TPTP Problem File: SLH0875^1.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 : Risk_Free_Lending/0000_Risk_Free_Lending/prob_00749_023086__5863910_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1409 ( 486 unt; 132 typ; 0 def)
% Number of atoms : 3860 (1074 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10641 ( 377 ~; 79 |; 259 &;8232 @)
% ( 0 <=>;1694 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 566 ( 566 >; 0 *; 0 +; 0 <<)
% Number of symbols : 115 ( 112 usr; 12 con; 0-4 aty)
% Number of variables : 3782 ( 353 ^;3290 !; 139 ?;3782 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:57:08.337
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J,type,
set_real_nat_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
set_real_real: $tType ).
thf(ty_n_t__Set__Oset_It__Risk____Free____Lending__Oaccount_J,type,
set_Ri1641125681238393385ccount: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
set_real_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Int__Oint_J_J,type,
set_real_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
set_nat_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mt__Real__Oreal_J_J,type,
set_int_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
set_nat_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
set_int_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_int_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__Risk____Free____Lending__Oaccount,type,
risk_Free_account: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (112)
thf(sy_c_Finite__Set_Ofinite_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
finite8160569183818053390nt_int: set_int_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
finite3115048166472474290nt_nat: set_int_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Int__Oint_Mt__Real__Oreal_J,type,
finite817158274246109198t_real: set_int_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
finite7161215471916998834at_int: set_nat_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite2115694454571419734at_nat: set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
finite7853608736407863218t_real: set_nat_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
finite8285584389518021885t_real: set_real_nat_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Real__Oreal_Mt__Int__Oint_J,type,
finite8924082062276562062al_int: set_real_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
finite3878561044930982962al_nat: set_real_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
finite2300945044744945038l_real: set_real_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Risk____Free____Lending__Oaccount,type,
finite1362240334998357386ccount: set_Ri1641125681238393385ccount > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
finite6197958912794628473et_int: set_set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
gcd_Gcd_int: set_int > int ).
thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
gcd_Gcd_nat: set_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Risk____Free____Lending__Oaccount,type,
minus_4846202936726426316ccount: risk_Free_account > risk_Free_account > risk_Free_account ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Risk____Free____Lending__Oaccount,type,
plus_p1863581527469039996ccount: risk_Free_account > risk_Free_account > risk_Free_account ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Risk____Free____Lending__Oaccount,type,
uminus3377898441596595772ccount: risk_Free_account > risk_Free_account ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Risk____Free____Lending__Oaccount,type,
zero_z1425366712893667068ccount: risk_Free_account ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
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__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_If_001t__Risk____Free____Lending__Oaccount,type,
if_Risk_Free_account: $o > risk_Free_account > risk_Free_account > risk_Free_account ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Lattices__Big_Olinorder_OMin_001t__Int__Oint,type,
lattices_Min_int: ( int > int > $o ) > set_int > int ).
thf(sy_c_Lattices__Big_Olinorder_OMin_001t__Nat__Onat,type,
lattices_Min_nat: ( nat > nat > $o ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder_OMin_001t__Real__Oreal,type,
lattices_Min_real: ( real > real > $o ) > set_real > real ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
lattic8263393255366662781ax_int: set_int > int ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Real__Oreal,type,
lattic4275903605611617917x_real: set_real > real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord_Omin_001t__Int__Oint,type,
min_int: ( int > int > $o ) > int > int > int ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Real__Oreal_J_M_Eo_J,type,
ord_less_nat_real_o: ( ( nat > real ) > $o ) > ( ( nat > real ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_real_o: ( real > $o ) > ( real > $o ) > $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__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Risk____Free____Lending__Oaccount,type,
ord_le2131251472502387783ccount: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
ord_le3527643927072297637t_real: set_nat_real > set_nat_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Real__Oreal_J_M_Eo_J,type,
ord_le7676461544873280788real_o: ( ( nat > real ) > $o ) > ( ( nat > real ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
ord_less_eq_nat_real: ( nat > real ) > ( nat > real ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $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__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Risk____Free____Lending__Oaccount,type,
ord_le4245800335709223507ccount: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
ord_le2908806416726583473t_real: set_nat_real > set_nat_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_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__Real__Oreal,type,
ord_max_real: real > real > real ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Risk____Free____Lending__Oaccount,type,
ord_ma8558070474537249246ccount: risk_Free_account > risk_Free_account > risk_Free_account ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
top_top_set_int: set_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Ordinal__Arithmetic_Ofin__support_001t__Real__Oreal_001t__Nat__Onat,type,
ordina1579063754167848977al_nat: real > set_nat > set_nat_real ).
thf(sy_c_Risk__Free__Lending_Oaccount_OAbs__account,type,
risk_F5458100604530014700ccount: ( nat > real ) > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Oaccount_ORep__account,type,
risk_F170160801229183585ccount: risk_Free_account > nat > real ).
thf(sy_c_Risk__Free__Lending_Ocash__reserve,type,
risk_F1914734008469130493eserve: risk_Free_account > real ).
thf(sy_c_Risk__Free__Lending_Ojust__cash,type,
risk_Free_just_cash: real > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Onet__asset__value,type,
risk_F2906766666041932210_value: risk_Free_account > real ).
thf(sy_c_Risk__Free__Lending_Oshortest__period,type,
risk_F4612863212915232279period: risk_Free_account > nat ).
thf(sy_c_Risk__Free__Lending_Ostrictly__solvent,type,
risk_F1636578016437888323olvent: risk_Free_account > $o ).
thf(sy_c_Risk__Free__Lending_Ovalid__transfer,type,
risk_F1023690899723030139ansfer: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Set_OCollect_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
collect_int_int: ( ( int > int ) > $o ) > set_int_int ).
thf(sy_c_Set_OCollect_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
collect_int_nat: ( ( int > nat ) > $o ) > set_int_nat ).
thf(sy_c_Set_OCollect_001_062_It__Int__Oint_Mt__Real__Oreal_J,type,
collect_int_real: ( ( int > real ) > $o ) > set_int_real ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
collect_nat_int: ( ( nat > int ) > $o ) > set_nat_int ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
collect_nat_real: ( ( nat > real ) > $o ) > set_nat_real ).
thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
collec336724336074912571t_real: ( ( real > nat > real ) > $o ) > set_real_nat_real ).
thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Int__Oint_J,type,
collect_real_int: ( ( real > int ) > $o ) > set_real_int ).
thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
collect_real_nat: ( ( real > nat ) > $o ) > set_real_nat ).
thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
collect_real_real: ( ( real > real ) > $o ) > set_real_real ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Typedef_Otype__definition_001t__Risk____Free____Lending__Oaccount_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
type_d8982087200295354172t_real: ( risk_Free_account > nat > real ) > ( ( nat > real ) > risk_Free_account ) > set_nat_real > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
member_nat_real: ( nat > real ) > set_nat_real > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Risk____Free____Lending__Oaccount,type,
member5612106785598075018ccount: risk_Free_account > set_Ri1641125681238393385ccount > $o ).
thf(sy_v__092_060alpha_062,type,
alpha: risk_Free_account ).
thf(sy_v__092_060beta_062,type,
beta: risk_Free_account ).
% Relevant facts (1267)
thf(fact_0_False,axiom,
~ ! [I: nat] :
( ( risk_F170160801229183585ccount @ ( plus_p1863581527469039996ccount @ alpha @ beta ) @ I )
= zero_zero_real ) ).
% False
thf(fact_1__092_060open_062finite_A_123i_O_A_092_060pi_062_A_I_092_060alpha_062_A_L_A_092_060beta_062_J_Ai_A_092_060noteq_062_A0_125_092_060close_062,axiom,
( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( risk_F170160801229183585ccount @ ( plus_p1863581527469039996ccount @ alpha @ beta ) @ I2 )
!= zero_zero_real ) ) ) ).
% \<open>finite {i. \<pi> (\<alpha> + \<beta>) i \<noteq> 0}\<close>
thf(fact_2_Rep__account__inject,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ( risk_F170160801229183585ccount @ X )
= ( risk_F170160801229183585ccount @ Y ) )
= ( X = Y ) ) ).
% Rep_account_inject
thf(fact_3__092_060open_062_092_060forall_062i_O_A_092_060pi_062_A_I_092_060alpha_062_A_L_A_092_060beta_062_J_Ai_A_092_060noteq_062_A0_A_092_060longrightarrow_062_Ai_A_092_060le_062_Amax_A_Ishortest__period_A_092_060alpha_062_J_A_Ishortest__period_A_092_060beta_062_J_092_060close_062,axiom,
! [I3: nat] :
( ( ( risk_F170160801229183585ccount @ ( plus_p1863581527469039996ccount @ alpha @ beta ) @ I3 )
!= zero_zero_real )
=> ( ord_less_eq_nat @ I3 @ ( ord_max_nat @ ( risk_F4612863212915232279period @ alpha ) @ ( risk_F4612863212915232279period @ beta ) ) ) ) ).
% \<open>\<forall>i. \<pi> (\<alpha> + \<beta>) i \<noteq> 0 \<longrightarrow> i \<le> max (shortest_period \<alpha>) (shortest_period \<beta>)\<close>
thf(fact_4__092_060open_062shortest__period_A_092_060alpha_062_A_092_060le_062_Amax_A_Ishortest__period_A_092_060alpha_062_J_A_Ishortest__period_A_092_060beta_062_J_092_060close_062,axiom,
ord_less_eq_nat @ ( risk_F4612863212915232279period @ alpha ) @ ( ord_max_nat @ ( risk_F4612863212915232279period @ alpha ) @ ( risk_F4612863212915232279period @ beta ) ) ).
% \<open>shortest_period \<alpha> \<le> max (shortest_period \<alpha>) (shortest_period \<beta>)\<close>
thf(fact_5__092_060open_062shortest__period_A_092_060beta_062_A_092_060le_062_Amax_A_Ishortest__period_A_092_060alpha_062_J_A_Ishortest__period_A_092_060beta_062_J_092_060close_062,axiom,
ord_less_eq_nat @ ( risk_F4612863212915232279period @ beta ) @ ( ord_max_nat @ ( risk_F4612863212915232279period @ alpha ) @ ( risk_F4612863212915232279period @ beta ) ) ).
% \<open>shortest_period \<beta> \<le> max (shortest_period \<alpha>) (shortest_period \<beta>)\<close>
thf(fact_6_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_7_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_8_add__le__same__cancel1,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ B @ A ) @ B )
= ( ord_le4245800335709223507ccount @ A @ zero_z1425366712893667068ccount ) ) ).
% add_le_same_cancel1
thf(fact_9_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_10_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_11_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_12_add__le__same__cancel2,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ B )
= ( ord_le4245800335709223507ccount @ A @ zero_z1425366712893667068ccount ) ) ).
% add_le_same_cancel2
thf(fact_13_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_14_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_15_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_16_le__add__same__cancel1,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ ( plus_p1863581527469039996ccount @ A @ B ) )
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ B ) ) ).
% le_add_same_cancel1
thf(fact_17_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_18_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_19_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_20_le__add__same__cancel2,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ ( plus_p1863581527469039996ccount @ B @ A ) )
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ B ) ) ).
% le_add_same_cancel2
thf(fact_21_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_22_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_23_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_24_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_25_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_26_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_27_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_28_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_29_add__right__cancel,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B @ A )
= ( plus_p1863581527469039996ccount @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_30_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_31_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_32_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_33_add__left__cancel,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A @ B )
= ( plus_p1863581527469039996ccount @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_34_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_35_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_36_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_37_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_38_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_39_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_40_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_41_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_42_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_43_add__le__cancel__right,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ C ) )
= ( ord_le4245800335709223507ccount @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_44_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_45_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_46_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_47_add__le__cancel__left,axiom,
! [C: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ C @ A ) @ ( plus_p1863581527469039996ccount @ C @ B ) )
= ( ord_le4245800335709223507ccount @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_48_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_49_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_50_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_51_add__0,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A )
= A ) ).
% add_0
thf(fact_52_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_53_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_54_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_55_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_56_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_57_add__cancel__right__right,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( A
= ( plus_p1863581527469039996ccount @ A @ B ) )
= ( B = zero_z1425366712893667068ccount ) ) ).
% add_cancel_right_right
thf(fact_58_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_59_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_60_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_61_add__cancel__right__left,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( A
= ( plus_p1863581527469039996ccount @ B @ A ) )
= ( B = zero_z1425366712893667068ccount ) ) ).
% add_cancel_right_left
thf(fact_62_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_63_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_64_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_65_add__cancel__left__right,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A @ B )
= A )
= ( B = zero_z1425366712893667068ccount ) ) ).
% add_cancel_left_right
thf(fact_66_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_67_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_68_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_69_add__cancel__left__left,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B @ A )
= A )
= ( B = zero_z1425366712893667068ccount ) ) ).
% add_cancel_left_left
thf(fact_70_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_71_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_72_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_73_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_74_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_75_add_Oright__neutral,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% add.right_neutral
thf(fact_76_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_77_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_78_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_79_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_80_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_81_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_82_Rep__account__zero,axiom,
( ( risk_F170160801229183585ccount @ zero_z1425366712893667068ccount )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% Rep_account_zero
thf(fact_83_Rep__account__plus,axiom,
! [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( plus_p1863581527469039996ccount @ Alpha_1 @ Alpha_2 ) )
= ( ^ [N2: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_1 @ N2 ) @ ( risk_F170160801229183585ccount @ Alpha_2 @ N2 ) ) ) ) ).
% Rep_account_plus
thf(fact_84_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_85_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_86_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_87_trans__le__add2,axiom,
! [I4: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_88_trans__le__add1,axiom,
! [I4: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_89_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_90_mem__Collect__eq,axiom,
! [A: nat > real,P: ( nat > real ) > $o] :
( ( member_nat_real @ A @ ( collect_nat_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_92_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_93_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_94_Collect__mem__eq,axiom,
! [A2: set_nat_real] :
( ( collect_nat_real
@ ^ [X2: nat > real] : ( member_nat_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_95_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_96_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_97_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_98_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_99_add__le__mono1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_100_add__le__mono,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_101_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_102_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_103_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_104_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_105_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_106_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_107_nat__add__max__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_108_nat__add__max__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
= ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_109_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_110_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_111_zero__reorient,axiom,
! [X: risk_Free_account] :
( ( zero_z1425366712893667068ccount = X )
= ( X = zero_z1425366712893667068ccount ) ) ).
% zero_reorient
thf(fact_112_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_113_add__right__imp__eq,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B @ A )
= ( plus_p1863581527469039996ccount @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_114_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_115_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_116_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_117_add__left__imp__eq,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A @ B )
= ( plus_p1863581527469039996ccount @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_118_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_119_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_120_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_121_add_Oleft__commute,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ B @ ( plus_p1863581527469039996ccount @ A @ C ) )
= ( plus_p1863581527469039996ccount @ A @ ( plus_p1863581527469039996ccount @ B @ C ) ) ) ).
% add.left_commute
thf(fact_122_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_123_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_124_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_125_add_Ocommute,axiom,
( plus_p1863581527469039996ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] : ( plus_p1863581527469039996ccount @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_126_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_127_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_128_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_129_add_Oright__cancel,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ B @ A )
= ( plus_p1863581527469039996ccount @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_130_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_131_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_132_add_Oleft__cancel,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ( plus_p1863581527469039996ccount @ A @ B )
= ( plus_p1863581527469039996ccount @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_133_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_134_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_135_add_Oassoc,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ C )
= ( plus_p1863581527469039996ccount @ A @ ( plus_p1863581527469039996ccount @ B @ C ) ) ) ).
% add.assoc
thf(fact_136_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_137_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_138_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_139_group__cancel_Oadd2,axiom,
! [B3: risk_Free_account,K: risk_Free_account,B: risk_Free_account,A: risk_Free_account] :
( ( B3
= ( plus_p1863581527469039996ccount @ K @ B ) )
=> ( ( plus_p1863581527469039996ccount @ A @ B3 )
= ( plus_p1863581527469039996ccount @ K @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_140_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_141_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_142_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_143_group__cancel_Oadd1,axiom,
! [A2: risk_Free_account,K: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( A2
= ( plus_p1863581527469039996ccount @ K @ A ) )
=> ( ( plus_p1863581527469039996ccount @ A2 @ B )
= ( plus_p1863581527469039996ccount @ K @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_144_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_145_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_146_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_147_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I4: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( I4 = J )
& ( K = L ) )
=> ( ( plus_p1863581527469039996ccount @ I4 @ K )
= ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_148_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( I4 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I4 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_149_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( I4 = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I4 @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_150_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( I4 = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I4 @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_151_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ C )
= ( plus_p1863581527469039996ccount @ A @ ( plus_p1863581527469039996ccount @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_152_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_153_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_154_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_155_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_156_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_157_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_158_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_159_le__trans,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I4 @ K ) ) ) ).
% le_trans
thf(fact_160_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_161_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_162_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_163_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_164_add__le__imp__le__right,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ C ) )
=> ( ord_le4245800335709223507ccount @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_165_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_166_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_167_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_168_add__le__imp__le__left,axiom,
! [C: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ C @ A ) @ ( plus_p1863581527469039996ccount @ C @ B ) )
=> ( ord_le4245800335709223507ccount @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_169_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_170_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_171_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_172_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_173_add__right__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ C ) ) ) ).
% add_right_mono
thf(fact_174_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_175_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_176_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_177_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_178_add__left__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ C @ A ) @ ( plus_p1863581527469039996ccount @ C @ B ) ) ) ).
% add_left_mono
thf(fact_179_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_180_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_181_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_182_add__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ C @ D )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ D ) ) ) ) ).
% add_mono
thf(fact_183_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_184_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_185_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I4 @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_186_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le4245800335709223507ccount @ I4 @ J )
& ( ord_le4245800335709223507ccount @ K @ L ) )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ I4 @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_187_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_188_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( I4 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_189_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( I4 = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_190_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( I4 = J )
& ( ord_le4245800335709223507ccount @ K @ L ) )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ I4 @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_191_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( I4 = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_192_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_193_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I4 @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_194_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le4245800335709223507ccount @ I4 @ J )
& ( K = L ) )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ I4 @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_195_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_196_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_197_add_Ogroup__left__neutral,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A )
= A ) ).
% add.group_left_neutral
thf(fact_198_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_199_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_200_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_201_add_Ocomm__neutral,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% add.comm_neutral
thf(fact_202_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_203_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_204_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_205_comm__monoid__add__class_Oadd__0,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_206_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_207_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_208_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_209_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_210_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_211_max__add__distrib__right,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ X @ ( ord_ma8558070474537249246ccount @ Y @ Z ) )
= ( ord_ma8558070474537249246ccount @ ( plus_p1863581527469039996ccount @ X @ Y ) @ ( plus_p1863581527469039996ccount @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_212_max__add__distrib__right,axiom,
! [X: real,Y: real,Z: real] :
( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z ) )
= ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_213_max__add__distrib__right,axiom,
! [X: int,Y: int,Z: int] :
( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
= ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_214_max__add__distrib__right,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_215_max__add__distrib__left,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ ( ord_ma8558070474537249246ccount @ X @ Y ) @ Z )
= ( ord_ma8558070474537249246ccount @ ( plus_p1863581527469039996ccount @ X @ Z ) @ ( plus_p1863581527469039996ccount @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_216_max__add__distrib__left,axiom,
! [X: real,Y: real,Z: real] :
( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z )
= ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_217_max__add__distrib__left,axiom,
! [X: int,Y: int,Z: int] :
( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
= ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_218_max__add__distrib__left,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_219_shortest__period__def,axiom,
( risk_F4612863212915232279period
= ( ^ [Alpha: risk_Free_account] :
( if_nat
@ ! [I2: nat] :
( ( risk_F170160801229183585ccount @ Alpha @ I2 )
= zero_zero_real )
@ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( risk_F170160801229183585ccount @ Alpha @ I2 )
!= zero_zero_real ) ) ) ) ) ) ).
% shortest_period_def
thf(fact_220_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_221_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_222_add__nonpos__eq__0__iff,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ Y @ zero_z1425366712893667068ccount )
=> ( ( ( plus_p1863581527469039996ccount @ X @ Y )
= zero_z1425366712893667068ccount )
= ( ( X = zero_z1425366712893667068ccount )
& ( Y = zero_z1425366712893667068ccount ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_223_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_224_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_225_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_226_add__nonneg__eq__0__iff,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ X )
=> ( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ Y )
=> ( ( ( plus_p1863581527469039996ccount @ X @ Y )
= zero_z1425366712893667068ccount )
= ( ( X = zero_z1425366712893667068ccount )
& ( Y = zero_z1425366712893667068ccount ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_227_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_228_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_229_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_230_add__nonpos__nonpos,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ B @ zero_z1425366712893667068ccount )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ zero_z1425366712893667068ccount ) ) ) ).
% add_nonpos_nonpos
thf(fact_231_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_232_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_233_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_234_add__nonneg__nonneg,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ B )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_235_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_236_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_237_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_238_add__increasing2,axiom,
! [C: risk_Free_account,B: risk_Free_account,A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ C )
=> ( ( ord_le4245800335709223507ccount @ B @ A )
=> ( ord_le4245800335709223507ccount @ B @ ( plus_p1863581527469039996ccount @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_239_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_240_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_241_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_242_add__decreasing2,axiom,
! [C: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ C @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_243_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_244_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_245_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_246_add__increasing,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ord_le4245800335709223507ccount @ B @ ( plus_p1863581527469039996ccount @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_247_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_248_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_249_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_250_add__decreasing,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ C @ B )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_251_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_252_finite__account__support,axiom,
! [Alpha2: risk_Free_account] :
( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( risk_F170160801229183585ccount @ Alpha2 @ I2 )
!= zero_zero_real ) ) ) ).
% finite_account_support
thf(fact_253__092_060open_062_092_060forall_062i_062max_A_Ishortest__period_A_092_060alpha_062_J_A_Ishortest__period_A_092_060beta_062_J_O_A_092_060pi_062_A_I_092_060alpha_062_A_L_A_092_060beta_062_J_Ai_A_061_A0_092_060close_062,axiom,
! [I3: nat] :
( ( ord_less_nat @ ( ord_max_nat @ ( risk_F4612863212915232279period @ alpha ) @ ( risk_F4612863212915232279period @ beta ) ) @ I3 )
=> ( ( risk_F170160801229183585ccount @ ( plus_p1863581527469039996ccount @ alpha @ beta ) @ I3 )
= zero_zero_real ) ) ).
% \<open>\<forall>i>max (shortest_period \<alpha>) (shortest_period \<beta>). \<pi> (\<alpha> + \<beta>) i = 0\<close>
thf(fact_254__092_060open_062_092_060forall_062i_062max_A_Ishortest__period_A_092_060alpha_062_J_A_Ishortest__period_A_092_060beta_062_J_O_A_092_060pi_062_A_092_060beta_062_Ai_A_061_A0_092_060close_062,axiom,
! [I3: nat] :
( ( ord_less_nat @ ( ord_max_nat @ ( risk_F4612863212915232279period @ alpha ) @ ( risk_F4612863212915232279period @ beta ) ) @ I3 )
=> ( ( risk_F170160801229183585ccount @ beta @ I3 )
= zero_zero_real ) ) ).
% \<open>\<forall>i>max (shortest_period \<alpha>) (shortest_period \<beta>). \<pi> \<beta> i = 0\<close>
thf(fact_255__092_060open_062_092_060forall_062i_062max_A_Ishortest__period_A_092_060alpha_062_J_A_Ishortest__period_A_092_060beta_062_J_O_A_092_060pi_062_A_092_060alpha_062_Ai_A_061_A0_092_060close_062,axiom,
! [I3: nat] :
( ( ord_less_nat @ ( ord_max_nat @ ( risk_F4612863212915232279period @ alpha ) @ ( risk_F4612863212915232279period @ beta ) ) @ I3 )
=> ( ( risk_F170160801229183585ccount @ alpha @ I3 )
= zero_zero_real ) ) ).
% \<open>\<forall>i>max (shortest_period \<alpha>) (shortest_period \<beta>). \<pi> \<alpha> i = 0\<close>
thf(fact_256_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_257_max_Obounded__iff,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.bounded_iff
thf(fact_258_max_Obounded__iff,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.bounded_iff
thf(fact_259_max_Obounded__iff,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.bounded_iff
thf(fact_260_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_261_max_Oabsorb2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_max_real @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_262_max_Oabsorb2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_263_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_264_max_Oabsorb1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_max_real @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_265_max_Oabsorb1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_266_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_267_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_268__092_060open_062_092_060forall_062i_062shortest__period_A_092_060beta_062_O_A_092_060pi_062_A_092_060beta_062_Ai_A_061_A0_092_060close_062,axiom,
! [I3: nat] :
( ( ord_less_nat @ ( risk_F4612863212915232279period @ beta ) @ I3 )
=> ( ( risk_F170160801229183585ccount @ beta @ I3 )
= zero_zero_real ) ) ).
% \<open>\<forall>i>shortest_period \<beta>. \<pi> \<beta> i = 0\<close>
thf(fact_269__092_060open_062_092_060forall_062i_062shortest__period_A_092_060alpha_062_O_A_092_060pi_062_A_092_060alpha_062_Ai_A_061_A0_092_060close_062,axiom,
! [I3: nat] :
( ( ord_less_nat @ ( risk_F4612863212915232279period @ alpha ) @ I3 )
=> ( ( risk_F170160801229183585ccount @ alpha @ I3 )
= zero_zero_real ) ) ).
% \<open>\<forall>i>shortest_period \<alpha>. \<pi> \<alpha> i = 0\<close>
thf(fact_270_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_271_finite__Collect__subsets,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [B4: set_int] : ( ord_less_eq_set_int @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_272_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_273_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_274_max_Oidem,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ A )
= A ) ).
% max.idem
thf(fact_275_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_276_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_277_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_278_finite__Collect__conjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_279_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_280_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_281_add__less__cancel__left,axiom,
! [C: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ C @ A ) @ ( plus_p1863581527469039996ccount @ C @ B ) )
= ( ord_le2131251472502387783ccount @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_282_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_283_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_284_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_285_add__less__cancel__right,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ C ) )
= ( ord_le2131251472502387783ccount @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_286_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_287_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_288_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_289_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_290_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_291_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_292_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_293_max_Oabsorb3,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_max_real @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_294_max_Oabsorb3,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_295_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_296_max_Oabsorb4,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_max_real @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_297_max_Oabsorb4,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_298_max__less__iff__conj,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 ) ) ) ).
% max_less_iff_conj
thf(fact_299_max__less__iff__conj,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 ) ) ) ).
% max_less_iff_conj
thf(fact_300_max__less__iff__conj,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 ) ) ) ).
% max_less_iff_conj
thf(fact_301_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_302_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_303_add__less__same__cancel1,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ B @ A ) @ B )
= ( ord_le2131251472502387783ccount @ A @ zero_z1425366712893667068ccount ) ) ).
% add_less_same_cancel1
thf(fact_304_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_305_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_306_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_307_add__less__same__cancel2,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ B )
= ( ord_le2131251472502387783ccount @ A @ zero_z1425366712893667068ccount ) ) ).
% add_less_same_cancel2
thf(fact_308_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_309_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_310_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_311_less__add__same__cancel1,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ ( plus_p1863581527469039996ccount @ A @ B ) )
= ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ B ) ) ).
% less_add_same_cancel1
thf(fact_312_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_313_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_314_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_315_less__add__same__cancel2,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ ( plus_p1863581527469039996ccount @ B @ A ) )
= ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ B ) ) ).
% less_add_same_cancel2
thf(fact_316_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_317_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_318_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_319_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_320_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_321_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_322_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_323_add__lessD1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J ) @ K )
=> ( ord_less_nat @ I4 @ K ) ) ).
% add_lessD1
thf(fact_324_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_325_add__less__mono,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_326_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_327_not__add__less1,axiom,
! [I4: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I4 @ J ) @ I4 ) ).
% not_add_less1
thf(fact_328_not__add__less2,axiom,
! [J: nat,I4: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I4 ) @ I4 ) ).
% not_add_less2
thf(fact_329_add__less__mono1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_330_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_331_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_332_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_333_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_334_trans__less__add1,axiom,
! [I4: nat,J: nat,M: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_335_trans__less__add2,axiom,
! [I4: nat,J: nat,M: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_336_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_337_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_338_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_339_less__max__iff__disj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z @ X )
| ( ord_less_nat @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_340_less__max__iff__disj,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y ) )
= ( ( ord_less_real @ Z @ X )
| ( ord_less_real @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_341_less__max__iff__disj,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
= ( ( ord_less_int @ Z @ X )
| ( ord_less_int @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_342_max_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_343_max_Ostrict__boundedE,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
=> ~ ( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_344_max_Ostrict__boundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
=> ~ ( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_345_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( A3
= ( ord_max_nat @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% max.strict_order_iff
thf(fact_346_max_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( A3
= ( ord_max_real @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% max.strict_order_iff
thf(fact_347_max_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( A3
= ( ord_max_int @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% max.strict_order_iff
thf(fact_348_max_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_349_max_Ostrict__coboundedI1,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ A )
=> ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_350_max_Ostrict__coboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ A )
=> ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_351_max_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_352_max_Ostrict__coboundedI2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_353_max_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_354_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_355_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_356_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_357_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_358_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_359_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le2131251472502387783ccount @ I4 @ J )
& ( ord_le2131251472502387783ccount @ K @ L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I4 @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_360_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I4 @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_361_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I4 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_362_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( I4 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_363_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( I4 = J )
& ( ord_le2131251472502387783ccount @ K @ L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I4 @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_364_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( I4 = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_365_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( I4 = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_366_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_367_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le2131251472502387783ccount @ I4 @ J )
& ( K = L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I4 @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_368_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I4 @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_369_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I4 @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_370_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_371_add__strict__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ C @ D )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_372_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_373_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_374_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_375_add__strict__left__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ C @ A ) @ ( plus_p1863581527469039996ccount @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_376_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_377_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_378_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_379_add__strict__right__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_380_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_381_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_382_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_383_add__less__imp__less__left,axiom,
! [C: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ C @ A ) @ ( plus_p1863581527469039996ccount @ C @ B ) )
=> ( ord_le2131251472502387783ccount @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_384_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_385_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_386_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_387_add__less__imp__less__right,axiom,
! [A: risk_Free_account,C: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ C ) )
=> ( ord_le2131251472502387783ccount @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_388_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_389_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_390_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_391_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_392_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_393_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_394_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_395_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_396_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_397_less__imp__add__positive,axiom,
! [I4: nat,J: nat] :
( ( ord_less_nat @ I4 @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I4 @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_398_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_399_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_400_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_401_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_402_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_403_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I4: nat,J: nat] :
( ! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_404_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_405_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_406_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I4 @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_407_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le4245800335709223507ccount @ I4 @ J )
& ( ord_le2131251472502387783ccount @ K @ L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I4 @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_408_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_409_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_410_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I4 @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_411_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le2131251472502387783ccount @ I4 @ J )
& ( ord_le4245800335709223507ccount @ K @ L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I4 @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_412_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I4 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_413_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_414_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_415_add__le__less__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ C @ D )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_416_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_417_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_418_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_419_add__less__le__mono,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ C @ D )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ C ) @ ( plus_p1863581527469039996ccount @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_420_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_421_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_422_add__neg__neg,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ zero_z1425366712893667068ccount )
=> ( ( ord_le2131251472502387783ccount @ B @ zero_z1425366712893667068ccount )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ zero_z1425366712893667068ccount ) ) ) ).
% add_neg_neg
thf(fact_423_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_424_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_425_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_426_add__pos__pos,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ B )
=> ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_427_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_428_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_429_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_430_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_431_pos__add__strict,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ B @ ( plus_p1863581527469039996ccount @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_432_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_433_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_434_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_435_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_436_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_437_add__neg__nonpos,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ zero_z1425366712893667068ccount )
=> ( ( ord_le4245800335709223507ccount @ B @ zero_z1425366712893667068ccount )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ zero_z1425366712893667068ccount ) ) ) ).
% add_neg_nonpos
thf(fact_438_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_439_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_440_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_441_add__nonneg__pos,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ B )
=> ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_442_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_443_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_444_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_445_add__nonpos__neg,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ zero_z1425366712893667068ccount )
=> ( ( ord_le2131251472502387783ccount @ B @ zero_z1425366712893667068ccount )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ A @ B ) @ zero_z1425366712893667068ccount ) ) ) ).
% add_nonpos_neg
thf(fact_446_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_447_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_448_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_449_add__pos__nonneg,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ B )
=> ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_450_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_451_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_452_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_453_add__strict__increasing,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ B @ ( plus_p1863581527469039996ccount @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_454_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_455_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_456_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_457_add__strict__increasing2,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ B @ ( plus_p1863581527469039996ccount @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_458_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_459_rev__finite__subset,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_460_rev__finite__subset,axiom,
! [B3: set_int,A2: set_int] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( finite_finite_int @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_461_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_462_infinite__super,axiom,
! [S2: set_int,T2: set_int] :
( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ~ ( finite_finite_int @ S2 )
=> ~ ( finite_finite_int @ T2 ) ) ) ).
% infinite_super
thf(fact_463_finite__subset,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_464_finite__subset,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( finite_finite_int @ B3 )
=> ( finite_finite_int @ A2 ) ) ) ).
% finite_subset
thf(fact_465_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_466_max_Ocommute,axiom,
( ord_max_nat
= ( ^ [A3: nat,B2: nat] : ( ord_max_nat @ B2 @ A3 ) ) ) ).
% max.commute
thf(fact_467_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_468_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_469_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_1: int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_470_pigeonhole__infinite__rel,axiom,
! [A2: set_real,B3: set_nat,R: real > nat > $o] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B3 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A3: real] :
( ( member_real @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_471_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_real,B3: set_nat,R: ( nat > real ) > nat > $o] :
( ~ ( finite7853608736407863218t_real @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B3 )
& ~ ( finite7853608736407863218t_real
@ ( collect_nat_real
@ ^ [A3: nat > real] :
( ( member_nat_real @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_472_pigeonhole__infinite__rel,axiom,
! [A2: set_real,B3: set_int,R: real > int > $o] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B3 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A3: real] :
( ( member_real @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_473_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_real,B3: set_int,R: ( nat > real ) > int > $o] :
( ~ ( finite7853608736407863218t_real @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B3 )
& ~ ( finite7853608736407863218t_real
@ ( collect_nat_real
@ ^ [A3: nat > real] :
( ( member_nat_real @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_474_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B3: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_475_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B3: set_int,R: nat > int > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_476_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B3: set_nat,R: int > nat > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B3 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_477_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B3: set_int,R: int > int > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B3 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_478_finite__has__maximal2,axiom,
! [A2: set_nat_real,A: nat > real] :
( ( finite7853608736407863218t_real @ A2 )
=> ( ( member_nat_real @ A @ A2 )
=> ? [X3: nat > real] :
( ( member_nat_real @ X3 @ A2 )
& ( ord_less_eq_nat_real @ A @ X3 )
& ! [Xa: nat > real] :
( ( member_nat_real @ Xa @ A2 )
=> ( ( ord_less_eq_nat_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_479_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_480_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ A @ X3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_481_finite__has__maximal2,axiom,
! [A2: set_Ri1641125681238393385ccount,A: risk_Free_account] :
( ( finite1362240334998357386ccount @ A2 )
=> ( ( member5612106785598075018ccount @ A @ A2 )
=> ? [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A2 )
& ( ord_le4245800335709223507ccount @ A @ X3 )
& ! [Xa: risk_Free_account] :
( ( member5612106785598075018ccount @ Xa @ A2 )
=> ( ( ord_le4245800335709223507ccount @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_482_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ A @ X3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_483_finite__has__minimal2,axiom,
! [A2: set_nat_real,A: nat > real] :
( ( finite7853608736407863218t_real @ A2 )
=> ( ( member_nat_real @ A @ A2 )
=> ? [X3: nat > real] :
( ( member_nat_real @ X3 @ A2 )
& ( ord_less_eq_nat_real @ X3 @ A )
& ! [Xa: nat > real] :
( ( member_nat_real @ Xa @ A2 )
=> ( ( ord_less_eq_nat_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_484_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_485_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_486_finite__has__minimal2,axiom,
! [A2: set_Ri1641125681238393385ccount,A: risk_Free_account] :
( ( finite1362240334998357386ccount @ A2 )
=> ( ( member5612106785598075018ccount @ A @ A2 )
=> ? [X3: risk_Free_account] :
( ( member5612106785598075018ccount @ X3 @ A2 )
& ( ord_le4245800335709223507ccount @ X3 @ A )
& ! [Xa: risk_Free_account] :
( ( member5612106785598075018ccount @ Xa @ A2 )
=> ( ( ord_le4245800335709223507ccount @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_487_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ X3 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_488_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_489_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_490_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_491_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_492_max_OorderE,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( A
= ( ord_max_real @ A @ B ) ) ) ).
% max.orderE
thf(fact_493_max_OorderE,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( A
= ( ord_max_int @ A @ B ) ) ) ).
% max.orderE
thf(fact_494_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_495_max_OorderI,axiom,
! [A: real,B: real] :
( ( A
= ( ord_max_real @ A @ B ) )
=> ( ord_less_eq_real @ B @ A ) ) ).
% max.orderI
thf(fact_496_max_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( ord_max_int @ A @ B ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
% max.orderI
thf(fact_497_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_498_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_499_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_500_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_501_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_502_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_503_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( A3
= ( ord_max_nat @ A3 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_504_max_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( A3
= ( ord_max_real @ A3 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_505_max_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( A3
= ( ord_max_int @ A3 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_506_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_507_max_Ocobounded1,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ A @ ( ord_max_real @ A @ B ) ) ).
% max.cobounded1
thf(fact_508_max_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded1
thf(fact_509_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_510_max_Ocobounded2,axiom,
! [B: real,A: real] : ( ord_less_eq_real @ B @ ( ord_max_real @ A @ B ) ) ).
% max.cobounded2
thf(fact_511_max_Ocobounded2,axiom,
! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded2
thf(fact_512_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_513_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_514_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_515_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_max_nat @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb_iff1
thf(fact_516_max_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( ( ord_max_real @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb_iff1
thf(fact_517_max_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( ord_max_int @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb_iff1
thf(fact_518_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_max_nat @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_519_max_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( ( ord_max_real @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_520_max_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( ord_max_int @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_521_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_522_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_523_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_524_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_525_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_526_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_527_finite__set__of__finite__funs,axiom,
! [A2: set_real,B3: set_real,D: real] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_real @ B3 )
=> ( finite2300945044744945038l_real
@ ( collect_real_real
@ ^ [F2: real > real] :
! [X2: real] :
( ( ( member_real @ X2 @ A2 )
=> ( member_real @ ( F2 @ X2 ) @ B3 ) )
& ( ~ ( member_real @ X2 @ A2 )
=> ( ( F2 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_528_finite__set__of__finite__funs,axiom,
! [A2: set_real,B3: set_nat,D: nat] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite3878561044930982962al_nat
@ ( collect_real_nat
@ ^ [F2: real > nat] :
! [X2: real] :
( ( ( member_real @ X2 @ A2 )
=> ( member_nat @ ( F2 @ X2 ) @ B3 ) )
& ( ~ ( member_real @ X2 @ A2 )
=> ( ( F2 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_529_finite__set__of__finite__funs,axiom,
! [A2: set_real,B3: set_int,D: int] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( finite8924082062276562062al_int
@ ( collect_real_int
@ ^ [F2: real > int] :
! [X2: real] :
( ( ( member_real @ X2 @ A2 )
=> ( member_int @ ( F2 @ X2 ) @ B3 ) )
& ( ~ ( member_real @ X2 @ A2 )
=> ( ( F2 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_530_finite__set__of__finite__funs,axiom,
! [A2: set_nat,B3: set_real,D: real] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_real @ B3 )
=> ( finite7853608736407863218t_real
@ ( collect_nat_real
@ ^ [F2: nat > real] :
! [X2: nat] :
( ( ( member_nat @ X2 @ A2 )
=> ( member_real @ ( F2 @ X2 ) @ B3 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( F2 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_531_finite__set__of__finite__funs,axiom,
! [A2: set_nat,B3: set_nat,D: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [F2: nat > nat] :
! [X2: nat] :
( ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F2 @ X2 ) @ B3 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( F2 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_532_finite__set__of__finite__funs,axiom,
! [A2: set_nat,B3: set_int,D: int] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( finite7161215471916998834at_int
@ ( collect_nat_int
@ ^ [F2: nat > int] :
! [X2: nat] :
( ( ( member_nat @ X2 @ A2 )
=> ( member_int @ ( F2 @ X2 ) @ B3 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( F2 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_533_finite__set__of__finite__funs,axiom,
! [A2: set_int,B3: set_real,D: real] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_real @ B3 )
=> ( finite817158274246109198t_real
@ ( collect_int_real
@ ^ [F2: int > real] :
! [X2: int] :
( ( ( member_int @ X2 @ A2 )
=> ( member_real @ ( F2 @ X2 ) @ B3 ) )
& ( ~ ( member_int @ X2 @ A2 )
=> ( ( F2 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_534_finite__set__of__finite__funs,axiom,
! [A2: set_int,B3: set_nat,D: nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite3115048166472474290nt_nat
@ ( collect_int_nat
@ ^ [F2: int > nat] :
! [X2: int] :
( ( ( member_int @ X2 @ A2 )
=> ( member_nat @ ( F2 @ X2 ) @ B3 ) )
& ( ~ ( member_int @ X2 @ A2 )
=> ( ( F2 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_535_finite__set__of__finite__funs,axiom,
! [A2: set_int,B3: set_int,D: int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( finite8160569183818053390nt_int
@ ( collect_int_int
@ ^ [F2: int > int] :
! [X2: int] :
( ( ( member_int @ X2 @ A2 )
=> ( member_int @ ( F2 @ X2 ) @ B3 ) )
& ( ~ ( member_int @ X2 @ A2 )
=> ( ( F2 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_536_finite__set__of__finite__funs,axiom,
! [A2: set_real,B3: set_nat_real,D: nat > real] :
( ( finite_finite_real @ A2 )
=> ( ( finite7853608736407863218t_real @ B3 )
=> ( finite8285584389518021885t_real
@ ( collec336724336074912571t_real
@ ^ [F2: real > nat > real] :
! [X2: real] :
( ( ( member_real @ X2 @ A2 )
=> ( member_nat_real @ ( F2 @ X2 ) @ B3 ) )
& ( ~ ( member_real @ X2 @ A2 )
=> ( ( F2 @ X2 )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_537_field__le__epsilon,axiom,
! [X: real,Y: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_538_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > real,Y: real > real] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I2: real] :
( ( member_real @ I2 @ I5 )
& ( ( X @ I2 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I2: real] :
( ( member_real @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I2: real] :
( ( member_real @ I2 @ I5 )
& ( ( plus_plus_real @ ( X @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_539_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > real,Y: nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( X @ I2 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( plus_plus_real @ ( X @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_540_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > real,Y: int > real] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( X @ I2 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( plus_plus_real @ ( X @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_541_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > nat,Y: real > nat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I2: real] :
( ( member_real @ I2 @ I5 )
& ( ( X @ I2 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I2: real] :
( ( member_real @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I2: real] :
( ( member_real @ I2 @ I5 )
& ( ( plus_plus_nat @ ( X @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_542_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( X @ I2 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( plus_plus_nat @ ( X @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_543_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > nat,Y: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( X @ I2 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( plus_plus_nat @ ( X @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_544_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > risk_Free_account,Y: real > risk_Free_account] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I2: real] :
( ( member_real @ I2 @ I5 )
& ( ( X @ I2 )
!= zero_z1425366712893667068ccount ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I2: real] :
( ( member_real @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_z1425366712893667068ccount ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I2: real] :
( ( member_real @ I2 @ I5 )
& ( ( plus_p1863581527469039996ccount @ ( X @ I2 ) @ ( Y @ I2 ) )
!= zero_z1425366712893667068ccount ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_545_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > risk_Free_account,Y: nat > risk_Free_account] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( X @ I2 )
!= zero_z1425366712893667068ccount ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_z1425366712893667068ccount ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( plus_p1863581527469039996ccount @ ( X @ I2 ) @ ( Y @ I2 ) )
!= zero_z1425366712893667068ccount ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_546_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > risk_Free_account,Y: int > risk_Free_account] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( X @ I2 )
!= zero_z1425366712893667068ccount ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_z1425366712893667068ccount ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( plus_p1863581527469039996ccount @ ( X @ I2 ) @ ( Y @ I2 ) )
!= zero_z1425366712893667068ccount ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_547_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > int,Y: real > int] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I2: real] :
( ( member_real @ I2 @ I5 )
& ( ( X @ I2 )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I2: real] :
( ( member_real @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_int ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I2: real] :
( ( member_real @ I2 @ I5 )
& ( ( plus_plus_int @ ( X @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_548_Max_Oin__idem,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ( ord_max_real @ X @ ( lattic4275903605611617917x_real @ A2 ) )
= ( lattic4275903605611617917x_real @ A2 ) ) ) ) ).
% Max.in_idem
thf(fact_549_Max_Oin__idem,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ X @ A2 )
=> ( ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ A2 ) )
= ( lattic8263393255366662781ax_int @ A2 ) ) ) ) ).
% Max.in_idem
thf(fact_550_Max_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) )
= ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ).
% Max.in_idem
thf(fact_551_Max__ge,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ord_less_eq_real @ X @ ( lattic4275903605611617917x_real @ A2 ) ) ) ) ).
% Max_ge
thf(fact_552_Max__ge,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ X @ A2 )
=> ( ord_less_eq_int @ X @ ( lattic8263393255366662781ax_int @ A2 ) ) ) ) ).
% Max_ge
thf(fact_553_Max__ge,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ).
% Max_ge
thf(fact_554_Max__eqI,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ! [Y2: real] :
( ( member_real @ Y2 @ A2 )
=> ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( ( lattic4275903605611617917x_real @ A2 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_555_Max__eqI,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ A2 )
=> ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( member_int @ X @ A2 )
=> ( ( lattic8263393255366662781ax_int @ A2 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_556_Max__eqI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_557_Max__eq__if,axiom,
! [A2: set_real,B3: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_real @ B3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [Xa: real] :
( ( member_real @ Xa @ B3 )
& ( ord_less_eq_real @ X3 @ Xa ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B3 )
=> ? [Xa: real] :
( ( member_real @ Xa @ A2 )
& ( ord_less_eq_real @ X3 @ Xa ) ) )
=> ( ( lattic4275903605611617917x_real @ A2 )
= ( lattic4275903605611617917x_real @ B3 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_558_Max__eq__if,axiom,
! [A2: set_int,B3: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B3 )
& ( ord_less_eq_int @ X3 @ Xa ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ A2 )
& ( ord_less_eq_int @ X3 @ Xa ) ) )
=> ( ( lattic8263393255366662781ax_int @ A2 )
= ( lattic8263393255366662781ax_int @ B3 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_559_Max__eq__if,axiom,
! [A2: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( ord_less_eq_nat @ X3 @ Xa ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ A2 )
& ( ord_less_eq_nat @ X3 @ Xa ) ) )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= ( lattic8265883725875713057ax_nat @ B3 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_560_Max_OcoboundedI,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ( ord_less_eq_real @ A @ ( lattic4275903605611617917x_real @ A2 ) ) ) ) ).
% Max.coboundedI
thf(fact_561_Max_OcoboundedI,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ( ord_less_eq_int @ A @ ( lattic8263393255366662781ax_int @ A2 ) ) ) ) ).
% Max.coboundedI
thf(fact_562_Max_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ A @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ).
% Max.coboundedI
thf(fact_563_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_564_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_565_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [B5: set_nat] :
( ( ord_less_set_nat @ B5 @ A4 )
=> ( P @ B5 ) )
=> ( P @ A4 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_566_finite__psubset__induct,axiom,
! [A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ A2 )
=> ( ! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ! [B5: set_int] :
( ( ord_less_set_int @ B5 @ A4 )
=> ( P @ B5 ) )
=> ( P @ A4 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_567_linordered__field__no__lb,axiom,
! [X4: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X4 ) ).
% linordered_field_no_lb
thf(fact_568_linordered__field__no__ub,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_569_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_570_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_571_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_572_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I4: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I4 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_573_subsetI,axiom,
! [A2: set_real,B3: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ X3 @ B3 ) )
=> ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% subsetI
thf(fact_574_subsetI,axiom,
! [A2: set_nat_real,B3: set_nat_real] :
( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A2 )
=> ( member_nat_real @ X3 @ B3 ) )
=> ( ord_le2908806416726583473t_real @ A2 @ B3 ) ) ).
% subsetI
thf(fact_575_max__def__raw,axiom,
( ord_max_nat
= ( ^ [A3: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% max_def_raw
thf(fact_576_max__def__raw,axiom,
( ord_max_real
= ( ^ [A3: real,B2: real] : ( if_real @ ( ord_less_eq_real @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% max_def_raw
thf(fact_577_max__def__raw,axiom,
( ord_ma8558070474537249246ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] : ( if_Risk_Free_account @ ( ord_le4245800335709223507ccount @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% max_def_raw
thf(fact_578_max__def__raw,axiom,
( ord_max_int
= ( ^ [A3: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% max_def_raw
thf(fact_579_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_580_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_581_dual__order_Orefl,axiom,
! [A: risk_Free_account] : ( ord_le4245800335709223507ccount @ A @ A ) ).
% dual_order.refl
thf(fact_582_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_583_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_584_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_585_order__refl,axiom,
! [X: risk_Free_account] : ( ord_le4245800335709223507ccount @ X @ X ) ).
% order_refl
thf(fact_586_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_587_less__set__def,axiom,
( ord_less_set_real
= ( ^ [A5: set_real,B4: set_real] :
( ord_less_real_o
@ ^ [X2: real] : ( member_real @ X2 @ A5 )
@ ^ [X2: real] : ( member_real @ X2 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_588_less__set__def,axiom,
( ord_le3527643927072297637t_real
= ( ^ [A5: set_nat_real,B4: set_nat_real] :
( ord_less_nat_real_o
@ ^ [X2: nat > real] : ( member_nat_real @ X2 @ A5 )
@ ^ [X2: nat > real] : ( member_nat_real @ X2 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_589_psubsetD,axiom,
! [A2: set_real,B3: set_real,C: real] :
( ( ord_less_set_real @ A2 @ B3 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_590_psubsetD,axiom,
! [A2: set_nat_real,B3: set_nat_real,C: nat > real] :
( ( ord_le3527643927072297637t_real @ A2 @ B3 )
=> ( ( member_nat_real @ C @ A2 )
=> ( member_nat_real @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_591_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_592_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_593_verit__comp__simplify1_I2_J,axiom,
! [A: risk_Free_account] : ( ord_le4245800335709223507ccount @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_594_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_595_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_596_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_597_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_598_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_599_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_600_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_601_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_602_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_603_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: risk_Free_account,Z2: risk_Free_account] : ( Y4 = Z2 ) )
= ( ^ [X2: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y5 )
& ( ord_le4245800335709223507ccount @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_604_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_605_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_606_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_607_ord__eq__le__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( A = B )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ord_le4245800335709223507ccount @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_608_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_609_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_610_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_611_ord__le__eq__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( B = C )
=> ( ord_le4245800335709223507ccount @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_612_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_613_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_614_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_615_order__antisym,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_616_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_617_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_618_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_619_order_Otrans,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ord_le4245800335709223507ccount @ A @ C ) ) ) ).
% order.trans
thf(fact_620_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_621_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_622_order__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_623_order__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ Z )
=> ( ord_le4245800335709223507ccount @ X @ Z ) ) ) ).
% order_trans
thf(fact_624_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_625_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_626_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A6: real,B6: real] :
( ( ord_less_eq_real @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: real,B6: real] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_627_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A6: int,B6: int] :
( ( ord_less_eq_int @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: int,B6: int] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_628_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_629_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_630_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: risk_Free_account,Z2: risk_Free_account] : ( Y4 = Z2 ) )
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A3 )
& ( ord_le4245800335709223507ccount @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_631_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_632_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_633_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_634_dual__order_Oantisym,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A )
=> ( ( ord_le4245800335709223507ccount @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_635_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_636_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_637_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_638_dual__order_Otrans,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A )
=> ( ( ord_le4245800335709223507ccount @ C @ B )
=> ( ord_le4245800335709223507ccount @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_639_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_640_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_641_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_642_antisym,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_643_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_644_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_645_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_646_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: risk_Free_account,Z2: risk_Free_account] : ( Y4 = Z2 ) )
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B2 )
& ( ord_le4245800335709223507ccount @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_647_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_648_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_649_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_650_order__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_651_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_652_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_653_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_654_order__subst1,axiom,
! [A: real,F: risk_Free_account > real,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_655_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_656_order__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_657_order__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B: real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_658_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_659_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_660_order__subst2,axiom,
! [A: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_661_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_662_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_663_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_664_order__subst2,axiom,
! [A: real,B: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_665_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_666_order__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_667_order__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_668_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_669_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_670_order__eq__refl,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( X = Y )
=> ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% order_eq_refl
thf(fact_671_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_672_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_673_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_674_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_675_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_676_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_677_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_678_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_679_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_680_ord__eq__le__subst,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_681_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_682_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_683_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_684_ord__eq__le__subst,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_685_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_686_ord__eq__le__subst,axiom,
! [A: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_687_ord__eq__le__subst,axiom,
! [A: real,F: risk_Free_account > real,B: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_688_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_689_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_690_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_691_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_692_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_693_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_694_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_695_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_696_ord__le__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_697_ord__le__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_698_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_699_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_700_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_701_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_702_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_703_order__antisym__conv,axiom,
! [Y: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y @ X )
=> ( ( ord_le4245800335709223507ccount @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_704_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_705_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_706_order__less__imp__not__less,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ~ ( ord_le2131251472502387783ccount @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_707_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_708_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_709_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_710_order__less__imp__not__eq2,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_711_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_712_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_713_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_714_order__less__imp__not__eq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_715_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_716_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_717_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_718_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_719_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_720_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_721_order__less__imp__triv,axiom,
! [X: risk_Free_account,Y: risk_Free_account,P: $o] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_722_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_723_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_724_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_725_order__less__not__sym,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ~ ( ord_le2131251472502387783ccount @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_726_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_727_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_728_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_729_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_730_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_731_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_732_order__less__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_733_order__less__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_734_order__less__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_735_order__less__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > int,C: int] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_736_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_737_order__less__subst2,axiom,
! [A: real,B: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_738_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_739_order__less__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_740_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_741_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_742_order__less__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_743_order__less__subst1,axiom,
! [A: risk_Free_account,F: risk_Free_account > risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_744_order__less__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B: real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_745_order__less__subst1,axiom,
! [A: risk_Free_account,F: int > risk_Free_account,B: int,C: int] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_746_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_747_order__less__subst1,axiom,
! [A: real,F: risk_Free_account > real,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_748_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_749_order__less__irrefl,axiom,
! [X: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ X @ X ) ).
% order_less_irrefl
thf(fact_750_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_751_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_752_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_753_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_754_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_755_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_756_ord__less__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_757_ord__less__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_758_ord__less__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_759_ord__less__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > int,C: int] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_760_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_761_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_762_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_763_ord__eq__less__subst,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_764_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_765_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_766_ord__eq__less__subst,axiom,
! [A: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_767_ord__eq__less__subst,axiom,
! [A: risk_Free_account,F: risk_Free_account > risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_768_ord__eq__less__subst,axiom,
! [A: real,F: risk_Free_account > real,B: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_769_ord__eq__less__subst,axiom,
! [A: int,F: risk_Free_account > int,B: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_770_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_771_ord__eq__less__subst,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_772_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_773_order__less__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ Z )
=> ( ord_le2131251472502387783ccount @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_774_order__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_775_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_776_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_777_order__less__asym_H,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ~ ( ord_le2131251472502387783ccount @ B @ A ) ) ).
% order_less_asym'
thf(fact_778_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_779_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_780_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_781_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_782_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_783_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_784_order__less__asym,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ~ ( ord_le2131251472502387783ccount @ Y @ X ) ) ).
% order_less_asym
thf(fact_785_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_786_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_787_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_788_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_789_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_790_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_791_dual__order_Ostrict__implies__not__eq,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_792_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_793_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_794_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_795_order_Ostrict__implies__not__eq,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_796_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_797_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_798_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_799_dual__order_Ostrict__trans,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A )
=> ( ( ord_le2131251472502387783ccount @ C @ B )
=> ( ord_le2131251472502387783ccount @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_800_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_801_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_802_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_803_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_804_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_805_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_806_order_Ostrict__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_807_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_808_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_809_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_810_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A6: real,B6: real] :
( ( ord_less_real @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: real] : ( P @ A6 @ A6 )
=> ( ! [A6: real,B6: real] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_811_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A6: int,B6: int] :
( ( ord_less_int @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: int] : ( P @ A6 @ A6 )
=> ( ! [A6: int,B6: int] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_812_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_813_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_814_dual__order_Oirrefl,axiom,
! [A: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A @ A ) ).
% dual_order.irrefl
thf(fact_815_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_816_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_817_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_818_dual__order_Oasym,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A )
=> ~ ( ord_le2131251472502387783ccount @ A @ B ) ) ).
% dual_order.asym
thf(fact_819_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_820_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_821_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_822_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_823_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_824_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_825_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_826_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_827_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X3 )
=> ( P @ Y3 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_828_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_829_ord__less__eq__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( B = C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_830_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_831_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_832_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_833_ord__eq__less__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( A = B )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_834_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_835_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_836_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_837_order_Oasym,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ~ ( ord_le2131251472502387783ccount @ B @ A ) ) ).
% order.asym
thf(fact_838_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_839_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_840_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_841_less__imp__neq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_842_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_843_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_844_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_845_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_846_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_847_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_848_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_849_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_850_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_851_verit__comp__simplify1_I1_J,axiom,
! [A: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_852_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_853_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_854_in__mono,axiom,
! [A2: set_real,B3: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B3 ) ) ) ).
% in_mono
thf(fact_855_in__mono,axiom,
! [A2: set_nat_real,B3: set_nat_real,X: nat > real] :
( ( ord_le2908806416726583473t_real @ A2 @ B3 )
=> ( ( member_nat_real @ X @ A2 )
=> ( member_nat_real @ X @ B3 ) ) ) ).
% in_mono
thf(fact_856_subsetD,axiom,
! [A2: set_real,B3: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B3 ) ) ) ).
% subsetD
thf(fact_857_subsetD,axiom,
! [A2: set_nat_real,B3: set_nat_real,C: nat > real] :
( ( ord_le2908806416726583473t_real @ A2 @ B3 )
=> ( ( member_nat_real @ C @ A2 )
=> ( member_nat_real @ C @ B3 ) ) ) ).
% subsetD
thf(fact_858_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B4: set_real] :
! [X2: real] :
( ( member_real @ X2 @ A5 )
=> ( member_real @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_859_subset__eq,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A5: set_nat_real,B4: set_nat_real] :
! [X2: nat > real] :
( ( member_nat_real @ X2 @ A5 )
=> ( member_nat_real @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_860_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B4: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A5 )
=> ( member_real @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_861_subset__iff,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A5: set_nat_real,B4: set_nat_real] :
! [T3: nat > real] :
( ( member_nat_real @ T3 @ A5 )
=> ( member_nat_real @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_862_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_863_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_864_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_865_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X2: int] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_866_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_867_Collect__subset,axiom,
! [A2: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_868_Collect__subset,axiom,
! [A2: set_nat_real,P: ( nat > real ) > $o] :
( ord_le2908806416726583473t_real
@ ( collect_nat_real
@ ^ [X2: nat > real] :
( ( member_nat_real @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_869_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_870_Collect__subset,axiom,
! [A2: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_871_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B4: set_real] :
( ord_less_eq_real_o
@ ^ [X2: real] : ( member_real @ X2 @ A5 )
@ ^ [X2: real] : ( member_real @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_872_less__eq__set__def,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A5: set_nat_real,B4: set_nat_real] :
( ord_le7676461544873280788real_o
@ ^ [X2: nat > real] : ( member_nat_real @ X2 @ A5 )
@ ^ [X2: nat > real] : ( member_nat_real @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_873_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_874_verit__comp__simplify1_I3_J,axiom,
! [B7: real,A7: real] :
( ( ~ ( ord_less_eq_real @ B7 @ A7 ) )
= ( ord_less_real @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_875_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A7: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A7 ) )
= ( ord_less_int @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_876_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_877_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_878_leD,axiom,
! [Y: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y @ X )
=> ~ ( ord_le2131251472502387783ccount @ X @ Y ) ) ).
% leD
thf(fact_879_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_880_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_881_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_882_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_883_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_884_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_885_nless__le,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ~ ( ord_le2131251472502387783ccount @ A @ B ) )
= ( ~ ( ord_le4245800335709223507ccount @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_886_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_887_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_888_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_889_antisym__conv1,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ~ ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_890_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_891_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_892_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_893_antisym__conv2,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ~ ( ord_le2131251472502387783ccount @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_894_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_895_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_896_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_897_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_898_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_899_less__le__not__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X2: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y5 )
& ~ ( ord_le4245800335709223507ccount @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_900_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_901_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_902_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_903_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_904_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_905_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_906_order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_907_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_908_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_909_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_910_order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_911_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_912_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_913_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_914_order_Ostrict__trans1,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_915_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_916_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_917_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_918_order_Ostrict__trans2,axiom,
! [A: risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_919_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_920_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_921_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_922_order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A3 @ B2 )
& ~ ( ord_le4245800335709223507ccount @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_923_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_924_dense__ge__bounded,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_925_dense__le__bounded,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_926_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_927_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_real @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_928_dual__order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [B2: risk_Free_account,A3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_929_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_930_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_931_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_932_dual__order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B2: risk_Free_account,A3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_933_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_934_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_935_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_936_dual__order_Ostrict__trans1,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A )
=> ( ( ord_le2131251472502387783ccount @ C @ B )
=> ( ord_le2131251472502387783ccount @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_937_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_938_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_939_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_940_dual__order_Ostrict__trans2,axiom,
! [B: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A )
=> ( ( ord_le4245800335709223507ccount @ C @ B )
=> ( ord_le2131251472502387783ccount @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_941_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_942_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_943_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_944_dual__order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B2: risk_Free_account,A3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A3 )
& ~ ( ord_le4245800335709223507ccount @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_945_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_946_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_947_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_948_order_Ostrict__implies__order,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ord_le4245800335709223507ccount @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_949_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_950_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_951_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_952_dual__order_Ostrict__implies__order,axiom,
! [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A )
=> ( ord_le4245800335709223507ccount @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_953_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_954_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_955_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_956_order__le__less,axiom,
( ord_le4245800335709223507ccount
= ( ^ [X2: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_957_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_int @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_958_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_959_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_960_order__less__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X2: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_961_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_962_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_963_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_964_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_965_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_966_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_967_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_968_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_969_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_970_order__less__imp__le,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_971_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_972_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_973_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_974_order__le__neq__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( A != B )
=> ( ord_le2131251472502387783ccount @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_975_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_976_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_977_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_978_order__neq__le__trans,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( A != B )
=> ( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ord_le2131251472502387783ccount @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_979_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_980_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_981_order__le__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_982_order__le__less__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ Z )
=> ( ord_le2131251472502387783ccount @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_983_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_984_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_985_order__less__le__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_986_order__less__le__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ Z )
=> ( ord_le2131251472502387783ccount @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_987_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_988_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_989_order__le__less__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_990_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_991_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_992_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_993_order__le__less__subst1,axiom,
! [A: real,F: risk_Free_account > real,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_994_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_995_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_996_order__le__less__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_997_order__le__less__subst1,axiom,
! [A: risk_Free_account,F: risk_Free_account > risk_Free_account,B: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B ) )
=> ( ( ord_le2131251472502387783ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_998_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_999_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1000_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1001_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1002_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1003_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1004_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1005_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1006_order__le__less__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1007_order__le__less__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1008_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1009_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1010_order__less__le__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B: nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1011_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1012_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1013_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1014_order__less__le__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B: real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1015_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1016_order__less__le__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1017_order__less__le__subst1,axiom,
! [A: real,F: risk_Free_account > real,B: risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1018_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1019_order__less__le__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1020_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1021_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1022_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1023_order__less__le__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1024_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1025_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1026_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1027_order__less__le__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B ) @ C )
=> ( ! [X3: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y2 )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1028_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1029_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1030_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1031_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1032_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1033_order__le__imp__less__or__eq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1034_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1035_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_1036_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1037_verit__sum__simplify,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% verit_sum_simplify
thf(fact_1038_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1039_max__def,axiom,
( ord_max_nat
= ( ^ [A3: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% max_def
thf(fact_1040_max__def,axiom,
( ord_max_real
= ( ^ [A3: real,B2: real] : ( if_real @ ( ord_less_eq_real @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% max_def
thf(fact_1041_max__def,axiom,
( ord_ma8558070474537249246ccount
= ( ^ [A3: risk_Free_account,B2: risk_Free_account] : ( if_Risk_Free_account @ ( ord_le4245800335709223507ccount @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% max_def
thf(fact_1042_max__def,axiom,
( ord_max_int
= ( ^ [A3: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% max_def
thf(fact_1043_max__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_max_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_1044_max__absorb1,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_max_real @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_1045_max__absorb1,axiom,
! [Y: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y @ X )
=> ( ( ord_ma8558070474537249246ccount @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_1046_max__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_max_int @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_1047_max__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_max_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_1048_max__absorb2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_max_real @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_1049_max__absorb2,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_ma8558070474537249246ccount @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_1050_max__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_max_int @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_1051_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1052_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1053_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1054_dual__Min,axiom,
( ( lattices_Min_real
@ ^ [X2: real,Y5: real] : ( ord_less_eq_real @ Y5 @ X2 ) )
= lattic4275903605611617917x_real ) ).
% dual_Min
thf(fact_1055_dual__Min,axiom,
( ( lattices_Min_int
@ ^ [X2: int,Y5: int] : ( ord_less_eq_int @ Y5 @ X2 ) )
= lattic8263393255366662781ax_int ) ).
% dual_Min
thf(fact_1056_dual__Min,axiom,
( ( lattices_Min_nat
@ ^ [X2: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X2 ) )
= lattic8265883725875713057ax_nat ) ).
% dual_Min
thf(fact_1057_cash__reserve__def,axiom,
( risk_F1914734008469130493eserve
= ( ^ [Alpha: risk_Free_account] : ( risk_F170160801229183585ccount @ Alpha @ zero_zero_nat ) ) ) ).
% cash_reserve_def
thf(fact_1058_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
@ ^ [X2: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X2 )
@ ^ [X2: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X2 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_1059_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
= ( P @ B6 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ zero_zero_nat )
=> ( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
=> ( P @ A6 @ ( plus_plus_nat @ A6 @ B6 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1060_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K3 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1061_plus__account__def,axiom,
( plus_p1863581527469039996ccount
= ( ^ [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N2: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_12 @ N2 ) @ ( risk_F170160801229183585ccount @ Alpha_22 @ N2 ) ) ) ) ) ).
% plus_account_def
thf(fact_1062_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_1063_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1064_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_1065_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [F: nat > nat > nat,Z: nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: nat,B: nat] :
( ( semila1623282765462674594er_nat @ F @ Z @ Less_eq @ Less )
=> ( ( Z
= ( F @ A @ B ) )
= ( ( A = Z )
& ( B = Z ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
thf(fact_1066_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [F: nat > nat > nat,Z: nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: nat,B: nat] :
( ( semila1623282765462674594er_nat @ F @ Z @ Less_eq @ Less )
=> ( ( ( F @ A @ B )
= Z )
= ( ( A = Z )
& ( B = Z ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
thf(fact_1067_Rep__account__inverse,axiom,
! [X: risk_Free_account] :
( ( risk_F5458100604530014700ccount @ ( risk_F170160801229183585ccount @ X ) )
= X ) ).
% Rep_account_inverse
thf(fact_1068_zero__account__def,axiom,
( zero_z1425366712893667068ccount
= ( risk_F5458100604530014700ccount
@ ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_account_def
thf(fact_1069_pred__subset__eq,axiom,
! [R: set_real,S2: set_real] :
( ( ord_less_eq_real_o
@ ^ [X2: real] : ( member_real @ X2 @ R )
@ ^ [X2: real] : ( member_real @ X2 @ S2 ) )
= ( ord_less_eq_set_real @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_1070_pred__subset__eq,axiom,
! [R: set_nat_real,S2: set_nat_real] :
( ( ord_le7676461544873280788real_o
@ ^ [X2: nat > real] : ( member_nat_real @ X2 @ R )
@ ^ [X2: nat > real] : ( member_nat_real @ X2 @ S2 ) )
= ( ord_le2908806416726583473t_real @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_1071_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1072_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1073_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_1074_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_1075_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1076_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: real] :
( ( ord_less_eq_real @ A @ C3 )
& ( ord_less_eq_real @ C3 @ B )
& ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D3: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1077_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: int] :
( ( ord_less_eq_int @ A @ C3 )
& ( ord_less_eq_int @ C3 @ B )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D3: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1078_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_1079_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ T ) ) ).
% pinf(6)
thf(fact_1080_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% pinf(6)
thf(fact_1081_ex__gt__or__lt,axiom,
! [A: real] :
? [B6: real] :
( ( ord_less_real @ A @ B6 )
| ( ord_less_real @ B6 @ A ) ) ).
% ex_gt_or_lt
thf(fact_1082_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1083_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1084_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1085_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1086_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1087_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1088_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_1089_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_1090_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_1091_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_1092_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_1093_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_1094_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_1095_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ~ ( ord_less_real @ X4 @ T ) ) ).
% pinf(5)
thf(fact_1096_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ~ ( ord_less_int @ X4 @ T ) ) ).
% pinf(5)
thf(fact_1097_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_1098_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( ord_less_real @ T @ X4 ) ) ).
% pinf(7)
thf(fact_1099_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ord_less_int @ T @ X4 ) ) ).
% pinf(7)
thf(fact_1100_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1101_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1102_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1103_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1104_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1105_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1106_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_1107_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_1108_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_1109_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_1110_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_1111_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_1112_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_1113_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z3 )
=> ( ord_less_real @ X4 @ T ) ) ).
% minf(5)
thf(fact_1114_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ord_less_int @ X4 @ T ) ) ).
% minf(5)
thf(fact_1115_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_1116_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z3 )
=> ~ ( ord_less_real @ T @ X4 ) ) ).
% minf(7)
thf(fact_1117_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ~ ( ord_less_int @ T @ X4 ) ) ).
% minf(7)
thf(fact_1118_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_1119_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_1120_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1121_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_1122_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1123_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_1124_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X4 ) ) ).
% minf(8)
thf(fact_1125_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% minf(8)
thf(fact_1126_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_1127_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z3 )
=> ( ord_less_eq_real @ X4 @ T ) ) ).
% minf(6)
thf(fact_1128_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ord_less_eq_int @ X4 @ T ) ) ).
% minf(6)
thf(fact_1129_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_1130_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( ord_less_eq_real @ T @ X4 ) ) ).
% pinf(8)
thf(fact_1131_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ord_less_eq_int @ T @ X4 ) ) ).
% pinf(8)
thf(fact_1132_dual__min,axiom,
( ( min_int
@ ^ [X2: int,Y5: int] : ( ord_less_eq_int @ Y5 @ X2 ) )
= ord_max_int ) ).
% dual_min
thf(fact_1133_Rep__account__just__cash,axiom,
! [C: real] :
( ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ C ) )
= ( ^ [N2: nat] : ( if_real @ ( N2 = zero_zero_nat ) @ C @ zero_zero_real ) ) ) ).
% Rep_account_just_cash
thf(fact_1134_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X2: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X2 )
@ ^ [X2: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X2 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1135_just__cash__def,axiom,
( risk_Free_just_cash
= ( ^ [C2: real] :
( risk_F5458100604530014700ccount
@ ^ [N2: nat] : ( if_real @ ( N2 = zero_zero_nat ) @ C2 @ zero_zero_real ) ) ) ) ).
% just_cash_def
thf(fact_1136_just__cash__plus,axiom,
! [A: real,B: real] :
( ( plus_p1863581527469039996ccount @ ( risk_Free_just_cash @ A ) @ ( risk_Free_just_cash @ B ) )
= ( risk_Free_just_cash @ ( plus_plus_real @ A @ B ) ) ) ).
% just_cash_plus
thf(fact_1137_just__cash__embed,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( risk_Free_just_cash @ A3 )
= ( risk_Free_just_cash @ B2 ) ) ) ) ).
% just_cash_embed
thf(fact_1138_just__cash__order__embed,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] : ( ord_le4245800335709223507ccount @ ( risk_Free_just_cash @ A3 ) @ ( risk_Free_just_cash @ B2 ) ) ) ) ).
% just_cash_order_embed
thf(fact_1139_zero__account__alt__def,axiom,
( ( risk_Free_just_cash @ zero_zero_real )
= zero_z1425366712893667068ccount ) ).
% zero_account_alt_def
thf(fact_1140_just__cash__valid__transfer,axiom,
! [C: real,T: real] :
( ( risk_F1023690899723030139ansfer @ ( risk_Free_just_cash @ C ) @ ( risk_Free_just_cash @ T ) )
= ( ( ord_less_eq_real @ zero_zero_real @ T )
& ( ord_less_eq_real @ T @ C ) ) ) ).
% just_cash_valid_transfer
thf(fact_1141_complete__real,axiom,
! [S2: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S2 )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Y2 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1142_less__account__def,axiom,
( ord_le2131251472502387783ccount
= ( ^ [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Alpha_12 @ Alpha_22 )
& ~ ( ord_le4245800335709223507ccount @ Alpha_22 @ Alpha_12 ) ) ) ) ).
% less_account_def
thf(fact_1143_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_1144_valid__transfer__alt__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha: risk_Free_account,Tau: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ Tau )
& ( ord_le4245800335709223507ccount @ Tau @ Alpha ) ) ) ) ).
% valid_transfer_alt_def
thf(fact_1145_strictly__solvent__non__negative__cash,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( risk_F1914734008469130493eserve @ Alpha2 ) ) ) ).
% strictly_solvent_non_negative_cash
thf(fact_1146_strictly__solvent__just__cash__equiv,axiom,
! [C: real] :
( ( risk_F1636578016437888323olvent @ ( risk_Free_just_cash @ C ) )
= ( ord_less_eq_real @ zero_zero_real @ C ) ) ).
% strictly_solvent_just_cash_equiv
thf(fact_1147_strictly__solvent__alt__def,axiom,
( risk_F1636578016437888323olvent
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount ) ) ).
% strictly_solvent_alt_def
thf(fact_1148_additive__strictly__solvent,axiom,
! [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha_1 )
=> ( ( risk_F1636578016437888323olvent @ Alpha_2 )
=> ( risk_F1636578016437888323olvent @ ( plus_p1863581527469039996ccount @ Alpha_1 @ Alpha_2 ) ) ) ) ).
% additive_strictly_solvent
thf(fact_1149_only__strictly__solvent__accounts__can__transfer,axiom,
! [Alpha2: risk_Free_account,Tau2: risk_Free_account] :
( ( risk_F1023690899723030139ansfer @ Alpha2 @ Tau2 )
=> ( risk_F1636578016437888323olvent @ Alpha2 ) ) ).
% only_strictly_solvent_accounts_can_transfer
thf(fact_1150_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1151_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1152_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1153_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1154_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1155_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1156_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1157_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_1158_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1159_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1160_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1161_finite__interval__int1,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A @ I2 )
& ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_1162_finite__interval__int4,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A @ I2 )
& ( ord_less_int @ I2 @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_1163_finite__interval__int3,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A @ I2 )
& ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_1164_finite__interval__int2,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A @ I2 )
& ( ord_less_int @ I2 @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_1165_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1166_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1167_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1168_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1169_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1170_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1171_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z5: int] :
? [N2: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1172_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1173_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1174_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P4: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1175_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P4: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1176_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1177_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_1178_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1179_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1180_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_1181_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1182_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1183_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1184_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1185_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1186_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1187_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1188_Rep__account__uminus,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( uminus3377898441596595772ccount @ Alpha2 ) )
= ( ^ [N2: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha2 @ N2 ) ) ) ) ).
% Rep_account_uminus
thf(fact_1189_just__cash__uminus,axiom,
! [A: real] :
( ( uminus3377898441596595772ccount @ ( risk_Free_just_cash @ A ) )
= ( risk_Free_just_cash @ ( uminus_uminus_real @ A ) ) ) ).
% just_cash_uminus
thf(fact_1190_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_1191_shortest__period__uminus,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F4612863212915232279period @ ( uminus3377898441596595772ccount @ Alpha2 ) )
= ( risk_F4612863212915232279period @ Alpha2 ) ) ).
% shortest_period_uminus
thf(fact_1192_uminus__account__def,axiom,
( uminus3377898441596595772ccount
= ( ^ [Alpha: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N2: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha @ N2 ) ) ) ) ) ).
% uminus_account_def
thf(fact_1193_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1194_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1195_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1196_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_1197_Gcd__int__greater__eq__0,axiom,
! [K4: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K4 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_1198_Gcd__remove0__nat,axiom,
! [M5: set_nat] :
( ( finite_finite_nat @ M5 )
=> ( ( gcd_Gcd_nat @ M5 )
= ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M5 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_1199_just__cash__subtract,axiom,
! [A: real,B: real] :
( ( minus_4846202936726426316ccount @ ( risk_Free_just_cash @ A ) @ ( risk_Free_just_cash @ B ) )
= ( risk_Free_just_cash @ ( minus_minus_real @ A @ B ) ) ) ).
% just_cash_subtract
thf(fact_1200_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1201_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1202_diff__diff__cancel,axiom,
! [I4: nat,N: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I4 ) )
= I4 ) ) ).
% diff_diff_cancel
thf(fact_1203_diff__diff__left,axiom,
! [I4: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J ) @ K )
= ( minus_minus_nat @ I4 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1204_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1205_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1206_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1207_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I4 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1208_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I4 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I4 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1209_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1210_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1211_diff__commute,axiom,
! [I4: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K ) @ J ) ) ).
% diff_commute
thf(fact_1212_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1213_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1214_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1215_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1216_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1217_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1218_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1219_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1220_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1221_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1222_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1223_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1224_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1225_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1226_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1227_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1228_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N3: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_1229_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1230_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1231_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1232_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1233_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1234_less__diff__conv,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1235_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1236_Nat_Ole__imp__diff__is__add,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ( minus_minus_nat @ J @ I4 )
= K )
= ( J
= ( plus_plus_nat @ K @ I4 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1237_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I4 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I4 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1238_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J ) @ K )
= ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1239_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1240_le__diff__conv,axiom,
! [J: nat,K: nat,I4: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I4 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I4 @ K ) ) ) ).
% le_diff_conv
thf(fact_1241_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1242_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X2: real,Y5: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y5 ) ) ) ) ).
% minus_real_def
thf(fact_1243_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
= ( ord_max_nat @ N @ M ) ) ).
% nat_minus_add_max
thf(fact_1244_valid__transfer__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha: risk_Free_account,Tau: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Tau )
& ( risk_F1636578016437888323olvent @ ( minus_4846202936726426316ccount @ Alpha @ Tau ) ) ) ) ) ).
% valid_transfer_def
thf(fact_1245_minus__account__def,axiom,
( minus_4846202936726426316ccount
= ( ^ [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] : ( plus_p1863581527469039996ccount @ Alpha_12 @ ( uminus3377898441596595772ccount @ Alpha_22 ) ) ) ) ).
% minus_account_def
thf(fact_1246_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1247_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1248_less__diff__conv2,axiom,
! [K: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I4 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I4 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1249_net__asset__value__def,axiom,
( risk_F2906766666041932210_value
= ( ^ [Alpha: risk_Free_account] :
( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha )
@ ( collect_nat
@ ^ [I2: nat] :
( ( risk_F170160801229183585ccount @ Alpha @ I2 )
!= zero_zero_real ) ) ) ) ) ).
% net_asset_value_def
thf(fact_1250_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A6: real,B6: real,C3: real] :
( ( P @ A6 @ B6 )
=> ( ( P @ B6 @ C3 )
=> ( ( ord_less_eq_real @ A6 @ B6 )
=> ( ( ord_less_eq_real @ B6 @ C3 )
=> ( P @ A6 @ C3 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A6: real,B6: real] :
( ( ( ord_less_eq_real @ A6 @ X3 )
& ( ord_less_eq_real @ X3 @ B6 )
& ( ord_less_real @ ( minus_minus_real @ B6 @ A6 ) @ D3 ) )
=> ( P @ A6 @ B6 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1251_eq__diff__eq_H,axiom,
! [X: real,Y: real,Z: real] :
( ( X
= ( minus_minus_real @ Y @ Z ) )
= ( Y
= ( plus_plus_real @ X @ Z ) ) ) ).
% eq_diff_eq'
thf(fact_1252_strictly__solvent__def,axiom,
( risk_F1636578016437888323olvent
= ( ^ [Alpha: risk_Free_account] :
! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha ) @ ( set_ord_atMost_nat @ N2 ) ) ) ) ) ).
% strictly_solvent_def
thf(fact_1253_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_1254_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_1255_partial__nav__just__cash,axiom,
! [A: real,N: nat] :
( ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ A ) ) @ ( set_ord_atMost_nat @ N ) )
= A ) ).
% partial_nav_just_cash
thf(fact_1256_Rep__account,axiom,
! [X: risk_Free_account] : ( member_nat_real @ ( risk_F170160801229183585ccount @ X ) @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) ) ).
% Rep_account
thf(fact_1257_Rep__account__cases,axiom,
! [Y: nat > real] :
( ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ~ ! [X3: risk_Free_account] :
( Y
!= ( risk_F170160801229183585ccount @ X3 ) ) ) ).
% Rep_account_cases
thf(fact_1258_Rep__account__induct,axiom,
! [Y: nat > real,P: ( nat > real ) > $o] :
( ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ! [X3: risk_Free_account] : ( P @ ( risk_F170160801229183585ccount @ X3 ) )
=> ( P @ Y ) ) ) ).
% Rep_account_induct
thf(fact_1259_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1260_infinite__UNIV__int,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_int
thf(fact_1261_Abs__account__inject,axiom,
! [X: nat > real,Y: nat > real] :
( ( member_nat_real @ X @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ( ( risk_F5458100604530014700ccount @ X )
= ( risk_F5458100604530014700ccount @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_account_inject
thf(fact_1262_Abs__account__induct,axiom,
! [P: risk_Free_account > $o,X: risk_Free_account] :
( ! [Y2: nat > real] :
( ( member_nat_real @ Y2 @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( P @ ( risk_F5458100604530014700ccount @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_account_induct
thf(fact_1263_Abs__account__cases,axiom,
! [X: risk_Free_account] :
~ ! [Y2: nat > real] :
( ( X
= ( risk_F5458100604530014700ccount @ Y2 ) )
=> ~ ( member_nat_real @ Y2 @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) ) ) ).
% Abs_account_cases
thf(fact_1264_Abs__account__inverse,axiom,
! [Y: nat > real] :
( ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ( risk_F170160801229183585ccount @ ( risk_F5458100604530014700ccount @ Y ) )
= Y ) ) ).
% Abs_account_inverse
thf(fact_1265_less__eq__account__def,axiom,
( ord_le4245800335709223507ccount
= ( ^ [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_12 ) @ ( set_ord_atMost_nat @ N2 ) ) @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_22 ) @ ( set_ord_atMost_nat @ N2 ) ) ) ) ) ).
% less_eq_account_def
thf(fact_1266_type__definition__account,axiom,
type_d8982087200295354172t_real @ risk_F170160801229183585ccount @ risk_F5458100604530014700ccount @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) ).
% type_definition_account
% Helper facts (9)
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__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_3_1_If_001t__Risk____Free____Lending__Oaccount_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Risk____Free____Lending__Oaccount_T,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( if_Risk_Free_account @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Risk____Free____Lending__Oaccount_T,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( if_Risk_Free_account @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ord_less_eq_nat
@ ( if_nat
@ ! [I2: nat] :
( ( risk_F170160801229183585ccount @ ( plus_p1863581527469039996ccount @ alpha @ beta ) @ I2 )
= zero_zero_real )
@ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( risk_F170160801229183585ccount @ ( plus_p1863581527469039996ccount @ alpha @ beta ) @ I2 )
!= zero_zero_real ) ) ) )
@ ( ord_max_nat
@ ( if_nat
@ ! [I2: nat] :
( ( risk_F170160801229183585ccount @ alpha @ I2 )
= zero_zero_real )
@ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( risk_F170160801229183585ccount @ alpha @ I2 )
!= zero_zero_real ) ) ) )
@ ( if_nat
@ ! [I2: nat] :
( ( risk_F170160801229183585ccount @ beta @ I2 )
= zero_zero_real )
@ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( risk_F170160801229183585ccount @ beta @ I2 )
!= zero_zero_real ) ) ) ) ) ) ).
%------------------------------------------------------------------------------