inverse_a @ A ) ) ) ) ).
% le_imp_inverse_le
thf(fact_1204_inverse__le__imp__le,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ A )
=> ( ord_less_eq_a @ B @ A ) ) ) ).
% inverse_le_imp_le
thf(fact_1205_inverse__le__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
= ( ( ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) )
=> ( ord_less_eq_a @ B @ A ) )
& ( ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a )
=> ( ord_less_eq_a @ A @ B ) ) ) ) ).
% inverse_le_iff
thf(fact_1206_inverse__less__iff,axiom,
! [A: a,B: a] :
( ( ord_less_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
= ( ( ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) )
=> ( ord_less_a @ B @ A ) )
& ( ( ord_less_eq_a @ ( times_times_a @ A @ B ) @ zero_zero_a )
=> ( ord_less_a @ A @ B ) ) ) ) ).
% inverse_less_iff
thf(fact_1207_inverse__less__imp__less,axiom,
! [A: a,B: a] :
( ( ord_less_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ A )
=> ( ord_less_a @ B @ A ) ) ) ).
% inverse_less_imp_less
thf(fact_1208_less__imp__inverse__less,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ zero_zero_a @ A )
=> ( ord_less_a @ ( inverse_inverse_a @ B ) @ ( inverse_inverse_a @ A ) ) ) ) ).
% less_imp_inverse_less
thf(fact_1209_inverse__less__imp__less__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ ( inverse_inverse_a @ A ) @ ( inverse_inverse_a @ B ) )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ B @ A ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_1210_less__imp__inverse__less__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( inverse_inverse_a @ B ) @ ( inverse_inverse_a @ A ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_1211_inverse__negative__imp__negative,axiom,
! [A: a] :
( ( ord_less_a @ ( inverse_inverse_a @ A ) @ zero_zero_a )
=> ( ( A != zero_zero_a )
=> ( ord_less_a @ A @ zero_zero_a ) ) ) ).
% inverse_negative_imp_negative
thf(fact_1212_inverse__positive__imp__positive,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ ( inverse_inverse_a @ A ) )
=> ( ( A != zero_zero_a )
=> ( ord_less_a @ zero_zero_a @ A ) ) ) ).
% inverse_positive_imp_positive
thf(fact_1213_negative__imp__inverse__negative,axiom,
! [A: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ord_less_a @ ( inverse_inverse_a @ A ) @ zero_zero_a ) ) ).
% negative_imp_inverse_negative
thf(fact_1214_positive__imp__inverse__positive,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ord_less_a @ zero_zero_a @ ( inverse_inverse_a @ A ) ) ) ).
% positive_imp_inverse_positive
thf(fact_1215_one__le__inverse__iff,axiom,
! [X2: a] :
( ( ord_less_eq_a @ one_one_a @ ( inverse_inverse_a @ X2 ) )
= ( ( ord_less_a @ zero_zero_a @ X2 )
& ( ord_less_eq_a @ X2 @ one_one_a ) ) ) ).
% one_le_inverse_iff
thf(fact_1216_inverse__less__1__iff,axiom,
! [X2: a] :
( ( ord_less_a @ ( inverse_inverse_a @ X2 ) @ one_one_a )
= ( ( ord_less_eq_a @ X2 @ zero_zero_a )
| ( ord_less_a @ one_one_a @ X2 ) ) ) ).
% inverse_less_1_iff
thf(fact_1217_one__le__inverse,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ A @ one_one_a )
=> ( ord_less_eq_a @ one_one_a @ ( inverse_inverse_a @ A ) ) ) ) ).
% one_le_inverse
thf(fact_1218_field__le__mult__one__interval,axiom,
! [X2: a,Y2: a] :
( ! [Z3: a] :
( ( ord_less_a @ zero_zero_a @ Z3 )
=> ( ( ord_less_a @ Z3 @ one_one_a )
=> ( ord_less_eq_a @ ( times_times_a @ Z3 @ X2 ) @ Y2 ) ) )
=> ( ord_less_eq_a @ X2 @ Y2 ) ) ).
% field_le_mult_one_interval
thf(fact_1219_real__mult__le__cancel__left__pos,axiom,
! [A: nat,B: nat,C: 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 @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% real_mult_le_cancel_left_pos
thf(fact_1220_real__mult__le__cancel__left__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ) ).
% real_mult_le_cancel_left_pos
thf(fact_1221_real__mult__le__cancel__right__pos,axiom,
! [A: nat,B: nat,C: 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 @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% real_mult_le_cancel_right_pos
thf(fact_1222_real__mult__le__cancel__right__pos,axiom,
! [A: a,B: a,C: a] :
( ( ( ord_less_a @ A @ zero_zero_a )
| ( A = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ A ) )
=> ( ( ( ord_less_a @ B @ zero_zero_a )
| ( B = zero_zero_a )
| ( ord_less_a @ zero_zero_a @ B ) )
=> ( ( ord_less_a @ zero_zero_a @ C )
=> ( ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
= ( ord_less_eq_a @ A @ B ) ) ) ) ) ).
% real_mult_le_cancel_right_pos
thf(fact_1223_mult__le__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ B ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1224_mult__le__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
= ( ( ( ord_less_a @ zero_zero_a @ C )
=> ( ord_less_eq_a @ A @ B ) )
& ( ( ord_less_a @ C @ zero_zero_a )
=> ( ord_less_eq_a @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1225_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1226_mult__left__less__imp__less,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1227_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_1228_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_1229_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1230_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_1231_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_1232_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_1233_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_1234_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_1235_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_1236_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_1237_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_1238_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_1239_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_1240_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1241_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1242_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1243_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_1244_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_1245_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_1246_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_1247_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_1248_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1249_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1250_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1251_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1252_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N3: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1253_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_1254_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_1255_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_1256_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_1257_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_1258_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_1259_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_1260_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1261_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1262_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1263_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1264_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1265_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1266_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1267_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1268_lam__def,axiom,
( lam
= ( vec_index_a @ l @ zero_zero_nat ) ) ).
% lam_def
thf(fact_1269_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M5: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1270__092_060open_062dim__vec_A_I0_092_060_094sub_062v_A_Inr_A_L_A1_A_L_A_Inc_A_L_Anc_J_A_L_Anr_J_J_A_061_Adim__vec_Aulv_A_092_060and_062_A_I_092_060forall_062i_060dim__vec_Aulv_O_A0_092_060_094sub_062v_A_Inr_A_L_A1_A_L_A_Inc_A_L_Anc_J_A_L_Anr_J_A_E_Ai_A_092_060le_062_Aulv_A_E_Ai_J_092_060close_062,axiom,
( ( ( dim_vec_a @ ( zero_vec_a @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) @ nr ) ) )
= ( dim_vec_a @ ulv ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( dim_vec_a @ ulv ) )
=> ( ord_less_eq_a @ ( vec_index_a @ ( zero_vec_a @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ nr @ one_one_nat ) @ ( plus_plus_nat @ nc @ nc ) ) @ nr ) ) @ I3 ) @ ( vec_index_a @ ulv @ I3 ) ) ) ) ).
% \<open>dim_vec (0\<^sub>v (nr + 1 + (nc + nc) + nr)) = dim_vec ulv \<and> (\<forall>i<dim_vec ulv. 0\<^sub>v (nr + 1 + (nc + nc) + nr) $ i \<le> ulv $ i)\<close>
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X2: a,Y2: a] :
( ( if_a @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X2: a,Y2: a] :
( ( if_a @ $true @ X2 @ Y2 )
= X2 ) ).
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,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_vec_a @ ( minus_minus_vec_a @ v @ w ) @ ( carrier_vec_a @ nc ) ).
%------------------------------------------------------------------------------