TPTP Problem File: SLH0173^1.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 : Frequency_Moments/0086_Frequency_Moment_2/prob_00105_003470__19956522_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 868 ( 443 unt; 94 typ; 0 def)
% Number of atoms : 1984 ( 719 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 5332 ( 209 ~; 42 |; 94 &;4247 @)
% ( 0 <=>; 740 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 215 ( 215 >; 0 *; 0 +; 0 <<)
% Number of symbols : 86 ( 83 usr; 21 con; 0-3 aty)
% Number of variables : 1772 ( 169 ^;1595 !; 8 ?;1772 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:17:32.527
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
numera4273646738625120315l_num1: $tType ).
thf(ty_n_t__Numeral____Type__Obit1_It__Numeral____Type__Onum1_J,type,
numera6367994245245682809l_num1: $tType ).
thf(ty_n_t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
numera2417102609627094330l_num1: $tType ).
thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (83)
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Othe__enat,type,
extended_the_enat: extended_enat > nat ).
thf(sy_c_Factorial__Ring_Onormalization__semidom__class_Oprime_001t__Int__Oint,type,
factor1798656936486255268me_int: int > $o ).
thf(sy_c_Factorial__Ring_Onormalization__semidom__class_Oprime_001t__Nat__Onat,type,
factor1801147406995305544me_nat: nat > $o ).
thf(sy_c_Frequency__Moment__2_Op,type,
frequency_Moment_p: nat > nat ).
thf(sy_c_Frequency__Moment__2_Os_092_060_094sub_0621,type,
frequency_Moment_s_1: rat > nat ).
thf(sy_c_Frequency__Moment__2_Os_092_060_094sub_0622,type,
frequency_Moment_s_2: rat > nat ).
thf(sy_c_Frequency__Moments__Preliminary__Results_Oprime__above,type,
freque8783664969267990145_above: nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
one_on2969667320475766781nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
one_on7795324986448017462l_num1: numera4273646738625120315l_num1 ).
thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
one_on3868389512446148991l_num1: numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit1_It__Numeral____Type__Onum1_J,type,
one_on7819281148064737470l_num1: numera6367994245245682809l_num1 ).
thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
one_one_rat: rat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
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__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
uminus1336558196688952754l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
uminus7224005126491068675l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
uminus_uminus_rat: rat > rat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
zero_zero_rat: rat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Extended____Nat__Oenat,type,
if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Num__Onum,type,
if_num: $o > num > num > num ).
thf(sy_c_If_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
if_Num3220014061592582145l_num1: $o > numera4273646738625120315l_num1 > numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).
thf(sy_c_If_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
if_Num9196306924077011444l_num1: $o > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nonnegative____Real__Oennreal,type,
numera4658534427948366547nnreal: num > extend8495563244428889912nnreal ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
numera7754357348821619680l_num1: num > numera4273646738625120315l_num1 ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
numera2161328050825114965l_num1: num > numera2417102609627094330l_num1 ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit1_It__Numeral____Type__Onum1_J,type,
numera6112219686443703444l_num1: num > numera6367994245245682809l_num1 ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
numeral_numeral_rat: num > rat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le7381754540660121996nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
ord_le9186584087594166503l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
ord_le8952004418993413518l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
ord_less_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
ord_le8929789700686245851l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
ord_le6858968202089213338l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
ord_less_eq_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
ord_max_int: int > int > int ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
ord_max_num: num > num > num ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
ord_ma5275733517906637200l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
ord_ma2828161993112673189l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
ord_max_rat: rat > rat > rat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
ord_max_real: real > real > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
collect_rat: ( rat > $o ) > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Rat__Orat,type,
set_or5199638295745620268an_rat: rat > rat > set_rat ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_v__092_060delta_062,type,
delta: rat ).
thf(sy_v__092_060epsilon_062,type,
epsilon: rat ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (758)
thf(fact_0_p__ge__3,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( frequency_Moment_p @ n ) ).
% p_ge_3
thf(fact_1_p__def,axiom,
( ( frequency_Moment_p @ n )
= ( freque8783664969267990145_above @ ( ord_max_nat @ n @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).
% p_def
thf(fact_2_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_le6858968202089213338l_num1 @ ( numera2161328050825114965l_num1 @ U ) @ ( numera2161328050825114965l_num1 @ V ) )
=> ( ( ord_ma2828161993112673189l_num1 @ ( numera2161328050825114965l_num1 @ U ) @ ( numera2161328050825114965l_num1 @ V ) )
= ( numera2161328050825114965l_num1 @ V ) ) )
& ( ~ ( ord_le6858968202089213338l_num1 @ ( numera2161328050825114965l_num1 @ U ) @ ( numera2161328050825114965l_num1 @ V ) )
=> ( ( ord_ma2828161993112673189l_num1 @ ( numera2161328050825114965l_num1 @ U ) @ ( numera2161328050825114965l_num1 @ V ) )
= ( numera2161328050825114965l_num1 @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_3_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_le8929789700686245851l_num1 @ ( numera7754357348821619680l_num1 @ U ) @ ( numera7754357348821619680l_num1 @ V ) )
=> ( ( ord_ma5275733517906637200l_num1 @ ( numera7754357348821619680l_num1 @ U ) @ ( numera7754357348821619680l_num1 @ V ) )
= ( numera7754357348821619680l_num1 @ V ) ) )
& ( ~ ( ord_le8929789700686245851l_num1 @ ( numera7754357348821619680l_num1 @ U ) @ ( numera7754357348821619680l_num1 @ V ) )
=> ( ( ord_ma5275733517906637200l_num1 @ ( numera7754357348821619680l_num1 @ U ) @ ( numera7754357348821619680l_num1 @ V ) )
= ( numera7754357348821619680l_num1 @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_4_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ V ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_5_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
=> ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
= ( numera1916890842035813515d_enat @ V ) ) )
& ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
=> ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
= ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_6_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ V ) ) )
& ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_7_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ V ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_8_pre__arith__simps_I1_J,axiom,
! [B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ ( ord_ma5275733517906637200l_num1 @ B @ C ) @ A )
= ( ( ord_le8929789700686245851l_num1 @ B @ A )
& ( ord_le8929789700686245851l_num1 @ C @ A ) ) ) ).
% pre_arith_simps(1)
thf(fact_9_pre__arith__simps_I1_J,axiom,
! [B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1,A: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ ( ord_ma2828161993112673189l_num1 @ B @ C ) @ A )
= ( ( ord_le6858968202089213338l_num1 @ B @ A )
& ( ord_le6858968202089213338l_num1 @ C @ A ) ) ) ).
% pre_arith_simps(1)
thf(fact_10_pre__arith__simps_I1_J,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% pre_arith_simps(1)
thf(fact_11_pre__arith__simps_I1_J,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
= ( ( ord_less_eq_num @ B @ A )
& ( ord_less_eq_num @ C @ A ) ) ) ).
% pre_arith_simps(1)
thf(fact_12_pre__arith__simps_I1_J,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
= ( ( ord_le2932123472753598470d_enat @ B @ A )
& ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% pre_arith_simps(1)
thf(fact_13_pre__arith__simps_I1_J,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
= ( ( ord_less_eq_real @ B @ A )
& ( ord_less_eq_real @ C @ A ) ) ) ).
% pre_arith_simps(1)
thf(fact_14_pre__arith__simps_I1_J,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ C @ A ) ) ) ).
% pre_arith_simps(1)
thf(fact_15_max_Oabsorb1,axiom,
! [B: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ B @ A )
=> ( ( ord_ma5275733517906637200l_num1 @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_16_max_Oabsorb1,axiom,
! [B: numera2417102609627094330l_num1,A: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ B @ A )
=> ( ( ord_ma2828161993112673189l_num1 @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_17_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_18_max_Oabsorb1,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_max_num @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_19_max_Oabsorb1,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_20_max_Oabsorb1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_max_real @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_21_max_Oabsorb1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_22_max_Oabsorb2,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ A @ B )
=> ( ( ord_ma5275733517906637200l_num1 @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_23_max_Oabsorb2,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ A @ B )
=> ( ( ord_ma2828161993112673189l_num1 @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_24_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_25_max_Oabsorb2,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_max_num @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_26_max_Oabsorb2,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_27_max_Oabsorb2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_max_real @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_28_max_Oabsorb2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_29_rel__simps_I19_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% rel_simps(19)
thf(fact_30_rel__simps_I17_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% rel_simps(17)
thf(fact_31_rel__simps_I24_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% rel_simps(24)
thf(fact_32_rel__simps_I24_J,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% rel_simps(24)
thf(fact_33_rel__simps_I24_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% rel_simps(24)
thf(fact_34_rel__simps_I24_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% rel_simps(24)
thf(fact_35_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera1916890842035813515d_enat @ M )
= ( numera1916890842035813515d_enat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_36_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_37_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_38_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_39_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_40_num_Oinject_I2_J,axiom,
! [X3: num,Y3: num] :
( ( ( bit1 @ X3 )
= ( bit1 @ Y3 ) )
= ( X3 = Y3 ) ) ).
% num.inject(2)
thf(fact_41_max_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
= ( ord_max_nat @ A @ B ) ) ).
% max.right_idem
thf(fact_42_max_Oright__idem,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_ma741700101516333627d_enat @ ( ord_ma741700101516333627d_enat @ A @ B ) @ B )
= ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% max.right_idem
thf(fact_43_max_Oright__idem,axiom,
! [A: int,B: int] :
( ( ord_max_int @ ( ord_max_int @ A @ B ) @ B )
= ( ord_max_int @ A @ B ) ) ).
% max.right_idem
thf(fact_44_max_Oright__idem,axiom,
! [A: real,B: real] :
( ( ord_max_real @ ( ord_max_real @ A @ B ) @ B )
= ( ord_max_real @ A @ B ) ) ).
% max.right_idem
thf(fact_45_max_Oright__idem,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( ord_ma5275733517906637200l_num1 @ ( ord_ma5275733517906637200l_num1 @ A @ B ) @ B )
= ( ord_ma5275733517906637200l_num1 @ A @ B ) ) ).
% max.right_idem
thf(fact_46_max_Oright__idem,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] :
( ( ord_ma2828161993112673189l_num1 @ ( ord_ma2828161993112673189l_num1 @ A @ B ) @ B )
= ( ord_ma2828161993112673189l_num1 @ A @ B ) ) ).
% max.right_idem
thf(fact_47_max_Oright__idem,axiom,
! [A: num,B: num] :
( ( ord_max_num @ ( ord_max_num @ A @ B ) @ B )
= ( ord_max_num @ A @ B ) ) ).
% max.right_idem
thf(fact_48_max_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
= ( ord_max_nat @ A @ B ) ) ).
% max.left_idem
thf(fact_49_max_Oleft__idem,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_ma741700101516333627d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) )
= ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% max.left_idem
thf(fact_50_max_Oleft__idem,axiom,
! [A: int,B: int] :
( ( ord_max_int @ A @ ( ord_max_int @ A @ B ) )
= ( ord_max_int @ A @ B ) ) ).
% max.left_idem
thf(fact_51_max_Oleft__idem,axiom,
! [A: real,B: real] :
( ( ord_max_real @ A @ ( ord_max_real @ A @ B ) )
= ( ord_max_real @ A @ B ) ) ).
% max.left_idem
thf(fact_52_max_Oleft__idem,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( ord_ma5275733517906637200l_num1 @ A @ ( ord_ma5275733517906637200l_num1 @ A @ B ) )
= ( ord_ma5275733517906637200l_num1 @ A @ B ) ) ).
% max.left_idem
thf(fact_53_max_Oleft__idem,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] :
( ( ord_ma2828161993112673189l_num1 @ A @ ( ord_ma2828161993112673189l_num1 @ A @ B ) )
= ( ord_ma2828161993112673189l_num1 @ A @ B ) ) ).
% max.left_idem
thf(fact_54_max_Oleft__idem,axiom,
! [A: num,B: num] :
( ( ord_max_num @ A @ ( ord_max_num @ A @ B ) )
= ( ord_max_num @ A @ B ) ) ).
% max.left_idem
thf(fact_55_max_Oidem,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ A )
= A ) ).
% max.idem
thf(fact_56_max_Oidem,axiom,
! [A: extended_enat] :
( ( ord_ma741700101516333627d_enat @ A @ A )
= A ) ).
% max.idem
thf(fact_57_max_Oidem,axiom,
! [A: int] :
( ( ord_max_int @ A @ A )
= A ) ).
% max.idem
thf(fact_58_max_Oidem,axiom,
! [A: real] :
( ( ord_max_real @ A @ A )
= A ) ).
% max.idem
thf(fact_59_max_Oidem,axiom,
! [A: numera4273646738625120315l_num1] :
( ( ord_ma5275733517906637200l_num1 @ A @ A )
= A ) ).
% max.idem
thf(fact_60_max_Oidem,axiom,
! [A: numera2417102609627094330l_num1] :
( ( ord_ma2828161993112673189l_num1 @ A @ A )
= A ) ).
% max.idem
thf(fact_61_max_Oidem,axiom,
! [A: num] :
( ( ord_max_num @ A @ A )
= A ) ).
% max.idem
thf(fact_62_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_63_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_64_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_65_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_66_max_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% max.left_commute
thf(fact_67_max_Oleft__commute,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_ma741700101516333627d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ C ) )
= ( ord_ma741700101516333627d_enat @ A @ ( ord_ma741700101516333627d_enat @ B @ C ) ) ) ).
% max.left_commute
thf(fact_68_max_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( ord_max_int @ B @ ( ord_max_int @ A @ C ) )
= ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).
% max.left_commute
thf(fact_69_max_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( ord_max_real @ B @ ( ord_max_real @ A @ C ) )
= ( ord_max_real @ A @ ( ord_max_real @ B @ C ) ) ) ).
% max.left_commute
thf(fact_70_max_Oleft__commute,axiom,
! [B: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( ord_ma5275733517906637200l_num1 @ B @ ( ord_ma5275733517906637200l_num1 @ A @ C ) )
= ( ord_ma5275733517906637200l_num1 @ A @ ( ord_ma5275733517906637200l_num1 @ B @ C ) ) ) ).
% max.left_commute
thf(fact_71_max_Oleft__commute,axiom,
! [B: numera2417102609627094330l_num1,A: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( ord_ma2828161993112673189l_num1 @ B @ ( ord_ma2828161993112673189l_num1 @ A @ C ) )
= ( ord_ma2828161993112673189l_num1 @ A @ ( ord_ma2828161993112673189l_num1 @ B @ C ) ) ) ).
% max.left_commute
thf(fact_72_max_Oleft__commute,axiom,
! [B: num,A: num,C: num] :
( ( ord_max_num @ B @ ( ord_max_num @ A @ C ) )
= ( ord_max_num @ A @ ( ord_max_num @ B @ C ) ) ) ).
% max.left_commute
thf(fact_73_max_Ocommute,axiom,
( ord_max_nat
= ( ^ [A2: nat,B2: nat] : ( ord_max_nat @ B2 @ A2 ) ) ) ).
% max.commute
thf(fact_74_max_Ocommute,axiom,
( ord_ma741700101516333627d_enat
= ( ^ [A2: extended_enat,B2: extended_enat] : ( ord_ma741700101516333627d_enat @ B2 @ A2 ) ) ) ).
% max.commute
thf(fact_75_max_Ocommute,axiom,
( ord_max_int
= ( ^ [A2: int,B2: int] : ( ord_max_int @ B2 @ A2 ) ) ) ).
% max.commute
thf(fact_76_max_Ocommute,axiom,
( ord_max_real
= ( ^ [A2: real,B2: real] : ( ord_max_real @ B2 @ A2 ) ) ) ).
% max.commute
thf(fact_77_max_Ocommute,axiom,
( ord_ma5275733517906637200l_num1
= ( ^ [A2: numera4273646738625120315l_num1,B2: numera4273646738625120315l_num1] : ( ord_ma5275733517906637200l_num1 @ B2 @ A2 ) ) ) ).
% max.commute
thf(fact_78_max_Ocommute,axiom,
( ord_ma2828161993112673189l_num1
= ( ^ [A2: numera2417102609627094330l_num1,B2: numera2417102609627094330l_num1] : ( ord_ma2828161993112673189l_num1 @ B2 @ A2 ) ) ) ).
% max.commute
thf(fact_79_max_Ocommute,axiom,
( ord_max_num
= ( ^ [A2: num,B2: num] : ( ord_max_num @ B2 @ A2 ) ) ) ).
% max.commute
thf(fact_80_mem__Collect__eq,axiom,
! [A: rat,P: rat > $o] :
( ( member_rat @ A @ ( collect_rat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A3: set_rat] :
( ( collect_rat
@ ^ [X2: rat] : ( member_rat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_82_max_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% max.assoc
thf(fact_83_max_Oassoc,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_ma741700101516333627d_enat @ ( ord_ma741700101516333627d_enat @ A @ B ) @ C )
= ( ord_ma741700101516333627d_enat @ A @ ( ord_ma741700101516333627d_enat @ B @ C ) ) ) ).
% max.assoc
thf(fact_84_max_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( ord_max_int @ ( ord_max_int @ A @ B ) @ C )
= ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).
% max.assoc
thf(fact_85_max_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( ord_max_real @ ( ord_max_real @ A @ B ) @ C )
= ( ord_max_real @ A @ ( ord_max_real @ B @ C ) ) ) ).
% max.assoc
thf(fact_86_max_Oassoc,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( ord_ma5275733517906637200l_num1 @ ( ord_ma5275733517906637200l_num1 @ A @ B ) @ C )
= ( ord_ma5275733517906637200l_num1 @ A @ ( ord_ma5275733517906637200l_num1 @ B @ C ) ) ) ).
% max.assoc
thf(fact_87_max_Oassoc,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( ord_ma2828161993112673189l_num1 @ ( ord_ma2828161993112673189l_num1 @ A @ B ) @ C )
= ( ord_ma2828161993112673189l_num1 @ A @ ( ord_ma2828161993112673189l_num1 @ B @ C ) ) ) ).
% max.assoc
thf(fact_88_max_Oassoc,axiom,
! [A: num,B: num,C: num] :
( ( ord_max_num @ ( ord_max_num @ A @ B ) @ C )
= ( ord_max_num @ A @ ( ord_max_num @ B @ C ) ) ) ).
% max.assoc
thf(fact_89_num_Odistinct_I3_J,axiom,
! [X3: num] :
( one
!= ( bit1 @ X3 ) ) ).
% num.distinct(3)
thf(fact_90_max_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_91_max_OcoboundedI2,axiom,
! [C: num,B: num,A: num] :
( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_92_max_OcoboundedI2,axiom,
! [C: extended_enat,B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_93_max_OcoboundedI2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_94_max_OcoboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_95_max_OcoboundedI2,axiom,
! [C: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ C @ B )
=> ( ord_le8929789700686245851l_num1 @ C @ ( ord_ma5275733517906637200l_num1 @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_96_max_OcoboundedI2,axiom,
! [C: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,A: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ C @ B )
=> ( ord_le6858968202089213338l_num1 @ C @ ( ord_ma2828161993112673189l_num1 @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_97_max_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_98_max_OcoboundedI1,axiom,
! [C: num,A: num,B: num] :
( ( ord_less_eq_num @ C @ A )
=> ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_99_max_OcoboundedI1,axiom,
! [C: extended_enat,A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ A )
=> ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_100_max_OcoboundedI1,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_101_max_OcoboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_102_max_OcoboundedI1,axiom,
! [C: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ C @ A )
=> ( ord_le8929789700686245851l_num1 @ C @ ( ord_ma5275733517906637200l_num1 @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_103_max_OcoboundedI1,axiom,
! [C: numera2417102609627094330l_num1,A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ C @ A )
=> ( ord_le6858968202089213338l_num1 @ C @ ( ord_ma2828161993112673189l_num1 @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_104_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_max_nat @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_105_max_Oabsorb__iff2,axiom,
( ord_less_eq_num
= ( ^ [A2: num,B2: num] :
( ( ord_max_num @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_106_max_Oabsorb__iff2,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A2: extended_enat,B2: extended_enat] :
( ( ord_ma741700101516333627d_enat @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_107_max_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] :
( ( ord_max_real @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_108_max_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] :
( ( ord_max_int @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_109_max_Oabsorb__iff2,axiom,
( ord_le8929789700686245851l_num1
= ( ^ [A2: numera4273646738625120315l_num1,B2: numera4273646738625120315l_num1] :
( ( ord_ma5275733517906637200l_num1 @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_110_max_Oabsorb__iff2,axiom,
( ord_le6858968202089213338l_num1
= ( ^ [A2: numera2417102609627094330l_num1,B2: numera2417102609627094330l_num1] :
( ( ord_ma2828161993112673189l_num1 @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_111_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_max_nat @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_112_max_Oabsorb__iff1,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A2: num] :
( ( ord_max_num @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_113_max_Oabsorb__iff1,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B2: extended_enat,A2: extended_enat] :
( ( ord_ma741700101516333627d_enat @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_114_max_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A2: real] :
( ( ord_max_real @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_115_max_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( ( ord_max_int @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_116_max_Oabsorb__iff1,axiom,
( ord_le8929789700686245851l_num1
= ( ^ [B2: numera4273646738625120315l_num1,A2: numera4273646738625120315l_num1] :
( ( ord_ma5275733517906637200l_num1 @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_117_max_Oabsorb__iff1,axiom,
( ord_le6858968202089213338l_num1
= ( ^ [B2: numera2417102609627094330l_num1,A2: numera2417102609627094330l_num1] :
( ( ord_ma2828161993112673189l_num1 @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_118_le__max__iff__disj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_eq_nat @ Z @ X )
| ( ord_less_eq_nat @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_119_le__max__iff__disj,axiom,
! [Z: num,X: num,Y: num] :
( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y ) )
= ( ( ord_less_eq_num @ Z @ X )
| ( ord_less_eq_num @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_120_le__max__iff__disj,axiom,
! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X @ Y ) )
= ( ( ord_le2932123472753598470d_enat @ Z @ X )
| ( ord_le2932123472753598470d_enat @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_121_le__max__iff__disj,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_eq_real @ Z @ ( ord_max_real @ X @ Y ) )
= ( ( ord_less_eq_real @ Z @ X )
| ( ord_less_eq_real @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_122_le__max__iff__disj,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
= ( ( ord_less_eq_int @ Z @ X )
| ( ord_less_eq_int @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_123_le__max__iff__disj,axiom,
! [Z: numera4273646738625120315l_num1,X: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ Z @ ( ord_ma5275733517906637200l_num1 @ X @ Y ) )
= ( ( ord_le8929789700686245851l_num1 @ Z @ X )
| ( ord_le8929789700686245851l_num1 @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_124_le__max__iff__disj,axiom,
! [Z: numera2417102609627094330l_num1,X: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ Z @ ( ord_ma2828161993112673189l_num1 @ X @ Y ) )
= ( ( ord_le6858968202089213338l_num1 @ Z @ X )
| ( ord_le6858968202089213338l_num1 @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_125_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_126_max_Ocobounded2,axiom,
! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded2
thf(fact_127_max_Ocobounded2,axiom,
! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% max.cobounded2
thf(fact_128_max_Ocobounded2,axiom,
! [B: real,A: real] : ( ord_less_eq_real @ B @ ( ord_max_real @ A @ B ) ) ).
% max.cobounded2
thf(fact_129_max_Ocobounded2,axiom,
! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded2
thf(fact_130_max_Ocobounded2,axiom,
! [B: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1] : ( ord_le8929789700686245851l_num1 @ B @ ( ord_ma5275733517906637200l_num1 @ A @ B ) ) ).
% max.cobounded2
thf(fact_131_max_Ocobounded2,axiom,
! [B: numera2417102609627094330l_num1,A: numera2417102609627094330l_num1] : ( ord_le6858968202089213338l_num1 @ B @ ( ord_ma2828161993112673189l_num1 @ A @ B ) ) ).
% max.cobounded2
thf(fact_132_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_133_max_Ocobounded1,axiom,
! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded1
thf(fact_134_max_Ocobounded1,axiom,
! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% max.cobounded1
thf(fact_135_max_Ocobounded1,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ A @ ( ord_max_real @ A @ B ) ) ).
% max.cobounded1
thf(fact_136_max_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded1
thf(fact_137_max_Ocobounded1,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] : ( ord_le8929789700686245851l_num1 @ A @ ( ord_ma5275733517906637200l_num1 @ A @ B ) ) ).
% max.cobounded1
thf(fact_138_max_Ocobounded1,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] : ( ord_le6858968202089213338l_num1 @ A @ ( ord_ma2828161993112673189l_num1 @ A @ B ) ) ).
% max.cobounded1
thf(fact_139_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( A2
= ( ord_max_nat @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_140_max_Oorder__iff,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A2: num] :
( A2
= ( ord_max_num @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_141_max_Oorder__iff,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B2: extended_enat,A2: extended_enat] :
( A2
= ( ord_ma741700101516333627d_enat @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_142_max_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A2: real] :
( A2
= ( ord_max_real @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_143_max_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( A2
= ( ord_max_int @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_144_max_Oorder__iff,axiom,
( ord_le8929789700686245851l_num1
= ( ^ [B2: numera4273646738625120315l_num1,A2: numera4273646738625120315l_num1] :
( A2
= ( ord_ma5275733517906637200l_num1 @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_145_max_Oorder__iff,axiom,
( ord_le6858968202089213338l_num1
= ( ^ [B2: numera2417102609627094330l_num1,A2: numera2417102609627094330l_num1] :
( A2
= ( ord_ma2828161993112673189l_num1 @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_146_max_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_147_max_OboundedI,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ A )
=> ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_148_max_OboundedI,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ C @ A )
=> ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_149_max_OboundedI,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_150_max_OboundedI,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_151_max_OboundedI,axiom,
! [B: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ B @ A )
=> ( ( ord_le8929789700686245851l_num1 @ C @ A )
=> ( ord_le8929789700686245851l_num1 @ ( ord_ma5275733517906637200l_num1 @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_152_max_OboundedI,axiom,
! [B: numera2417102609627094330l_num1,A: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ B @ A )
=> ( ( ord_le6858968202089213338l_num1 @ C @ A )
=> ( ord_le6858968202089213338l_num1 @ ( ord_ma2828161993112673189l_num1 @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_153_max_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_154_max_OboundedE,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_num @ B @ A )
=> ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% max.boundedE
thf(fact_155_max_OboundedE,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
=> ~ ( ( ord_le2932123472753598470d_enat @ B @ A )
=> ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_156_max_OboundedE,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_real @ B @ A )
=> ~ ( ord_less_eq_real @ C @ A ) ) ) ).
% max.boundedE
thf(fact_157_max_OboundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_int @ B @ A )
=> ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% max.boundedE
thf(fact_158_max_OboundedE,axiom,
! [B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ ( ord_ma5275733517906637200l_num1 @ B @ C ) @ A )
=> ~ ( ( ord_le8929789700686245851l_num1 @ B @ A )
=> ~ ( ord_le8929789700686245851l_num1 @ C @ A ) ) ) ).
% max.boundedE
thf(fact_159_max_OboundedE,axiom,
! [B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1,A: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ ( ord_ma2828161993112673189l_num1 @ B @ C ) @ A )
=> ~ ( ( ord_le6858968202089213338l_num1 @ B @ A )
=> ~ ( ord_le6858968202089213338l_num1 @ C @ A ) ) ) ).
% max.boundedE
thf(fact_160_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_161_max_OorderI,axiom,
! [A: num,B: num] :
( ( A
= ( ord_max_num @ A @ B ) )
=> ( ord_less_eq_num @ B @ A ) ) ).
% max.orderI
thf(fact_162_max_OorderI,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A
= ( ord_ma741700101516333627d_enat @ A @ B ) )
=> ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% max.orderI
thf(fact_163_max_OorderI,axiom,
! [A: real,B: real] :
( ( A
= ( ord_max_real @ A @ B ) )
=> ( ord_less_eq_real @ B @ A ) ) ).
% max.orderI
thf(fact_164_max_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( ord_max_int @ A @ B ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
% max.orderI
thf(fact_165_max_OorderI,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( A
= ( ord_ma5275733517906637200l_num1 @ A @ B ) )
=> ( ord_le8929789700686245851l_num1 @ B @ A ) ) ).
% max.orderI
thf(fact_166_max_OorderI,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] :
( ( A
= ( ord_ma2828161993112673189l_num1 @ A @ B ) )
=> ( ord_le6858968202089213338l_num1 @ B @ A ) ) ).
% max.orderI
thf(fact_167_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_168_max_OorderE,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( A
= ( ord_max_num @ A @ B ) ) ) ).
% max.orderE
thf(fact_169_max_OorderE,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( A
= ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.orderE
thf(fact_170_max_OorderE,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( A
= ( ord_max_real @ A @ B ) ) ) ).
% max.orderE
thf(fact_171_max_OorderE,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( A
= ( ord_max_int @ A @ B ) ) ) ).
% max.orderE
thf(fact_172_max_OorderE,axiom,
! [B: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ B @ A )
=> ( A
= ( ord_ma5275733517906637200l_num1 @ A @ B ) ) ) ).
% max.orderE
thf(fact_173_max_OorderE,axiom,
! [B: numera2417102609627094330l_num1,A: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ B @ A )
=> ( A
= ( ord_ma2828161993112673189l_num1 @ A @ B ) ) ) ).
% max.orderE
thf(fact_174_max_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_175_max_Omono,axiom,
! [C: num,A: num,D: num,B: num] :
( ( ord_less_eq_num @ C @ A )
=> ( ( ord_less_eq_num @ D @ B )
=> ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% max.mono
thf(fact_176_max_Omono,axiom,
! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ A )
=> ( ( ord_le2932123472753598470d_enat @ D @ B )
=> ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_177_max_Omono,axiom,
! [C: real,A: real,D: real,B: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ( ord_less_eq_real @ D @ B )
=> ( ord_less_eq_real @ ( ord_max_real @ C @ D ) @ ( ord_max_real @ A @ B ) ) ) ) ).
% max.mono
thf(fact_178_max_Omono,axiom,
! [C: int,A: int,D: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ( ord_less_eq_int @ D @ B )
=> ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% max.mono
thf(fact_179_max_Omono,axiom,
! [C: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1,D: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ C @ A )
=> ( ( ord_le8929789700686245851l_num1 @ D @ B )
=> ( ord_le8929789700686245851l_num1 @ ( ord_ma5275733517906637200l_num1 @ C @ D ) @ ( ord_ma5275733517906637200l_num1 @ A @ B ) ) ) ) ).
% max.mono
thf(fact_180_max_Omono,axiom,
! [C: numera2417102609627094330l_num1,A: numera2417102609627094330l_num1,D: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ C @ A )
=> ( ( ord_le6858968202089213338l_num1 @ D @ B )
=> ( ord_le6858968202089213338l_num1 @ ( ord_ma2828161993112673189l_num1 @ C @ D ) @ ( ord_ma2828161993112673189l_num1 @ A @ B ) ) ) ) ).
% max.mono
thf(fact_181_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_182_p__prime,axiom,
factor1801147406995305544me_nat @ ( frequency_Moment_p @ n ) ).
% p_prime
thf(fact_183_prime__above__lower__bound,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( freque8783664969267990145_above @ N ) ) ).
% prime_above_lower_bound
thf(fact_184_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_185_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_186_dual__order_Orefl,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).
% dual_order.refl
thf(fact_187_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_188_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_189_dual__order_Orefl,axiom,
! [A: numera4273646738625120315l_num1] : ( ord_le8929789700686245851l_num1 @ A @ A ) ).
% dual_order.refl
thf(fact_190_dual__order_Orefl,axiom,
! [A: numera2417102609627094330l_num1] : ( ord_le6858968202089213338l_num1 @ A @ A ) ).
% dual_order.refl
thf(fact_191_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_192_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_193_order__refl,axiom,
! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ X @ X ) ).
% order_refl
thf(fact_194_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_195_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_196_order__refl,axiom,
! [X: numera4273646738625120315l_num1] : ( ord_le8929789700686245851l_num1 @ X @ X ) ).
% order_refl
thf(fact_197_order__refl,axiom,
! [X: numera2417102609627094330l_num1] : ( ord_le6858968202089213338l_num1 @ X @ X ) ).
% order_refl
thf(fact_198_prime__above__prime,axiom,
! [N: nat] : ( factor1801147406995305544me_nat @ ( freque8783664969267990145_above @ N ) ) ).
% prime_above_prime
thf(fact_199_verit__eq__simplify_I6_J,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_200_verit__eq__simplify_I6_J,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_201_verit__eq__simplify_I6_J,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( X = Y )
=> ( ord_le2932123472753598470d_enat @ X @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_202_verit__eq__simplify_I6_J,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_203_verit__eq__simplify_I6_J,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_204_verit__eq__simplify_I6_J,axiom,
! [X: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1] :
( ( X = Y )
=> ( ord_le8929789700686245851l_num1 @ X @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_205_verit__eq__simplify_I6_J,axiom,
! [X: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1] :
( ( X = Y )
=> ( ord_le6858968202089213338l_num1 @ X @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_206_verit__comp__simplify_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_207_verit__comp__simplify_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_208_verit__comp__simplify_I2_J,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_209_verit__comp__simplify_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_210_verit__comp__simplify_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_211_verit__comp__simplify_I2_J,axiom,
! [A: numera4273646738625120315l_num1] : ( ord_le8929789700686245851l_num1 @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_212_verit__comp__simplify_I2_J,axiom,
! [A: numera2417102609627094330l_num1] : ( ord_le6858968202089213338l_num1 @ A @ A ) ).
% verit_comp_simplify(2)
thf(fact_213_order__trans__rules_I26_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_214_order__trans__rules_I26_J,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_215_order__trans__rules_I26_J,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( A = B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_216_order__trans__rules_I26_J,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_217_order__trans__rules_I26_J,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_218_order__trans__rules_I26_J,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( A = B )
=> ( ( ord_le8929789700686245851l_num1 @ B @ C )
=> ( ord_le8929789700686245851l_num1 @ A @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_219_order__trans__rules_I26_J,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( A = B )
=> ( ( ord_le6858968202089213338l_num1 @ B @ C )
=> ( ord_le6858968202089213338l_num1 @ A @ C ) ) ) ).
% order_trans_rules(26)
thf(fact_220_order__trans__rules_I25_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_221_order__trans__rules_I25_J,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_222_order__trans__rules_I25_J,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( B = C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_223_order__trans__rules_I25_J,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_224_order__trans__rules_I25_J,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_225_order__trans__rules_I25_J,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ A @ B )
=> ( ( B = C )
=> ( ord_le8929789700686245851l_num1 @ A @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_226_order__trans__rules_I25_J,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ A @ B )
=> ( ( B = C )
=> ( ord_le6858968202089213338l_num1 @ A @ C ) ) ) ).
% order_trans_rules(25)
thf(fact_227_order__trans__rules_I24_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_trans_rules(24)
thf(fact_228_order__trans__rules_I24_J,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% order_trans_rules(24)
thf(fact_229_order__trans__rules_I24_J,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( A = B ) ) ) ).
% order_trans_rules(24)
thf(fact_230_order__trans__rules_I24_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% order_trans_rules(24)
thf(fact_231_order__trans__rules_I24_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% order_trans_rules(24)
thf(fact_232_order__trans__rules_I24_J,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ A @ B )
=> ( ( ord_le8929789700686245851l_num1 @ B @ A )
=> ( A = B ) ) ) ).
% order_trans_rules(24)
thf(fact_233_order__trans__rules_I24_J,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ A @ B )
=> ( ( ord_le6858968202089213338l_num1 @ B @ A )
=> ( A = B ) ) ) ).
% order_trans_rules(24)
thf(fact_234_order__trans__rules_I23_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans_rules(23)
thf(fact_235_order__trans__rules_I23_J,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_eq_num @ X @ Z ) ) ) ).
% order_trans_rules(23)
thf(fact_236_order__trans__rules_I23_J,axiom,
! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ Y @ Z )
=> ( ord_le2932123472753598470d_enat @ X @ Z ) ) ) ).
% order_trans_rules(23)
thf(fact_237_order__trans__rules_I23_J,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans_rules(23)
thf(fact_238_order__trans__rules_I23_J,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans_rules(23)
thf(fact_239_order__trans__rules_I23_J,axiom,
! [X: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1,Z: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ X @ Y )
=> ( ( ord_le8929789700686245851l_num1 @ Y @ Z )
=> ( ord_le8929789700686245851l_num1 @ X @ Z ) ) ) ).
% order_trans_rules(23)
thf(fact_240_order__trans__rules_I23_J,axiom,
! [X: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1,Z: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ X @ Y )
=> ( ( ord_le6858968202089213338l_num1 @ Y @ Z )
=> ( ord_le6858968202089213338l_num1 @ X @ Z ) ) ) ).
% order_trans_rules(23)
thf(fact_241_order__trans__rules_I10_J,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_242_order__trans__rules_I10_J,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_243_order__trans__rules_I10_J,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_244_order__trans__rules_I10_J,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_245_order__trans__rules_I10_J,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_246_order__trans__rules_I10_J,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_247_order__trans__rules_I10_J,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_248_order__trans__rules_I10_J,axiom,
! [A: extended_enat,F: num > extended_enat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_249_order__trans__rules_I10_J,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_250_order__trans__rules_I10_J,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(10)
thf(fact_251_order__trans__rules_I9_J,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_252_order__trans__rules_I9_J,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_253_order__trans__rules_I9_J,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_254_order__trans__rules_I9_J,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_255_order__trans__rules_I9_J,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_256_order__trans__rules_I9_J,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_257_order__trans__rules_I9_J,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_258_order__trans__rules_I9_J,axiom,
! [A: num,B: num,F: num > extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_259_order__trans__rules_I9_J,axiom,
! [A: num,B: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_260_order__trans__rules_I9_J,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(9)
thf(fact_261_order__trans__rules_I8_J,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_262_order__trans__rules_I8_J,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_263_order__trans__rules_I8_J,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X4: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_264_order__trans__rules_I8_J,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_265_order__trans__rules_I8_J,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_266_order__trans__rules_I8_J,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_267_order__trans__rules_I8_J,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_268_order__trans__rules_I8_J,axiom,
! [A: num,F: extended_enat > num,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X4: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X4 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_269_order__trans__rules_I8_J,axiom,
! [A: num,F: real > num,B: real,C: real] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_270_order__trans__rules_I8_J,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(8)
thf(fact_271_order__trans__rules_I7_J,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_272_order__trans__rules_I7_J,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_273_order__trans__rules_I7_J,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_274_order__trans__rules_I7_J,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_275_order__trans__rules_I7_J,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_276_order__trans__rules_I7_J,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_277_order__trans__rules_I7_J,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_278_order__trans__rules_I7_J,axiom,
! [A: num,B: num,F: num > extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_279_order__trans__rules_I7_J,axiom,
! [A: num,B: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_280_order__trans__rules_I7_J,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(7)
thf(fact_281_linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linear
thf(fact_282_linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linear
thf(fact_283_linear,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
| ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% linear
thf(fact_284_linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linear
thf(fact_285_linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linear
thf(fact_286_linear,axiom,
! [X: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ X @ Y )
| ( ord_le8929789700686245851l_num1 @ Y @ X ) ) ).
% linear
thf(fact_287_linear,axiom,
! [X: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ X @ Y )
| ( ord_le6858968202089213338l_num1 @ Y @ X ) ) ).
% linear
thf(fact_288_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_289_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_290_nle__le,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ A @ B ) )
= ( ( ord_le2932123472753598470d_enat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_291_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_292_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_293_nle__le,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( ~ ( ord_le8929789700686245851l_num1 @ A @ B ) )
= ( ( ord_le8929789700686245851l_num1 @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_294_nle__le,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] :
( ( ~ ( ord_le6858968202089213338l_num1 @ A @ B ) )
= ( ( ord_le6858968202089213338l_num1 @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_295_le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% le_cases
thf(fact_296_le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% le_cases
thf(fact_297_le__cases,axiom,
! [X: extended_enat,Y: extended_enat] :
( ~ ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% le_cases
thf(fact_298_le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% le_cases
thf(fact_299_le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% le_cases
thf(fact_300_le__cases,axiom,
! [X: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1] :
( ~ ( ord_le8929789700686245851l_num1 @ X @ Y )
=> ( ord_le8929789700686245851l_num1 @ Y @ X ) ) ).
% le_cases
thf(fact_301_le__cases,axiom,
! [X: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1] :
( ~ ( ord_le6858968202089213338l_num1 @ X @ Y )
=> ( ord_le6858968202089213338l_num1 @ Y @ X ) ) ).
% le_cases
thf(fact_302_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_303_le__cases3,axiom,
! [X: num,Y: num,Z: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z ) )
=> ( ( ( ord_less_eq_num @ X @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y ) )
=> ( ( ( ord_less_eq_num @ Z @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z )
=> ~ ( ord_less_eq_num @ Z @ X ) )
=> ~ ( ( ord_less_eq_num @ Z @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_304_le__cases3,axiom,
! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ~ ( ord_le2932123472753598470d_enat @ Y @ Z ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ~ ( ord_le2932123472753598470d_enat @ X @ Z ) )
=> ( ( ( ord_le2932123472753598470d_enat @ X @ Z )
=> ~ ( ord_le2932123472753598470d_enat @ Z @ Y ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Z @ Y )
=> ~ ( ord_le2932123472753598470d_enat @ Y @ X ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Y @ Z )
=> ~ ( ord_le2932123472753598470d_enat @ Z @ X ) )
=> ~ ( ( ord_le2932123472753598470d_enat @ Z @ X )
=> ~ ( ord_le2932123472753598470d_enat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_305_le__cases3,axiom,
! [X: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_306_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_307_le__cases3,axiom,
! [X: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1,Z: numera4273646738625120315l_num1] :
( ( ( ord_le8929789700686245851l_num1 @ X @ Y )
=> ~ ( ord_le8929789700686245851l_num1 @ Y @ Z ) )
=> ( ( ( ord_le8929789700686245851l_num1 @ Y @ X )
=> ~ ( ord_le8929789700686245851l_num1 @ X @ Z ) )
=> ( ( ( ord_le8929789700686245851l_num1 @ X @ Z )
=> ~ ( ord_le8929789700686245851l_num1 @ Z @ Y ) )
=> ( ( ( ord_le8929789700686245851l_num1 @ Z @ Y )
=> ~ ( ord_le8929789700686245851l_num1 @ Y @ X ) )
=> ( ( ( ord_le8929789700686245851l_num1 @ Y @ Z )
=> ~ ( ord_le8929789700686245851l_num1 @ Z @ X ) )
=> ~ ( ( ord_le8929789700686245851l_num1 @ Z @ X )
=> ~ ( ord_le8929789700686245851l_num1 @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_308_le__cases3,axiom,
! [X: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1,Z: numera2417102609627094330l_num1] :
( ( ( ord_le6858968202089213338l_num1 @ X @ Y )
=> ~ ( ord_le6858968202089213338l_num1 @ Y @ Z ) )
=> ( ( ( ord_le6858968202089213338l_num1 @ Y @ X )
=> ~ ( ord_le6858968202089213338l_num1 @ X @ Z ) )
=> ( ( ( ord_le6858968202089213338l_num1 @ X @ Z )
=> ~ ( ord_le6858968202089213338l_num1 @ Z @ Y ) )
=> ( ( ( ord_le6858968202089213338l_num1 @ Z @ Y )
=> ~ ( ord_le6858968202089213338l_num1 @ Y @ X ) )
=> ( ( ( ord_le6858968202089213338l_num1 @ Y @ Z )
=> ~ ( ord_le6858968202089213338l_num1 @ Z @ X ) )
=> ~ ( ( ord_le6858968202089213338l_num1 @ Z @ X )
=> ~ ( ord_le6858968202089213338l_num1 @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_309_antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_310_antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_311_antisym__conv,axiom,
! [Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ( ( ord_le2932123472753598470d_enat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_312_antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_313_antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_314_antisym__conv,axiom,
! [Y: numera4273646738625120315l_num1,X: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ Y @ X )
=> ( ( ord_le8929789700686245851l_num1 @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_315_antisym__conv,axiom,
! [Y: numera2417102609627094330l_num1,X: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ Y @ X )
=> ( ( ord_le6858968202089213338l_num1 @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_316_order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.eq_iff
thf(fact_317_order_Oeq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).
% order.eq_iff
thf(fact_318_order_Oeq__iff,axiom,
( ( ^ [Y4: extended_enat,Z2: extended_enat] : ( Y4 = Z2 ) )
= ( ^ [A2: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
& ( ord_le2932123472753598470d_enat @ B2 @ A2 ) ) ) ) ).
% order.eq_iff
thf(fact_319_order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% order.eq_iff
thf(fact_320_order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% order.eq_iff
thf(fact_321_order_Oeq__iff,axiom,
( ( ^ [Y4: numera4273646738625120315l_num1,Z2: numera4273646738625120315l_num1] : ( Y4 = Z2 ) )
= ( ^ [A2: numera4273646738625120315l_num1,B2: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ A2 @ B2 )
& ( ord_le8929789700686245851l_num1 @ B2 @ A2 ) ) ) ) ).
% order.eq_iff
thf(fact_322_order_Oeq__iff,axiom,
( ( ^ [Y4: numera2417102609627094330l_num1,Z2: numera2417102609627094330l_num1] : ( Y4 = Z2 ) )
= ( ^ [A2: numera2417102609627094330l_num1,B2: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ A2 @ B2 )
& ( ord_le6858968202089213338l_num1 @ B2 @ A2 ) ) ) ) ).
% order.eq_iff
thf(fact_323_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_324_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
= ( ^ [X2: num,Y5: num] :
( ( ord_less_eq_num @ X2 @ Y5 )
& ( ord_less_eq_num @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_325_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: extended_enat,Z2: extended_enat] : ( Y4 = Z2 ) )
= ( ^ [X2: extended_enat,Y5: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X2 @ Y5 )
& ( ord_le2932123472753598470d_enat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_326_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_327_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_328_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: numera4273646738625120315l_num1,Z2: numera4273646738625120315l_num1] : ( Y4 = Z2 ) )
= ( ^ [X2: numera4273646738625120315l_num1,Y5: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ X2 @ Y5 )
& ( ord_le8929789700686245851l_num1 @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_329_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: numera2417102609627094330l_num1,Z2: numera2417102609627094330l_num1] : ( Y4 = Z2 ) )
= ( ^ [X2: numera2417102609627094330l_num1,Y5: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ X2 @ Y5 )
& ( ord_le6858968202089213338l_num1 @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_330_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_331_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_332_order__antisym,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_333_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_334_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_335_order__antisym,axiom,
! [X: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ X @ Y )
=> ( ( ord_le8929789700686245851l_num1 @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_336_order__antisym,axiom,
! [X: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ X @ Y )
=> ( ( ord_le6858968202089213338l_num1 @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_337_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_338_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_339_order_Otrans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% order.trans
thf(fact_340_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_341_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_342_order_Otrans,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ A @ B )
=> ( ( ord_le8929789700686245851l_num1 @ B @ C )
=> ( ord_le8929789700686245851l_num1 @ A @ C ) ) ) ).
% order.trans
thf(fact_343_order_Otrans,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ A @ B )
=> ( ( ord_le6858968202089213338l_num1 @ B @ C )
=> ( ord_le6858968202089213338l_num1 @ A @ C ) ) ) ).
% order.trans
thf(fact_344_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_345_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: num,B3: num] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_346_linorder__wlog,axiom,
! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
( ! [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: extended_enat,B3: extended_enat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_347_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_348_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_349_linorder__wlog,axiom,
! [P: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 > $o,A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ! [A4: numera4273646738625120315l_num1,B3: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: numera4273646738625120315l_num1,B3: numera4273646738625120315l_num1] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_350_linorder__wlog,axiom,
! [P: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > $o,A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] :
( ! [A4: numera2417102609627094330l_num1,B3: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: numera2417102609627094330l_num1,B3: numera2417102609627094330l_num1] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_351_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_352_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_353_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: extended_enat,Z2: extended_enat] : ( Y4 = Z2 ) )
= ( ^ [A2: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
& ( ord_le2932123472753598470d_enat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_354_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_355_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_356_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: numera4273646738625120315l_num1,Z2: numera4273646738625120315l_num1] : ( Y4 = Z2 ) )
= ( ^ [A2: numera4273646738625120315l_num1,B2: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ B2 @ A2 )
& ( ord_le8929789700686245851l_num1 @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_357_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: numera2417102609627094330l_num1,Z2: numera2417102609627094330l_num1] : ( Y4 = Z2 ) )
= ( ^ [A2: numera2417102609627094330l_num1,B2: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ B2 @ A2 )
& ( ord_le6858968202089213338l_num1 @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_358_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_359_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_360_dual__order_Oantisym,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_361_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_362_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_363_dual__order_Oantisym,axiom,
! [B: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ B @ A )
=> ( ( ord_le8929789700686245851l_num1 @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_364_dual__order_Oantisym,axiom,
! [B: numera2417102609627094330l_num1,A: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ B @ A )
=> ( ( ord_le6858968202089213338l_num1 @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_365_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_366_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_367_dual__order_Otrans,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_368_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_369_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_370_dual__order_Otrans,axiom,
! [B: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ B @ A )
=> ( ( ord_le8929789700686245851l_num1 @ C @ B )
=> ( ord_le8929789700686245851l_num1 @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_371_dual__order_Otrans,axiom,
! [B: numera2417102609627094330l_num1,A: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ B @ A )
=> ( ( ord_le6858968202089213338l_num1 @ C @ B )
=> ( ord_le6858968202089213338l_num1 @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_372_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_373_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_374_verit__la__disequality,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A = B )
| ~ ( ord_le2932123472753598470d_enat @ A @ B )
| ~ ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_375_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_376_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_377_verit__la__disequality,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( A = B )
| ~ ( ord_le8929789700686245851l_num1 @ A @ B )
| ~ ( ord_le8929789700686245851l_num1 @ B @ A ) ) ).
% verit_la_disequality
thf(fact_378_verit__la__disequality,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] :
( ( A = B )
| ~ ( ord_le6858968202089213338l_num1 @ A @ B )
| ~ ( ord_le6858968202089213338l_num1 @ B @ A ) ) ).
% verit_la_disequality
thf(fact_379_max__def,axiom,
( ord_max_nat
= ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_380_max__def,axiom,
( ord_max_num
= ( ^ [A2: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_381_max__def,axiom,
( ord_ma741700101516333627d_enat
= ( ^ [A2: extended_enat,B2: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_382_max__def,axiom,
( ord_max_real
= ( ^ [A2: real,B2: real] : ( if_real @ ( ord_less_eq_real @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_383_max__def,axiom,
( ord_max_int
= ( ^ [A2: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_384_max__def,axiom,
( ord_ma5275733517906637200l_num1
= ( ^ [A2: numera4273646738625120315l_num1,B2: numera4273646738625120315l_num1] : ( if_Num3220014061592582145l_num1 @ ( ord_le8929789700686245851l_num1 @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_385_max__def,axiom,
( ord_ma2828161993112673189l_num1
= ( ^ [A2: numera2417102609627094330l_num1,B2: numera2417102609627094330l_num1] : ( if_Num9196306924077011444l_num1 @ ( ord_le6858968202089213338l_num1 @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_386_max__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_max_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_387_max__absorb1,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_max_num @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_388_max__absorb1,axiom,
! [Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ( ( ord_ma741700101516333627d_enat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_389_max__absorb1,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_max_real @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_390_max__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_max_int @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_391_max__absorb1,axiom,
! [Y: numera4273646738625120315l_num1,X: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ Y @ X )
=> ( ( ord_ma5275733517906637200l_num1 @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_392_max__absorb1,axiom,
! [Y: numera2417102609627094330l_num1,X: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ Y @ X )
=> ( ( ord_ma2828161993112673189l_num1 @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_393_max__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_max_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_394_max__absorb2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_max_num @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_395_max__absorb2,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_ma741700101516333627d_enat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_396_max__absorb2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_max_real @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_397_max__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_max_int @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_398_max__absorb2,axiom,
! [X: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1] :
( ( ord_le8929789700686245851l_num1 @ X @ Y )
=> ( ( ord_ma5275733517906637200l_num1 @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_399_max__absorb2,axiom,
! [X: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1] :
( ( ord_le6858968202089213338l_num1 @ X @ Y )
=> ( ( ord_ma2828161993112673189l_num1 @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_400_verit__comp__simplify_I6_J,axiom,
! [N: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( ord_less_eq_num @ N @ one ) ) ).
% verit_comp_simplify(6)
thf(fact_401_verit__comp__simplify_I6_J,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% verit_comp_simplify(6)
thf(fact_402_verit__comp__simplify_I6_J,axiom,
! [N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
= ( ord_less_eq_num @ N @ one ) ) ).
% verit_comp_simplify(6)
thf(fact_403_verit__comp__simplify_I6_J,axiom,
! [N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% verit_comp_simplify(6)
thf(fact_404_verit__comp__simplify_I6_J,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% verit_comp_simplify(6)
thf(fact_405_numeral__le__enat__iff,axiom,
! [M: num,N: nat] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% numeral_le_enat_iff
thf(fact_406_prime__above__min,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( freque8783664969267990145_above @ N ) ) ).
% prime_above_min
thf(fact_407_rel__simps_I29_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% rel_simps(29)
thf(fact_408_rel__simps_I29_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% rel_simps(29)
thf(fact_409_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_410_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_411_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_412_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_413_verit__eq__simplify_I25_J,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% verit_eq_simplify(25)
thf(fact_414_verit__eq__simplify_I25_J,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% verit_eq_simplify(25)
thf(fact_415_more__arith__simps_I10_J,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% more_arith_simps(10)
thf(fact_416_more__arith__simps_I10_J,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% more_arith_simps(10)
thf(fact_417_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_418_more__arith__simps_I1_J,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% more_arith_simps(1)
thf(fact_419_more__arith__simps_I1_J,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% more_arith_simps(1)
thf(fact_420_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_421_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_422_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_423_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_424_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_425_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_426_rel__simps_I4_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% rel_simps(4)
thf(fact_427_max__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_ma741700101516333627d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( ord_max_nat @ M @ N ) ) ) ).
% max_enat_simps(1)
thf(fact_428_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_rat
= ( numeral_numeral_rat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_429_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_on7984719198319812577d_enat
= ( numera1916890842035813515d_enat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_430_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_431_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_432_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_433_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_rat @ N )
= one_one_rat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_434_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera1916890842035813515d_enat @ N )
= one_on7984719198319812577d_enat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_435_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_436_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_437_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_438_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(5)
thf(fact_439_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(5)
thf(fact_440_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(5)
thf(fact_441_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(5)
thf(fact_442_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(5)
thf(fact_443_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(6)
thf(fact_444_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(6)
thf(fact_445_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(6)
thf(fact_446_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(6)
thf(fact_447_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(6)
thf(fact_448_rel__simps_I2_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% rel_simps(2)
thf(fact_449_rel__simps_I5_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% rel_simps(5)
thf(fact_450_enat__ord__code_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% enat_ord_code(1)
thf(fact_451_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_452_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ one_one_real ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_453_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_454_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_rat @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_455_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_456_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_457_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
& ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( numeral_numeral_real @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_458_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( numeral_numeral_int @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_459_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_le8929789700686245851l_num1 @ ( numera7754357348821619680l_num1 @ U ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) )
=> ( ( ord_ma5275733517906637200l_num1 @ ( numera7754357348821619680l_num1 @ U ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) ) )
& ( ~ ( ord_le8929789700686245851l_num1 @ ( numera7754357348821619680l_num1 @ U ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) )
=> ( ( ord_ma5275733517906637200l_num1 @ ( numera7754357348821619680l_num1 @ U ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) )
= ( numera7754357348821619680l_num1 @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_460_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_le6858968202089213338l_num1 @ ( numera2161328050825114965l_num1 @ U ) @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ V ) ) )
=> ( ( ord_ma2828161993112673189l_num1 @ ( numera2161328050825114965l_num1 @ U ) @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ V ) ) )
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ V ) ) ) )
& ( ~ ( ord_le6858968202089213338l_num1 @ ( numera2161328050825114965l_num1 @ U ) @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ V ) ) )
=> ( ( ord_ma2828161993112673189l_num1 @ ( numera2161328050825114965l_num1 @ U ) @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ V ) ) )
= ( numera2161328050825114965l_num1 @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_461_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ V ) ) )
& ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_462_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ V ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_463_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_le8929789700686245851l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ U ) ) @ ( numera7754357348821619680l_num1 @ V ) )
=> ( ( ord_ma5275733517906637200l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ U ) ) @ ( numera7754357348821619680l_num1 @ V ) )
= ( numera7754357348821619680l_num1 @ V ) ) )
& ( ~ ( ord_le8929789700686245851l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ U ) ) @ ( numera7754357348821619680l_num1 @ V ) )
=> ( ( ord_ma5275733517906637200l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ U ) ) @ ( numera7754357348821619680l_num1 @ V ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_464_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_le6858968202089213338l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ U ) ) @ ( numera2161328050825114965l_num1 @ V ) )
=> ( ( ord_ma2828161993112673189l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ U ) ) @ ( numera2161328050825114965l_num1 @ V ) )
= ( numera2161328050825114965l_num1 @ V ) ) )
& ( ~ ( ord_le6858968202089213338l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ U ) ) @ ( numera2161328050825114965l_num1 @ V ) )
=> ( ( ord_ma2828161993112673189l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ U ) ) @ ( numera2161328050825114965l_num1 @ V ) )
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_465_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
& ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_466_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_467_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_le8929789700686245851l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ U ) ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) )
=> ( ( ord_ma5275733517906637200l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ U ) ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) ) )
& ( ~ ( ord_le8929789700686245851l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ U ) ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) )
=> ( ( ord_ma5275733517906637200l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ U ) ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_468_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_le6858968202089213338l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ U ) ) @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ V ) ) )
=> ( ( ord_ma2828161993112673189l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ U ) ) @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ V ) ) )
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ V ) ) ) )
& ( ~ ( ord_le6858968202089213338l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ U ) ) @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ V ) ) )
=> ( ( ord_ma2828161993112673189l_num1 @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ U ) ) @ ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ V ) ) )
= ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_469_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_470_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_471_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_472_one__enat__def,axiom,
( one_on7984719198319812577d_enat
= ( extended_enat2 @ one_one_nat ) ) ).
% one_enat_def
thf(fact_473_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_474_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_475_enat__1__iff_I1_J,axiom,
! [X: nat] :
( ( ( extended_enat2 @ X )
= one_on7984719198319812577d_enat )
= ( X = one_one_nat ) ) ).
% enat_1_iff(1)
thf(fact_476_enat__1__iff_I2_J,axiom,
! [X: nat] :
( ( one_on7984719198319812577d_enat
= ( extended_enat2 @ X ) )
= ( X = one_one_nat ) ) ).
% enat_1_iff(2)
thf(fact_477_le__minus__one__simps_I2_J,axiom,
ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% le_minus_one_simps(2)
thf(fact_478_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_479_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_480_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% le_minus_one_simps(4)
thf(fact_481_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_482_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_483_one__neq__neg__one,axiom,
( one_one_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% one_neq_neg_one
thf(fact_484_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_485_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_486_rel__simps_I85_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ N )
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% rel_simps(85)
thf(fact_487_rel__simps_I85_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ N )
!= ( uminus_uminus_real @ one_one_real ) ) ).
% rel_simps(85)
thf(fact_488_rel__simps_I85_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% rel_simps(85)
thf(fact_489_rel__simps_I81_J,axiom,
! [N: num] :
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
!= one_one_rat ) ).
% rel_simps(81)
thf(fact_490_rel__simps_I81_J,axiom,
! [N: num] :
( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
!= one_one_real ) ).
% rel_simps(81)
thf(fact_491_rel__simps_I81_J,axiom,
! [N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
!= one_one_int ) ).
% rel_simps(81)
thf(fact_492_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_493_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_494_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_495_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% not_numeral_le_neg_one
thf(fact_496_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_le_neg_one
thf(fact_497_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_498_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% neg_numeral_le_neg_one
thf(fact_499_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% neg_numeral_le_neg_one
thf(fact_500_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_501_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% neg_one_le_numeral
thf(fact_502_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% neg_one_le_numeral
thf(fact_503_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_le_numeral
thf(fact_504_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% neg_numeral_le_one
thf(fact_505_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% neg_numeral_le_one
thf(fact_506_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_507_uminus__numeral__One,axiom,
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% uminus_numeral_One
thf(fact_508_uminus__numeral__One,axiom,
( ( uminus7224005126491068675l_num1 @ ( numera2161328050825114965l_num1 @ one ) )
= ( uminus7224005126491068675l_num1 @ one_on3868389512446148991l_num1 ) ) ).
% uminus_numeral_One
thf(fact_509_uminus__numeral__One,axiom,
( ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ one ) )
= ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) ) ).
% uminus_numeral_One
thf(fact_510_uminus__numeral__One,axiom,
( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% uminus_numeral_One
thf(fact_511_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_512_verit__negate__coefficient_I1_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(1)
thf(fact_513_verit__negate__coefficient_I1_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(1)
thf(fact_514_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_real @ M )
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_515_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_516_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
!= ( numeral_numeral_real @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_517_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_518_num_Odistinct_I1_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% num.distinct(1)
thf(fact_519_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_520_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_521_le__numeral__extra_I4_J,axiom,
ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ).
% le_numeral_extra(4)
thf(fact_522_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_523_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_524_enat__ile,axiom,
! [N: extended_enat,M: nat] :
( ( ord_le2932123472753598470d_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_ile
thf(fact_525_num_Odistinct_I5_J,axiom,
! [X22: num,X3: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X3 ) ) ).
% num.distinct(5)
thf(fact_526_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_527_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_528_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_529_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_530_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% one_le_numeral
thf(fact_531_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_532_one__le__numeral,axiom,
! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% one_le_numeral
thf(fact_533_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% one_le_numeral
thf(fact_534_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_535_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X32: num] :
( Y
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_536_numeral__One,axiom,
( ( numeral_numeral_rat @ one )
= one_one_rat ) ).
% numeral_One
thf(fact_537_numeral__One,axiom,
( ( numera2161328050825114965l_num1 @ one )
= one_on3868389512446148991l_num1 ) ).
% numeral_One
thf(fact_538_numeral__One,axiom,
( ( numera7754357348821619680l_num1 @ one )
= one_on7795324986448017462l_num1 ) ).
% numeral_One
thf(fact_539_numeral__One,axiom,
( ( numera1916890842035813515d_enat @ one )
= one_on7984719198319812577d_enat ) ).
% numeral_One
thf(fact_540_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_541_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_542_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_543_enat__numeral,axiom,
! [K2: num] :
( ( extended_enat2 @ ( numeral_numeral_nat @ K2 ) )
= ( numera1916890842035813515d_enat @ K2 ) ) ).
% enat_numeral
thf(fact_544_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_545_prime__ge__2__nat,axiom,
! [P2: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P2 ) ) ).
% prime_ge_2_nat
thf(fact_546_two__is__prime__nat,axiom,
factor1801147406995305544me_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% two_is_prime_nat
thf(fact_547_p__gt__1,axiom,
ord_less_nat @ one_one_nat @ ( frequency_Moment_p @ n ) ).
% p_gt_1
thf(fact_548_exhaust__4,axiom,
! [X: numera4273646738625120315l_num1] :
( ( X = one_on7795324986448017462l_num1 )
| ( X
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
| ( X
= ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
| ( X
= ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% exhaust_4
thf(fact_549_forall__4,axiom,
( ( ^ [P3: numera4273646738625120315l_num1 > $o] :
! [X5: numera4273646738625120315l_num1] : ( P3 @ X5 ) )
= ( ^ [P4: numera4273646738625120315l_num1 > $o] :
( ( P4 @ one_on7795324986448017462l_num1 )
& ( P4 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
& ( P4 @ ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
& ( P4 @ ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ).
% forall_4
thf(fact_550_exhaust__3,axiom,
! [X: numera6367994245245682809l_num1] :
( ( X = one_on7819281148064737470l_num1 )
| ( X
= ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
| ( X
= ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ).
% exhaust_3
thf(fact_551_forall__3,axiom,
( ( ^ [P3: numera6367994245245682809l_num1 > $o] :
! [X5: numera6367994245245682809l_num1] : ( P3 @ X5 ) )
= ( ^ [P4: numera6367994245245682809l_num1 > $o] :
( ( P4 @ one_on7819281148064737470l_num1 )
& ( P4 @ ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
& ( P4 @ ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ) ) ).
% forall_3
thf(fact_552_rel__simps_I48_J,axiom,
! [M: num,N: num] :
( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% rel_simps(48)
thf(fact_553_rel__simps_I48_J,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% rel_simps(48)
thf(fact_554_rel__simps_I48_J,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% rel_simps(48)
thf(fact_555_rel__simps_I48_J,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% rel_simps(48)
thf(fact_556_rel__simps_I48_J,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% rel_simps(48)
thf(fact_557_neg__less__iff__less,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_558_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_559_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_560_pre__arith__simps_I2_J,axiom,
! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X @ Y ) @ Z )
= ( ( ord_le72135733267957522d_enat @ X @ Z )
& ( ord_le72135733267957522d_enat @ Y @ Z ) ) ) ).
% pre_arith_simps(2)
thf(fact_561_pre__arith__simps_I2_J,axiom,
! [X: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1,Z: numera4273646738625120315l_num1] :
( ( ord_le9186584087594166503l_num1 @ ( ord_ma5275733517906637200l_num1 @ X @ Y ) @ Z )
= ( ( ord_le9186584087594166503l_num1 @ X @ Z )
& ( ord_le9186584087594166503l_num1 @ Y @ Z ) ) ) ).
% pre_arith_simps(2)
thf(fact_562_pre__arith__simps_I2_J,axiom,
! [X: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1,Z: numera2417102609627094330l_num1] :
( ( ord_le8952004418993413518l_num1 @ ( ord_ma2828161993112673189l_num1 @ X @ Y ) @ Z )
= ( ( ord_le8952004418993413518l_num1 @ X @ Z )
& ( ord_le8952004418993413518l_num1 @ Y @ Z ) ) ) ).
% pre_arith_simps(2)
thf(fact_563_pre__arith__simps_I2_J,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z )
= ( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Y @ Z ) ) ) ).
% pre_arith_simps(2)
thf(fact_564_pre__arith__simps_I2_J,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z )
= ( ( ord_less_rat @ X @ Z )
& ( ord_less_rat @ Y @ Z ) ) ) ).
% pre_arith_simps(2)
thf(fact_565_pre__arith__simps_I2_J,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z )
= ( ( ord_less_num @ X @ Z )
& ( ord_less_num @ Y @ Z ) ) ) ).
% pre_arith_simps(2)
thf(fact_566_pre__arith__simps_I2_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
= ( ( ord_less_nat @ X @ Z )
& ( ord_less_nat @ Y @ Z ) ) ) ).
% pre_arith_simps(2)
thf(fact_567_pre__arith__simps_I2_J,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
= ( ( ord_less_int @ X @ Z )
& ( ord_less_int @ Y @ Z ) ) ) ).
% pre_arith_simps(2)
thf(fact_568_max_Oabsorb3,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_569_max_Oabsorb3,axiom,
! [B: numera4273646738625120315l_num1,A: numera4273646738625120315l_num1] :
( ( ord_le9186584087594166503l_num1 @ B @ A )
=> ( ( ord_ma5275733517906637200l_num1 @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_570_max_Oabsorb3,axiom,
! [B: numera2417102609627094330l_num1,A: numera2417102609627094330l_num1] :
( ( ord_le8952004418993413518l_num1 @ B @ A )
=> ( ( ord_ma2828161993112673189l_num1 @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_571_max_Oabsorb3,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_max_real @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_572_max_Oabsorb3,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_max_rat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_573_max_Oabsorb3,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_max_num @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_574_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_575_max_Oabsorb3,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_576_max_Oabsorb4,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_577_max_Oabsorb4,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( ord_le9186584087594166503l_num1 @ A @ B )
=> ( ( ord_ma5275733517906637200l_num1 @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_578_max_Oabsorb4,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] :
( ( ord_le8952004418993413518l_num1 @ A @ B )
=> ( ( ord_ma2828161993112673189l_num1 @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_579_max_Oabsorb4,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_max_real @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_580_max_Oabsorb4,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_max_rat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_581_max_Oabsorb4,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_max_num @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_582_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_583_max_Oabsorb4,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_584_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_585_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_586_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_587_verit__comp__simplify_I17_J,axiom,
! [N: num] :
( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% verit_comp_simplify(17)
thf(fact_588_verit__comp__simplify_I17_J,axiom,
! [N: num] :
( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% verit_comp_simplify(17)
thf(fact_589_verit__comp__simplify_I17_J,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% verit_comp_simplify(17)
thf(fact_590_verit__comp__simplify_I17_J,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% verit_comp_simplify(17)
thf(fact_591_verit__comp__simplify_I17_J,axiom,
! [N: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% verit_comp_simplify(17)
thf(fact_592_verit__comp__simplify_I39_J,axiom,
! [M: num] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( M != one ) ) ).
% verit_comp_simplify(39)
thf(fact_593_verit__comp__simplify_I39_J,axiom,
! [M: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( M != one ) ) ).
% verit_comp_simplify(39)
thf(fact_594_verit__comp__simplify_I39_J,axiom,
! [M: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M != one ) ) ).
% verit_comp_simplify(39)
thf(fact_595_less__minus__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% less_minus_iff
thf(fact_596_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_597_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_598_minus__less__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_599_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_600_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_601_verit__comp__simplify_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify(1)
thf(fact_602_verit__comp__simplify_I1_J,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% verit_comp_simplify(1)
thf(fact_603_verit__comp__simplify_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify(1)
thf(fact_604_verit__comp__simplify_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify(1)
thf(fact_605_verit__comp__simplify_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify(1)
thf(fact_606_order__trans__rules_I28_J,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order_trans_rules(28)
thf(fact_607_order__trans__rules_I28_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order_trans_rules(28)
thf(fact_608_order__trans__rules_I28_J,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order_trans_rules(28)
thf(fact_609_order__trans__rules_I28_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order_trans_rules(28)
thf(fact_610_order__trans__rules_I28_J,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order_trans_rules(28)
thf(fact_611_order__trans__rules_I27_J,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order_trans_rules(27)
thf(fact_612_order__trans__rules_I27_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( B = C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order_trans_rules(27)
thf(fact_613_order__trans__rules_I27_J,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order_trans_rules(27)
thf(fact_614_order__trans__rules_I27_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order_trans_rules(27)
thf(fact_615_order__trans__rules_I27_J,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order_trans_rules(27)
thf(fact_616_order__trans__rules_I20_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_trans_rules(20)
thf(fact_617_order__trans__rules_I20_J,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order_trans_rules(20)
thf(fact_618_order__trans__rules_I20_J,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_trans_rules(20)
thf(fact_619_order__trans__rules_I20_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_trans_rules(20)
thf(fact_620_order__trans__rules_I20_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_trans_rules(20)
thf(fact_621_order__trans__rules_I19_J,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_trans_rules(19)
thf(fact_622_order__trans__rules_I19_J,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_trans_rules(19)
thf(fact_623_order__trans__rules_I19_J,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_trans_rules(19)
thf(fact_624_order__trans__rules_I19_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_trans_rules(19)
thf(fact_625_order__trans__rules_I19_J,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_trans_rules(19)
thf(fact_626_order__trans__rules_I12_J,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_627_order__trans__rules_I12_J,axiom,
! [A: rat,F: real > rat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_628_order__trans__rules_I12_J,axiom,
! [A: num,F: real > num,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_629_order__trans__rules_I12_J,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_630_order__trans__rules_I12_J,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_631_order__trans__rules_I12_J,axiom,
! [A: real,F: rat > real,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X4: rat,Y2: rat] :
( ( ord_less_rat @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_632_order__trans__rules_I12_J,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X4: rat,Y2: rat] :
( ( ord_less_rat @ X4 @ Y2 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_633_order__trans__rules_I12_J,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X4: rat,Y2: rat] :
( ( ord_less_rat @ X4 @ Y2 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_634_order__trans__rules_I12_J,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X4: rat,Y2: rat] :
( ( ord_less_rat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_635_order__trans__rules_I12_J,axiom,
! [A: int,F: rat > int,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X4: rat,Y2: rat] :
( ( ord_less_rat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_636_order__trans__rules_I11_J,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(11)
thf(fact_637_order__trans__rules_I11_J,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(11)
thf(fact_638_order__trans__rules_I11_J,axiom,
! [A: int,B: int,F: int > rat,C: rat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(11)
thf(fact_639_order__trans__rules_I11_J,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(11)
thf(fact_640_order__trans__rules_I11_J,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(11)
thf(fact_641_order__trans__rules_I11_J,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_trans_rules(11)
thf(fact_642_bigger__prime,axiom,
! [N: nat] :
? [P5: nat] :
( ( factor1801147406995305544me_nat @ P5 )
& ( ord_less_nat @ N @ P5 ) ) ).
% bigger_prime
thf(fact_643_prime__gt__1__nat,axiom,
! [P2: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ord_less_nat @ one_one_nat @ P2 ) ) ).
% prime_gt_1_nat
thf(fact_644_exhaust__2,axiom,
! [X: numera2417102609627094330l_num1] :
( ( X = one_on3868389512446148991l_num1 )
| ( X
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).
% exhaust_2
thf(fact_645_forall__2,axiom,
( ( ^ [P3: numera2417102609627094330l_num1 > $o] :
! [X5: numera2417102609627094330l_num1] : ( P3 @ X5 ) )
= ( ^ [P4: numera2417102609627094330l_num1 > $o] :
( ( P4 @ one_on3868389512446148991l_num1 )
& ( P4 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).
% forall_2
thf(fact_646_prime__ge__1__nat,axiom,
! [P2: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ord_less_eq_nat @ one_one_nat @ P2 ) ) ).
% prime_ge_1_nat
thf(fact_647_p__gt__0,axiom,
ord_less_nat @ zero_zero_nat @ ( frequency_Moment_p @ n ) ).
% p_gt_0
thf(fact_648_enat__ord__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% enat_ord_simps(2)
thf(fact_649_verit__comp__simplify_I24_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% verit_comp_simplify(24)
thf(fact_650_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_651_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_652_rel__simps_I9_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% rel_simps(9)
thf(fact_653_rel__simps_I14_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% rel_simps(14)
thf(fact_654_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_655_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_656_rel__simps_I12_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% rel_simps(12)
thf(fact_657_rel__simps_I7_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% rel_simps(7)
thf(fact_658_numeral__less__enat__iff,axiom,
! [M: num,N: nat] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% numeral_less_enat_iff
thf(fact_659_enat__iless,axiom,
! [N: extended_enat,M: nat] :
( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_iless
thf(fact_660_less__enatE,axiom,
! [N: extended_enat,M: nat] :
( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
=> ~ ! [K: nat] :
( ( N
= ( extended_enat2 @ K ) )
=> ~ ( ord_less_nat @ K @ M ) ) ) ).
% less_enatE
thf(fact_661_prime__gt__0__nat,axiom,
! [P2: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ord_less_nat @ zero_zero_nat @ P2 ) ) ).
% prime_gt_0_nat
thf(fact_662_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_663_one__less__numeral,axiom,
! [N: num] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral
thf(fact_664_max__nat_Ocomm__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.comm_neutral
thf(fact_665_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_666__092_060delta_062__range,axiom,
ord_less_rat @ zero_zero_rat @ delta ).
% \<delta>_range
thf(fact_667_i0__less,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_668_max__enat__simps_I2_J,axiom,
! [Q: extended_enat] :
( ( ord_ma741700101516333627d_enat @ Q @ zero_z5237406670263579293d_enat )
= Q ) ).
% max_enat_simps(2)
thf(fact_669_max__enat__simps_I3_J,axiom,
! [Q: extended_enat] :
( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q )
= Q ) ).
% max_enat_simps(3)
thf(fact_670_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_671_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_672_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_673_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_674_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_675_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_676_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_677_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_678_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_679_ennreal__zero__less__one,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% ennreal_zero_less_one
thf(fact_680_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_681_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N2: extended_enat] :
( ! [M2: extended_enat] :
( ( ord_le72135733267957522d_enat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_682_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_683_i0__lb,axiom,
! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% i0_lb
thf(fact_684_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
thf(fact_685_zero__enat__def,axiom,
( zero_z5237406670263579293d_enat
= ( extended_enat2 @ zero_zero_nat ) ) ).
% zero_enat_def
thf(fact_686_enat__0__iff_I1_J,axiom,
! [X: nat] :
( ( ( extended_enat2 @ X )
= zero_z5237406670263579293d_enat )
= ( X = zero_zero_nat ) ) ).
% enat_0_iff(1)
thf(fact_687_enat__0__iff_I2_J,axiom,
! [X: nat] :
( ( zero_z5237406670263579293d_enat
= ( extended_enat2 @ X ) )
= ( X = zero_zero_nat ) ) ).
% enat_0_iff(2)
thf(fact_688_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_689_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_690_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_691_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_692_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_693_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_694_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_695_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_696_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_697_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_698_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_699_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_700_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_701_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_702_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_703_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_704_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_705_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_706_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_707_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_708_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_709_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_710_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_711_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_712_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_713_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_714_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_715_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_716_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_717_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_718_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_719_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ~ ( P @ I3 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_720_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_721_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q2: nat > $o] :
( ! [X4: nat > real] :
( ( P @ X4 )
=> ( P @ ( F @ X4 ) ) )
=> ( ! [X4: nat > real] :
( ( P @ X4 )
=> ! [I2: nat] :
( ( Q2 @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I2 ) )
& ( ord_less_eq_real @ ( X4 @ I2 ) @ one_one_real ) ) ) )
=> ? [L: ( nat > real ) > nat > nat] :
( ! [X6: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L @ X6 @ I3 ) @ one_one_nat )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q2 @ I3 )
& ( ( X6 @ I3 )
= zero_zero_real ) )
=> ( ( L @ X6 @ I3 )
= zero_zero_nat ) )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q2 @ I3 )
& ( ( X6 @ I3 )
= one_one_real ) )
=> ( ( L @ X6 @ I3 )
= one_one_nat ) )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q2 @ I3 )
& ( ( L @ X6 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X6 @ I3 ) @ ( F @ X6 @ I3 ) ) )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q2 @ I3 )
& ( ( L @ X6 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X6 @ I3 ) @ ( X6 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_722_s1__gt__0,axiom,
ord_less_nat @ zero_zero_nat @ ( frequency_Moment_s_1 @ delta ) ).
% s1_gt_0
thf(fact_723_real__eq__0__iff__le__ge__0,axiom,
! [X: real] :
( ( X = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_724_the__enat_Osimps,axiom,
! [N: nat] :
( ( extended_the_enat @ ( extended_enat2 @ N ) )
= N ) ).
% the_enat.simps
thf(fact_725_prime__int__numeral__eq,axiom,
! [M: num] :
( ( factor1798656936486255268me_int @ ( numeral_numeral_int @ M ) )
= ( factor1801147406995305544me_nat @ ( numeral_numeral_nat @ M ) ) ) ).
% prime_int_numeral_eq
thf(fact_726_prime__ge__2__int,axiom,
! [P2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ord_less_eq_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ P2 ) ) ).
% prime_ge_2_int
thf(fact_727_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N @ K )
=> ( P @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K @ I3 )
=> ( P @ I3 ) )
=> ( P @ K ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_728_prime__ge__0__int,axiom,
! [P2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ord_less_eq_int @ zero_zero_int @ P2 ) ) ).
% prime_ge_0_int
thf(fact_729_prime__ge__1__int,axiom,
! [P2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ord_less_eq_int @ one_one_int @ P2 ) ) ).
% prime_ge_1_int
thf(fact_730_prime__gt__1__int,axiom,
! [P2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ord_less_int @ one_one_int @ P2 ) ) ).
% prime_gt_1_int
thf(fact_731_prime__gt__0__int,axiom,
! [P2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ord_less_int @ zero_zero_int @ P2 ) ) ).
% prime_gt_0_int
thf(fact_732_seq__mono__lemma,axiom,
! [M: nat,D: nat > real,E: nat > real] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_real @ ( D @ N2 ) @ ( E @ N2 ) ) )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_real @ ( E @ N2 ) @ ( E @ M ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ord_less_real @ ( D @ N4 ) @ ( E @ M ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_733_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_734_int_Ominus__zero,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% int.minus_zero
thf(fact_735_int_Ozero__not__one,axiom,
zero_zero_int != one_one_int ).
% int.zero_not_one
thf(fact_736_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_737_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_738_int_Olless__eq,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% int.lless_eq
thf(fact_739_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P6: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P6 ) ) ) ) ).
% imp_le_cong
thf(fact_740_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P6: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P6 ) ) ) ) ).
% conj_le_cong
thf(fact_741_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_742_s2__gt__0,axiom,
ord_less_nat @ zero_zero_nat @ ( frequency_Moment_s_2 @ epsilon ) ).
% s2_gt_0
thf(fact_743__092_060epsilon_062__range,axiom,
member_rat @ epsilon @ ( set_or5199638295745620268an_rat @ zero_zero_rat @ one_one_rat ) ).
% \<epsilon>_range
thf(fact_744_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_745_prime__nat__naiveI,axiom,
! [P2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P2 )
=> ( ! [N2: nat] :
( ( dvd_dvd_nat @ N2 @ P2 )
=> ( ( N2 = one_one_nat )
| ( N2 = P2 ) ) )
=> ( factor1801147406995305544me_nat @ P2 ) ) ) ).
% prime_nat_naiveI
thf(fact_746_arith__simps_I5_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% arith_simps(5)
thf(fact_747_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_748_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_749_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_750_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_751_plus__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( plus_p3455044024723400733d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( plus_plus_nat @ M @ N ) ) ) ).
% plus_enat_simps(1)
thf(fact_752_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_753_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_754_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_755_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_756_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X = A ) ) ).
% real_add_minus_iff
thf(fact_757_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
% Helper facts (15)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( if_Extended_enat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( if_Extended_enat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_T,axiom,
! [X: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1] :
( ( if_Num9196306924077011444l_num1 @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_T,axiom,
! [X: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1] :
( ( if_Num9196306924077011444l_num1 @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_T,axiom,
! [X: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1] :
( ( if_Num3220014061592582145l_num1 @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_T,axiom,
! [X: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1] :
( ( if_Num3220014061592582145l_num1 @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_nat @ n @ ( freque8783664969267990145_above @ ( ord_max_nat @ n @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
%------------------------------------------------------------------------------