TPTP Problem File: SLH0515^1.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_00957_029253__5920938_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1381 ( 575 unt; 105 typ; 0 def)
% Number of atoms : 3548 (1175 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10054 ( 295 ~; 60 |; 200 &;8032 @)
% ( 0 <=>;1467 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 574 ( 574 >; 0 *; 0 +; 0 <<)
% Number of symbols : 97 ( 94 usr; 15 con; 0-4 aty)
% Number of variables : 3527 ( 295 ^;3174 !; 58 ?;3527 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:57:26.833
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
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__Nat__Onat_Mt__Real__Oreal_J_J,type,
set_nat_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $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__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (94)
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
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_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_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__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Risk____Free____Lending__Oaccount,type,
plus_p1863581527469039996ccount: risk_Free_account > risk_Free_account > risk_Free_account ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_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_001_062_It__Nat__Onat_Mt__Real__Oreal_J_001t__Int__Oint,type,
groups777517501785750147al_int: ( ( nat > real ) > int ) > set_nat_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_062_It__Nat__Onat_Mt__Real__Oreal_J_001t__Nat__Onat,type,
groups780007972294800423al_nat: ( ( nat > real ) > nat ) > set_nat_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_062_It__Nat__Onat_Mt__Real__Oreal_J_001t__Real__Oreal,type,
groups4253619806861319043l_real: ( ( nat > real ) > real ) > set_nat_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_062_It__Nat__Onat_Mt__Real__Oreal_J_001t__Risk____Free____Lending__Oaccount,type,
groups383684539861946442ccount: ( ( nat > real ) > risk_Free_account ) > set_nat_real > 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_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Risk____Free____Lending__Oaccount,type,
groups8516999891779824987ccount: ( real > risk_Free_account ) > set_real > risk_Free_account ).
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_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
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_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $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_Oless__eq_001t__Set__Oset_It__Risk____Free____Lending__Oaccount_J,type,
ord_le4487465848215339657ccount: set_Ri1641125681238393385ccount > set_Ri1641125681238393385ccount > $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_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
ord_max_set_nat: set_nat > set_nat > set_nat ).
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_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
top_top_set_nat_real: set_nat_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_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__Nat__Onat_Mt__Real__Oreal_J,type,
collect_nat_real: ( ( nat > real ) > $o ) > set_nat_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__Risk____Free____Lending__Oaccount,type,
collec1856553087948576712ccount: ( risk_Free_account > $o ) > set_Ri1641125681238393385ccount ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
set_or1122926678442080148t_real: ( nat > real ) > set_nat_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Risk____Free____Lending__Oaccount,type,
set_or3854930313887350124ccount: risk_Free_account > set_Ri1641125681238393385ccount ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_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 (1266)
thf(fact_0_assms,axiom,
ord_le4245800335709223507ccount @ alpha @ beta ).
% assms
thf(fact_1_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_2_order__refl,axiom,
! [X: risk_Free_account] : ( ord_le4245800335709223507ccount @ X @ X ) ).
% order_refl
thf(fact_3_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_4_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_5_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_6_dual__order_Orefl,axiom,
! [A: risk_Free_account] : ( ord_le4245800335709223507ccount @ A @ A ) ).
% dual_order.refl
thf(fact_7_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_8_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_9__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_10__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_11_complete__real,axiom,
! [S: set_real] :
( ? [X2: real] : ( member_real @ X2 @ S )
=> ( ? [Z: real] :
! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ? [Y: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Y ) )
& ! [Z: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ) ) ).
% complete_real
thf(fact_12__092_060open_062net__asset__value_A_092_060alpha_062_A_061_Asum_A_I_092_060pi_062_A_092_060alpha_062_J_A_123_O_Omax_A_Ishortest__period_A_092_060alpha_062_J_A_Ishortest__period_A_092_060beta_062_J_125_092_060close_062,axiom,
( ( risk_F2906766666041932210_value @ alpha )
= ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ alpha ) @ ( set_ord_atMost_nat @ ( ord_max_nat @ ( risk_F4612863212915232279period @ alpha ) @ ( risk_F4612863212915232279period @ beta ) ) ) ) ) ).
% \<open>net_asset_value \<alpha> = sum (\<pi> \<alpha>) {..max (shortest_period \<alpha>) (shortest_period \<beta>)}\<close>
thf(fact_13__092_060open_062net__asset__value_A_092_060beta_062_A_061_Asum_A_I_092_060pi_062_A_092_060beta_062_J_A_123_O_Omax_A_Ishortest__period_A_092_060alpha_062_J_A_Ishortest__period_A_092_060beta_062_J_125_092_060close_062,axiom,
( ( risk_F2906766666041932210_value @ beta )
= ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ beta ) @ ( set_ord_atMost_nat @ ( ord_max_nat @ ( risk_F4612863212915232279period @ alpha ) @ ( risk_F4612863212915232279period @ beta ) ) ) ) ) ).
% \<open>net_asset_value \<beta> = sum (\<pi> \<beta>) {..max (shortest_period \<alpha>) (shortest_period \<beta>)}\<close>
thf(fact_14_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_15_verit__comp__simplify1_I2_J,axiom,
! [A: risk_Free_account] : ( ord_le4245800335709223507ccount @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_16_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_17_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_18_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_19_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_20_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_21_le__cases3,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_22_le__cases3,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_23_le__cases3,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_24_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_25_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: risk_Free_account,Z3: risk_Free_account] : ( Y3 = Z3 ) )
= ( ^ [X4: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y4 )
& ( ord_le4245800335709223507ccount @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_26_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_27_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_28_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_29_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_30_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_31_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_32_max__def__raw,axiom,
( ord_max_real
= ( ^ [A2: real,B2: real] : ( if_real @ ( ord_less_eq_real @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def_raw
thf(fact_33_max__def__raw,axiom,
( ord_ma8558070474537249246ccount
= ( ^ [A2: risk_Free_account,B2: risk_Free_account] : ( if_Risk_Free_account @ ( ord_le4245800335709223507ccount @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def_raw
thf(fact_34_max__def__raw,axiom,
( ord_max_nat
= ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def_raw
thf(fact_35_max__def__raw,axiom,
( ord_max_int
= ( ^ [A2: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def_raw
thf(fact_36_less__eq__account__def,axiom,
( ord_le4245800335709223507ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
! [N: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_1 ) @ ( set_ord_atMost_nat @ N ) ) @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_2 ) @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% less_eq_account_def
thf(fact_37_net__asset__value__shortest__period__ge,axiom,
! [Alpha: risk_Free_account,N2: nat] :
( ( ord_less_eq_nat @ ( risk_F4612863212915232279period @ Alpha ) @ N2 )
=> ( ( risk_F2906766666041932210_value @ Alpha )
= ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha ) @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% net_asset_value_shortest_period_ge
thf(fact_38_net__asset__value__alt__def,axiom,
( risk_F2906766666041932210_value
= ( ^ [Alpha2: risk_Free_account] : ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha2 ) @ ( set_ord_atMost_nat @ ( risk_F4612863212915232279period @ Alpha2 ) ) ) ) ) ).
% net_asset_value_alt_def
thf(fact_39_max__absorb2,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_max_real @ X @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_40_max__absorb2,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y2 )
=> ( ( ord_ma8558070474537249246ccount @ X @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_41_max__absorb2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_max_nat @ X @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_42_max__absorb2,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_max_int @ X @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_43_max__absorb1,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ( ( ord_max_real @ X @ Y2 )
= X ) ) ).
% max_absorb1
thf(fact_44_max__absorb1,axiom,
! [Y2: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y2 @ X )
=> ( ( ord_ma8558070474537249246ccount @ X @ Y2 )
= X ) ) ).
% max_absorb1
thf(fact_45_max__absorb1,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_max_nat @ X @ Y2 )
= X ) ) ).
% max_absorb1
thf(fact_46_max__absorb1,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( ord_max_int @ X @ Y2 )
= X ) ) ).
% max_absorb1
thf(fact_47_max__def,axiom,
( ord_max_real
= ( ^ [A2: real,B2: real] : ( if_real @ ( ord_less_eq_real @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_48_max__def,axiom,
( ord_ma8558070474537249246ccount
= ( ^ [A2: risk_Free_account,B2: risk_Free_account] : ( if_Risk_Free_account @ ( ord_le4245800335709223507ccount @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_49_max__def,axiom,
( ord_max_nat
= ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_50_max__def,axiom,
( ord_max_int
= ( ^ [A2: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_51_Rep__account__inject,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( ( risk_F170160801229183585ccount @ X )
= ( risk_F170160801229183585ccount @ Y2 ) )
= ( X = Y2 ) ) ).
% Rep_account_inject
thf(fact_52_order__antisym__conv,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_53_order__antisym__conv,axiom,
! [Y2: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y2 @ X )
=> ( ( ord_le4245800335709223507ccount @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_54_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_55_order__antisym__conv,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_56_linorder__le__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_57_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_58_linorder__le__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_59_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_60_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_61_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_62_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_63_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,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_64_ord__le__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_65_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,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_66_ord__le__eq__subst,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > int,C: int] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_67_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,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_68_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,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_69_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_70_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_71_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_72_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_73_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,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_74_ord__eq__le__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_le4245800335709223507ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_75_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,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_76_ord__eq__le__subst,axiom,
! [A: int,F: risk_Free_account > int,B: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B ) )
=> ( ( ord_le4245800335709223507ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_77_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,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_78_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,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_79_linorder__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_80_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_81_linorder__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_82_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_83_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_84_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_85_order__eq__refl,axiom,
! [X: real,Y2: real] :
( ( X = Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_86_order__eq__refl,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( X = Y2 )
=> ( ord_le4245800335709223507ccount @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_87_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_88_order__eq__refl,axiom,
! [X: int,Y2: int] :
( ( X = Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_89_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_90_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_91_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_92_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_93_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,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_94_order__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B ) @ C )
=> ( ! [X3: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_95_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,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_96_order__subst2,axiom,
! [A: risk_Free_account,B: risk_Free_account,F: risk_Free_account > int,C: int] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_97_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,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_98_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,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_99_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_100_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,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_101_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,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_102_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,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_103_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_104_order__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_le4245800335709223507ccount @ B @ C )
=> ( ! [X3: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_105_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,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_106_order__subst1,axiom,
! [A: risk_Free_account,F: int > risk_Free_account,B: int,C: int] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_107_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,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_108_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,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_109_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_110_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: risk_Free_account,Z3: risk_Free_account] : ( Y3 = Z3 ) )
= ( ^ [A2: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B2 )
& ( ord_le4245800335709223507ccount @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_111_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_112_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_113_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_114_antisym,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
=> ( ( ord_le4245800335709223507ccount @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_115_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_116_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_117_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_118_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_119_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_120_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_121_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_122_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_123_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_124_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_125_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_126_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: risk_Free_account,Z3: risk_Free_account] : ( Y3 = Z3 ) )
= ( ^ [A2: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A2 )
& ( ord_le4245800335709223507ccount @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_127_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_128_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_129_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_130_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_131_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_132_order__trans,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_133_order__trans,axiom,
! [X: risk_Free_account,Y2: risk_Free_account,Z2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y2 )
=> ( ( ord_le4245800335709223507ccount @ Y2 @ Z2 )
=> ( ord_le4245800335709223507ccount @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_134_order__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_135_order__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_136_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_137_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_138_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_139_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_140_order__antisym,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_141_order__antisym,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y2 )
=> ( ( ord_le4245800335709223507ccount @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_142_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_143_order__antisym,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_152_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_153_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_154_Collect__mem__eq,axiom,
! [A4: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_155_Collect__mem__eq,axiom,
! [A4: set_nat_real] :
( ( collect_nat_real
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_156_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_157_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_158_max_Oabsorb2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_max_real @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_159_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_160_max_Oabsorb2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_161_max_Oabsorb1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_max_real @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_162_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_163_max_Oabsorb1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_164_atMost__subset__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X ) @ ( set_ord_atMost_real @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_165_atMost__subset__iff,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4487465848215339657ccount @ ( set_or3854930313887350124ccount @ X ) @ ( set_or3854930313887350124ccount @ Y2 ) )
= ( ord_le4245800335709223507ccount @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_166_atMost__subset__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y2 ) )
= ( ord_less_eq_int @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_167_atMost__subset__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_168_atMost__iff,axiom,
! [I: nat > real,K: nat > real] :
( ( member_nat_real @ I @ ( set_or1122926678442080148t_real @ K ) )
= ( ord_less_eq_nat_real @ I @ K ) ) ).
% atMost_iff
thf(fact_169_atMost__iff,axiom,
! [I: real,K: real] :
( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I @ K ) ) ).
% atMost_iff
thf(fact_170_atMost__iff,axiom,
! [I: risk_Free_account,K: risk_Free_account] :
( ( member5612106785598075018ccount @ I @ ( set_or3854930313887350124ccount @ K ) )
= ( ord_le4245800335709223507ccount @ I @ K ) ) ).
% atMost_iff
thf(fact_171_atMost__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I @ K ) ) ).
% atMost_iff
thf(fact_172_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_173_max_Oidem,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ A )
= A ) ).
% max.idem
thf(fact_174_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_175_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_176_atMost__eq__iff,axiom,
! [X: nat,Y2: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y2 ) )
= ( X = Y2 ) ) ).
% atMost_eq_iff
thf(fact_177_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U: real] :
( collect_real
@ ^ [X4: real] : ( ord_less_eq_real @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_178_atMost__def,axiom,
( set_or3854930313887350124ccount
= ( ^ [U: risk_Free_account] :
( collec1856553087948576712ccount
@ ^ [X4: risk_Free_account] : ( ord_le4245800335709223507ccount @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_179_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U: int] :
( collect_int
@ ^ [X4: int] : ( ord_less_eq_int @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_180_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_181_sum__mono,axiom,
! [K2: set_real,F: real > real,G: real > real] :
( ! [I2: real] :
( ( member_real @ I2 @ K2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K2 ) @ ( groups8097168146408367636l_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_182_sum__mono,axiom,
! [K2: set_nat_real,F: ( nat > real ) > real,G: ( nat > real ) > real] :
( ! [I2: nat > real] :
( ( member_nat_real @ I2 @ K2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups4253619806861319043l_real @ F @ K2 ) @ ( groups4253619806861319043l_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_183_sum__mono,axiom,
! [K2: set_real,F: real > risk_Free_account,G: real > risk_Free_account] :
( ! [I2: real] :
( ( member_real @ I2 @ K2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups8516999891779824987ccount @ F @ K2 ) @ ( groups8516999891779824987ccount @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_184_sum__mono,axiom,
! [K2: set_nat_real,F: ( nat > real ) > risk_Free_account,G: ( nat > real ) > risk_Free_account] :
( ! [I2: nat > real] :
( ( member_nat_real @ I2 @ K2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups383684539861946442ccount @ F @ K2 ) @ ( groups383684539861946442ccount @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_185_sum__mono,axiom,
! [K2: set_real,F: real > nat,G: real > nat] :
( ! [I2: real] :
( ( member_real @ I2 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K2 ) @ ( groups1935376822645274424al_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_186_sum__mono,axiom,
! [K2: set_nat_real,F: ( nat > real ) > nat,G: ( nat > real ) > nat] :
( ! [I2: nat > real] :
( ( member_nat_real @ I2 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups780007972294800423al_nat @ F @ K2 ) @ ( groups780007972294800423al_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_187_sum__mono,axiom,
! [K2: set_real,F: real > int,G: real > int] :
( ! [I2: real] :
( ( member_real @ I2 @ K2 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K2 ) @ ( groups1932886352136224148al_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_188_sum__mono,axiom,
! [K2: set_nat_real,F: ( nat > real ) > int,G: ( nat > real ) > int] :
( ! [I2: nat > real] :
( ( member_nat_real @ I2 @ K2 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups777517501785750147al_int @ F @ K2 ) @ ( groups777517501785750147al_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_189_sum__mono,axiom,
! [K2: set_nat,F: nat > real,G: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K2 ) @ ( groups6591440286371151544t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_190_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_191_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_192_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_193_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_194_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I: nat > real,J: real > nat,T: set_nat,H: nat > real,G: real > real] :
( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_real @ ( I @ B3 ) @ S ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_195_sum_Oreindex__bij__witness,axiom,
! [S: set_nat_real,I: nat > nat > real,J: ( nat > real ) > nat,T: set_nat,H: nat > real,G: ( nat > real ) > real] :
( ! [A3: nat > real] :
( ( member_nat_real @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat > real] :
( ( member_nat_real @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat_real @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat > real] :
( ( member_nat_real @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4253619806861319043l_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_196_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: real > nat,J: nat > real,T: set_real,H: real > real,G: nat > real] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_real @ ( J @ A3 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_197_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: ( nat > real ) > nat,J: nat > nat > real,T: set_nat_real,H: ( nat > real ) > real,G: nat > real] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_nat_real @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat > real] :
( ( member_nat_real @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat > real] :
( ( member_nat_real @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups4253619806861319043l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_198_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: nat > nat,J: nat > nat,T: set_nat,H: nat > real,G: nat > real] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_199_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K: nat > real,A4: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ( ( member_real @ ( K @ Y ) @ A4 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A4 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_200_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K: nat > nat > real,A4: set_nat_real,H: ( nat > real ) > nat,Gamma: nat > real,Phi: ( nat > real ) > real] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ( ( member_nat_real @ ( K @ Y ) @ A4 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A4 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups4253619806861319043l_real @ Phi @ A4 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_201_sum_Oeq__general__inverses,axiom,
! [B4: set_real,K: real > nat,A4: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y: real] :
( ( member_real @ Y @ B4 )
=> ( ( member_nat @ ( K @ Y ) @ A4 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_real @ ( H @ X3 ) @ B4 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A4 )
= ( groups8097168146408367636l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_202_sum_Oeq__general__inverses,axiom,
! [B4: set_nat_real,K: ( nat > real ) > nat,A4: set_nat,H: nat > nat > real,Gamma: ( nat > real ) > real,Phi: nat > real] :
( ! [Y: nat > real] :
( ( member_nat_real @ Y @ B4 )
=> ( ( member_nat @ ( K @ Y ) @ A4 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_nat_real @ ( H @ X3 ) @ B4 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A4 )
= ( groups4253619806861319043l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_203_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K: nat > nat,A4: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ( ( member_nat @ ( K @ Y ) @ A4 )
& ( ( H @ ( K @ Y ) )
= Y ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A4 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_204_sum_Oeq__general,axiom,
! [B4: set_nat,A4: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ? [X2: real] :
( ( member_real @ X2 @ A4 )
& ( ( H @ X2 )
= Y )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A4 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A4 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_205_sum_Oeq__general,axiom,
! [B4: set_nat,A4: set_nat_real,H: ( nat > real ) > nat,Gamma: nat > real,Phi: ( nat > real ) > real] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ? [X2: nat > real] :
( ( member_nat_real @ X2 @ A4 )
& ( ( H @ X2 )
= Y )
& ! [Ya: nat > real] :
( ( ( member_nat_real @ Ya @ A4 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A4 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups4253619806861319043l_real @ Phi @ A4 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_206_sum_Oeq__general,axiom,
! [B4: set_real,A4: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y: real] :
( ( member_real @ Y @ B4 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ( ( H @ X2 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A4 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_real @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A4 )
= ( groups8097168146408367636l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_207_sum_Oeq__general,axiom,
! [B4: set_nat_real,A4: set_nat,H: nat > nat > real,Gamma: ( nat > real ) > real,Phi: nat > real] :
( ! [Y: nat > real] :
( ( member_nat_real @ Y @ B4 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ( ( H @ X2 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A4 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_nat_real @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A4 )
= ( groups4253619806861319043l_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_208_sum_Oeq__general,axiom,
! [B4: set_nat,A4: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y: nat] :
( ( member_nat @ Y @ B4 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ( ( H @ X2 )
= Y )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A4 )
& ( ( H @ Ya )
= Y ) )
=> ( Ya = X2 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A4 )
= ( groups6591440286371151544t_real @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_209_sum_Ocong,axiom,
! [A4: set_nat,B4: set_nat,G: nat > real,H: nat > real] :
( ( A4 = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A4 )
= ( groups6591440286371151544t_real @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_210_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_211_max_Ocommute,axiom,
( ord_max_nat
= ( ^ [A2: nat,B2: nat] : ( ord_max_nat @ B2 @ A2 ) ) ) ).
% max.commute
thf(fact_212_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_213_sum_Oswap,axiom,
! [G: nat > nat > real,B4: set_nat,A4: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( G @ I3 ) @ B4 )
@ A4 )
= ( groups6591440286371151544t_real
@ ^ [J2: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ I3 @ J2 )
@ A4 )
@ B4 ) ) ).
% sum.swap
thf(fact_214_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_215_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_216_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_217_max_OorderE,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( A
= ( ord_max_real @ A @ B ) ) ) ).
% max.orderE
thf(fact_218_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_219_max_OorderE,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( A
= ( ord_max_int @ A @ B ) ) ) ).
% max.orderE
thf(fact_220_max_OorderI,axiom,
! [A: real,B: real] :
( ( A
= ( ord_max_real @ A @ B ) )
=> ( ord_less_eq_real @ B @ A ) ) ).
% max.orderI
thf(fact_221_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_222_max_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( ord_max_int @ A @ B ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
% max.orderI
thf(fact_223_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_224_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_225_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_226_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_227_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_228_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_229_max_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A2: real] :
( A2
= ( ord_max_real @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_230_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( A2
= ( ord_max_nat @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_231_max_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( A2
= ( ord_max_int @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_232_max_Ocobounded1,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ A @ ( ord_max_real @ A @ B ) ) ).
% max.cobounded1
thf(fact_233_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_234_max_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded1
thf(fact_235_max_Ocobounded2,axiom,
! [B: real,A: real] : ( ord_less_eq_real @ B @ ( ord_max_real @ A @ B ) ) ).
% max.cobounded2
thf(fact_236_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_237_max_Ocobounded2,axiom,
! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded2
thf(fact_238_le__max__iff__disj,axiom,
! [Z2: real,X: real,Y2: real] :
( ( ord_less_eq_real @ Z2 @ ( ord_max_real @ X @ Y2 ) )
= ( ( ord_less_eq_real @ Z2 @ X )
| ( ord_less_eq_real @ Z2 @ Y2 ) ) ) ).
% le_max_iff_disj
thf(fact_239_le__max__iff__disj,axiom,
! [Z2: nat,X: nat,Y2: nat] :
( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X @ Y2 ) )
= ( ( ord_less_eq_nat @ Z2 @ X )
| ( ord_less_eq_nat @ Z2 @ Y2 ) ) ) ).
% le_max_iff_disj
thf(fact_240_le__max__iff__disj,axiom,
! [Z2: int,X: int,Y2: int] :
( ( ord_less_eq_int @ Z2 @ ( ord_max_int @ X @ Y2 ) )
= ( ( ord_less_eq_int @ Z2 @ X )
| ( ord_less_eq_int @ Z2 @ Y2 ) ) ) ).
% le_max_iff_disj
thf(fact_241_max_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A2: real] :
( ( ord_max_real @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_242_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_max_nat @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_243_max_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( ( ord_max_int @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_244_max_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] :
( ( ord_max_real @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_245_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_max_nat @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_246_max_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] :
( ( ord_max_int @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_247_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_248_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_249_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_250_subsetI,axiom,
! [A4: set_real,B4: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( member_real @ X3 @ B4 ) )
=> ( ord_less_eq_set_real @ A4 @ B4 ) ) ).
% subsetI
thf(fact_251_subsetI,axiom,
! [A4: set_nat_real,B4: set_nat_real] :
( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A4 )
=> ( member_nat_real @ X3 @ B4 ) )
=> ( ord_le2908806416726583473t_real @ A4 @ B4 ) ) ).
% subsetI
thf(fact_252_partial__nav__just__cash,axiom,
! [A: real,N2: nat] :
( ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ A ) ) @ ( set_ord_atMost_nat @ N2 ) )
= A ) ).
% partial_nav_just_cash
thf(fact_253_net__asset__value__def,axiom,
( risk_F2906766666041932210_value
= ( ^ [Alpha2: risk_Free_account] :
( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha2 )
@ ( collect_nat
@ ^ [I3: nat] :
( ( risk_F170160801229183585ccount @ Alpha2 @ I3 )
!= zero_zero_real ) ) ) ) ) ).
% net_asset_value_def
thf(fact_254_Collect__subset,axiom,
! [A4: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A4 )
& ( P @ X4 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_255_Collect__subset,axiom,
! [A4: set_nat_real,P: ( nat > real ) > $o] :
( ord_le2908806416726583473t_real
@ ( collect_nat_real
@ ^ [X4: nat > real] :
( ( member_nat_real @ X4 @ A4 )
& ( P @ X4 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_256_Collect__subset,axiom,
! [A4: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P @ X4 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_257_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
( ord_less_eq_real_o
@ ^ [X4: real] : ( member_real @ X4 @ A5 )
@ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_258_less__eq__set__def,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A5: set_nat_real,B5: set_nat_real] :
( ord_le7676461544873280788real_o
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ A5 )
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_259_pred__subset__eq,axiom,
! [R: set_real,S: set_real] :
( ( ord_less_eq_real_o
@ ^ [X4: real] : ( member_real @ X4 @ R )
@ ^ [X4: real] : ( member_real @ X4 @ S ) )
= ( ord_less_eq_set_real @ R @ S ) ) ).
% pred_subset_eq
thf(fact_260_pred__subset__eq,axiom,
! [R: set_nat_real,S: set_nat_real] :
( ( ord_le7676461544873280788real_o
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ R )
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ S ) )
= ( ord_le2908806416726583473t_real @ R @ S ) ) ).
% pred_subset_eq
thf(fact_261_conj__subset__def,axiom,
! [A4: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A4
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq_set_nat @ A4 @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A4 @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_262_prop__restrict,axiom,
! [X: real,Z4: set_real,X5: set_real,P: real > $o] :
( ( member_real @ X @ Z4 )
=> ( ( ord_less_eq_set_real @ Z4
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_263_prop__restrict,axiom,
! [X: nat > real,Z4: set_nat_real,X5: set_nat_real,P: ( nat > real ) > $o] :
( ( member_nat_real @ X @ Z4 )
=> ( ( ord_le2908806416726583473t_real @ Z4
@ ( collect_nat_real
@ ^ [X4: nat > real] :
( ( member_nat_real @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_264_prop__restrict,axiom,
! [X: nat,Z4: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_265_Collect__restrict,axiom,
! [X5: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_266_Collect__restrict,axiom,
! [X5: set_nat_real,P: ( nat > real ) > $o] :
( ord_le2908806416726583473t_real
@ ( collect_nat_real
@ ^ [X4: nat > real] :
( ( member_nat_real @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_267_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_268_subset__CollectI,axiom,
! [B4: set_real,A4: set_real,Q: real > $o,P: real > $o] :
( ( ord_less_eq_set_real @ B4 @ A4 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A4 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_269_subset__CollectI,axiom,
! [B4: set_nat_real,A4: set_nat_real,Q: ( nat > real ) > $o,P: ( nat > real ) > $o] :
( ( ord_le2908806416726583473t_real @ B4 @ A4 )
=> ( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le2908806416726583473t_real
@ ( collect_nat_real
@ ^ [X4: nat > real] :
( ( member_nat_real @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect_nat_real
@ ^ [X4: nat > real] :
( ( member_nat_real @ X4 @ A4 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_270_subset__CollectI,axiom,
! [B4: set_nat,A4: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_271_sum_Oneutral__const,axiom,
! [A4: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A4 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_272_just__cash__embed,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A2: real,B2: real] :
( ( risk_Free_just_cash @ A2 )
= ( risk_Free_just_cash @ B2 ) ) ) ) ).
% just_cash_embed
thf(fact_273_sum_Oneutral,axiom,
! [A4: set_nat,G: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A4 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_274_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A4: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A4 )
!= zero_zero_real )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_275_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: ( nat > real ) > real,A4: set_nat_real] :
( ( ( groups4253619806861319043l_real @ G @ A4 )
!= zero_zero_real )
=> ~ ! [A3: nat > real] :
( ( member_nat_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_276_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A4: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A4 )
!= zero_zero_nat )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_277_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: ( nat > real ) > nat,A4: set_nat_real] :
( ( ( groups780007972294800423al_nat @ G @ A4 )
!= zero_zero_nat )
=> ~ ! [A3: nat > real] :
( ( member_nat_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_278_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > risk_Free_account,A4: set_real] :
( ( ( groups8516999891779824987ccount @ G @ A4 )
!= zero_z1425366712893667068ccount )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_279_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: ( nat > real ) > risk_Free_account,A4: set_nat_real] :
( ( ( groups383684539861946442ccount @ G @ A4 )
!= zero_z1425366712893667068ccount )
=> ~ ! [A3: nat > real] :
( ( member_nat_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_280_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > int,A4: set_real] :
( ( ( groups1932886352136224148al_int @ G @ A4 )
!= zero_zero_int )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_281_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: ( nat > real ) > int,A4: set_nat_real] :
( ( ( groups777517501785750147al_int @ G @ A4 )
!= zero_zero_int )
=> ~ ! [A3: nat > real] :
( ( member_nat_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_282_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A4: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A4 )
!= zero_zero_real )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_283_sum__nonpos,axiom,
! [A4: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_284_sum__nonpos,axiom,
! [A4: set_nat_real,F: ( nat > real ) > real] :
( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups4253619806861319043l_real @ F @ A4 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_285_sum__nonpos,axiom,
! [A4: set_real,F: real > risk_Free_account] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ zero_z1425366712893667068ccount ) )
=> ( ord_le4245800335709223507ccount @ ( groups8516999891779824987ccount @ F @ A4 ) @ zero_z1425366712893667068ccount ) ) ).
% sum_nonpos
thf(fact_286_sum__nonpos,axiom,
! [A4: set_nat_real,F: ( nat > real ) > risk_Free_account] :
( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A4 )
=> ( ord_le4245800335709223507ccount @ ( F @ X3 ) @ zero_z1425366712893667068ccount ) )
=> ( ord_le4245800335709223507ccount @ ( groups383684539861946442ccount @ F @ A4 ) @ zero_z1425366712893667068ccount ) ) ).
% sum_nonpos
thf(fact_287_sum__nonpos,axiom,
! [A4: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_288_sum__nonpos,axiom,
! [A4: set_nat_real,F: ( nat > real ) > nat] :
( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups780007972294800423al_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_289_sum__nonpos,axiom,
! [A4: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_290_sum__nonpos,axiom,
! [A4: set_nat_real,F: ( nat > real ) > int] :
( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups777517501785750147al_int @ F @ A4 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_291_sum__nonpos,axiom,
! [A4: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_292_sum__nonneg,axiom,
! [A4: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_293_sum__nonneg,axiom,
! [A4: set_nat_real,F: ( nat > real ) > real] :
( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups4253619806861319043l_real @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_294_sum__nonneg,axiom,
! [A4: set_real,F: real > risk_Free_account] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X3 ) ) )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( groups8516999891779824987ccount @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_295_sum__nonneg,axiom,
! [A4: set_nat_real,F: ( nat > real ) > risk_Free_account] :
( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A4 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X3 ) ) )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( groups383684539861946442ccount @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_296_sum__nonneg,axiom,
! [A4: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_297_sum__nonneg,axiom,
! [A4: set_nat_real,F: ( nat > real ) > nat] :
( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups780007972294800423al_nat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_298_sum__nonneg,axiom,
! [A4: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_299_sum__nonneg,axiom,
! [A4: set_nat_real,F: ( nat > real ) > int] :
( ! [X3: nat > real] :
( ( member_nat_real @ X3 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups777517501785750147al_int @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_300_sum__nonneg,axiom,
! [A4: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_301_shortest__period___092_060pi_062,axiom,
! [Alpha: risk_Free_account,I: nat] :
( ( ( risk_F170160801229183585ccount @ Alpha @ I )
!= zero_zero_real )
=> ( ( risk_F170160801229183585ccount @ Alpha @ ( risk_F4612863212915232279period @ Alpha ) )
!= zero_zero_real ) ) ).
% shortest_period_\<pi>
thf(fact_302_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_303_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_304_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A5 )
=> ( member_real @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_305_subset__iff,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A5: set_nat_real,B5: set_nat_real] :
! [T2: nat > real] :
( ( member_nat_real @ T2 @ A5 )
=> ( member_nat_real @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_306_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [X4: real] :
( ( member_real @ X4 @ A5 )
=> ( member_real @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_307_subset__eq,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A5: set_nat_real,B5: set_nat_real] :
! [X4: nat > real] :
( ( member_nat_real @ X4 @ A5 )
=> ( member_nat_real @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_308_subsetD,axiom,
! [A4: set_real,B4: set_real,C: real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
=> ( ( member_real @ C @ A4 )
=> ( member_real @ C @ B4 ) ) ) ).
% subsetD
thf(fact_309_subsetD,axiom,
! [A4: set_nat_real,B4: set_nat_real,C: nat > real] :
( ( ord_le2908806416726583473t_real @ A4 @ B4 )
=> ( ( member_nat_real @ C @ A4 )
=> ( member_nat_real @ C @ B4 ) ) ) ).
% subsetD
thf(fact_310_in__mono,axiom,
! [A4: set_real,B4: set_real,X: real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
=> ( ( member_real @ X @ A4 )
=> ( member_real @ X @ B4 ) ) ) ).
% in_mono
thf(fact_311_in__mono,axiom,
! [A4: set_nat_real,B4: set_nat_real,X: nat > real] :
( ( ord_le2908806416726583473t_real @ A4 @ B4 )
=> ( ( member_nat_real @ X @ A4 )
=> ( member_nat_real @ X @ B4 ) ) ) ).
% in_mono
thf(fact_312_just__cash__order__embed,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] : ( ord_le4245800335709223507ccount @ ( risk_Free_just_cash @ A2 ) @ ( risk_Free_just_cash @ B2 ) ) ) ) ).
% just_cash_order_embed
thf(fact_313_shortest__period__bound,axiom,
! [Alpha: risk_Free_account,I: nat] :
( ( ( risk_F170160801229183585ccount @ Alpha @ I )
!= zero_zero_real )
=> ( ord_less_eq_nat @ I @ ( risk_F4612863212915232279period @ Alpha ) ) ) ).
% shortest_period_bound
thf(fact_314_subset__Collect__iff,axiom,
! [B4: set_real,A4: set_real,P: real > $o] :
( ( ord_less_eq_set_real @ B4 @ A4 )
=> ( ( ord_less_eq_set_real @ B4
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A4 )
& ( P @ X4 ) ) ) )
= ( ! [X4: real] :
( ( member_real @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_315_subset__Collect__iff,axiom,
! [B4: set_nat_real,A4: set_nat_real,P: ( nat > real ) > $o] :
( ( ord_le2908806416726583473t_real @ B4 @ A4 )
=> ( ( ord_le2908806416726583473t_real @ B4
@ ( collect_nat_real
@ ^ [X4: nat > real] :
( ( member_nat_real @ X4 @ A4 )
& ( P @ X4 ) ) ) )
= ( ! [X4: nat > real] :
( ( member_nat_real @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_316_subset__Collect__iff,axiom,
! [B4: set_nat,A4: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( ( ord_less_eq_set_nat @ B4
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P @ X4 ) ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_317_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_318_strictly__solvent__def,axiom,
( risk_F1636578016437888323olvent
= ( ^ [Alpha2: risk_Free_account] :
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha2 ) @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% strictly_solvent_def
thf(fact_319_just__cash__valid__transfer,axiom,
! [C: real,T3: real] :
( ( risk_F1023690899723030139ansfer @ ( risk_Free_just_cash @ C ) @ ( risk_Free_just_cash @ T3 ) )
= ( ( ord_less_eq_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ C ) ) ) ).
% just_cash_valid_transfer
thf(fact_320_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_321_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_322_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_323_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_324_strictly__solvent__net__asset__value,axiom,
! [Alpha: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha )
=> ( ord_less_eq_real @ zero_zero_real @ ( risk_F2906766666041932210_value @ Alpha ) ) ) ).
% strictly_solvent_net_asset_value
thf(fact_325_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_326_Rep__account__just__cash,axiom,
! [C: real] :
( ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ C ) )
= ( ^ [N: nat] : ( if_real @ ( N = zero_zero_nat ) @ C @ zero_zero_real ) ) ) ).
% Rep_account_just_cash
thf(fact_327_Rep__account__zero,axiom,
( ( risk_F170160801229183585ccount @ zero_z1425366712893667068ccount )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% Rep_account_zero
thf(fact_328_net__asset__value__zero,axiom,
( ( risk_F2906766666041932210_value @ zero_z1425366712893667068ccount )
= zero_zero_real ) ).
% net_asset_value_zero
thf(fact_329_only__strictly__solvent__accounts__can__transfer,axiom,
! [Alpha: risk_Free_account,Tau: risk_Free_account] :
( ( risk_F1023690899723030139ansfer @ Alpha @ Tau )
=> ( risk_F1636578016437888323olvent @ Alpha ) ) ).
% only_strictly_solvent_accounts_can_transfer
thf(fact_330_strictly__solvent__alt__def,axiom,
( risk_F1636578016437888323olvent
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount ) ) ).
% strictly_solvent_alt_def
thf(fact_331_valid__transfer__alt__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha2: risk_Free_account,Tau2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ Tau2 )
& ( ord_le4245800335709223507ccount @ Tau2 @ Alpha2 ) ) ) ) ).
% valid_transfer_alt_def
thf(fact_332_zero__account__alt__def,axiom,
( ( risk_Free_just_cash @ zero_zero_real )
= zero_z1425366712893667068ccount ) ).
% zero_account_alt_def
thf(fact_333_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_334_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_335_zero__reorient,axiom,
! [X: risk_Free_account] :
( ( zero_z1425366712893667068ccount = X )
= ( X = zero_z1425366712893667068ccount ) ) ).
% zero_reorient
thf(fact_336_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_337_max__0R,axiom,
! [N2: nat] :
( ( ord_max_nat @ N2 @ zero_zero_nat )
= N2 ) ).
% max_0R
thf(fact_338_max__0L,axiom,
! [N2: nat] :
( ( ord_max_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% max_0L
thf(fact_339_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_340_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_341_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_342_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_343_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_344_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_345_strictly__solvent__non__negative__cash,axiom,
! [Alpha: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha )
=> ( ord_less_eq_real @ zero_zero_real @ ( risk_F1914734008469130493eserve @ Alpha ) ) ) ).
% strictly_solvent_non_negative_cash
thf(fact_346_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_347_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_348_eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( M3 = N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% eq_imp_le
thf(fact_349_le__antisym,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M3 )
=> ( M3 = N2 ) ) ) ).
% le_antisym
thf(fact_350_nat__le__linear,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
| ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% nat_le_linear
thf(fact_351_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_352_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_353_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_354_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_355_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_356_cash__reserve__def,axiom,
( risk_F1914734008469130493eserve
= ( ^ [Alpha2: risk_Free_account] : ( risk_F170160801229183585ccount @ Alpha2 @ zero_zero_nat ) ) ) ).
% cash_reserve_def
thf(fact_357_just__cash__def,axiom,
( risk_Free_just_cash
= ( ^ [C2: real] :
( risk_F5458100604530014700ccount
@ ^ [N: nat] : ( if_real @ ( N = zero_zero_nat ) @ C2 @ zero_zero_real ) ) ) ) ).
% just_cash_def
thf(fact_358_zero__account__def,axiom,
( zero_z1425366712893667068ccount
= ( risk_F5458100604530014700ccount
@ ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_account_def
thf(fact_359_sum__abs__ge__zero,axiom,
! [F: nat > real,A4: set_nat] :
( ord_less_eq_real @ zero_zero_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
@ A4 ) ) ).
% sum_abs_ge_zero
thf(fact_360_greater__than__shortest__period__zero,axiom,
! [Alpha: risk_Free_account,M3: nat] :
( ( ord_less_nat @ ( risk_F4612863212915232279period @ Alpha ) @ M3 )
=> ( ( risk_F170160801229183585ccount @ Alpha @ M3 )
= zero_zero_real ) ) ).
% greater_than_shortest_period_zero
thf(fact_361_shortest__period__plus,axiom,
! [Alpha: risk_Free_account,Beta: risk_Free_account] : ( ord_less_eq_nat @ ( risk_F4612863212915232279period @ ( plus_p1863581527469039996ccount @ Alpha @ Beta ) ) @ ( ord_max_nat @ ( risk_F4612863212915232279period @ Alpha ) @ ( risk_F4612863212915232279period @ Beta ) ) ) ).
% shortest_period_plus
thf(fact_362_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_363_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_364_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_365_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_366_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_367_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_368_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_369_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_370_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_371_UNIV__I,axiom,
! [X: nat > real] : ( member_nat_real @ X @ top_top_set_nat_real ) ).
% UNIV_I
thf(fact_372_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_373_atMost__UNIV__triv,axiom,
( ( set_or4236626031148496127et_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% atMost_UNIV_triv
thf(fact_374_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_375_abs__idempotent,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_idempotent
thf(fact_376_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_377_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_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_386_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_387_add__0,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A )
= A ) ).
% add_0
thf(fact_388_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_389_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y2 ) )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_390_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_391_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_392_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_393_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_394_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_395_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_396_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_397_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_398_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_399_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_400_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_401_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_402_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_403_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_404_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_405_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_406_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_407_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_408_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_409_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_410_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_411_add_Oright__neutral,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% add.right_neutral
thf(fact_412_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_413_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_414_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_415_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_416_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_417_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_418_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_419_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_420_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_421_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_422_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_423_abs__eq__0,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_424_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_425_abs__0__eq,axiom,
! [A: real] :
( ( zero_zero_real
= ( abs_abs_real @ A ) )
= ( A = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_426_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_427_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_428_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_429_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_430_abs__add__abs,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
= ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_add_abs
thf(fact_431_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_432_max__less__iff__conj,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X @ Y2 ) @ Z2 )
= ( ( ord_less_nat @ X @ Z2 )
& ( ord_less_nat @ Y2 @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_433_max__less__iff__conj,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_real @ ( ord_max_real @ X @ Y2 ) @ Z2 )
= ( ( ord_less_real @ X @ Z2 )
& ( ord_less_real @ Y2 @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_434_max__less__iff__conj,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_int @ ( ord_max_int @ X @ Y2 ) @ Z2 )
= ( ( ord_less_int @ X @ Z2 )
& ( ord_less_int @ Y2 @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_435_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_436_max_Oabsorb4,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_max_real @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_437_max_Oabsorb4,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_438_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_439_max_Oabsorb3,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_max_real @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_440_max_Oabsorb3,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_441_max__top2,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% max_top2
thf(fact_442_max__top,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% max_top
thf(fact_443_abs__sum__abs,axiom,
! [F: nat > real,A4: set_nat] :
( ( abs_abs_real
@ ( groups6591440286371151544t_real
@ ^ [A2: nat] : ( abs_abs_real @ ( F @ A2 ) )
@ A4 ) )
= ( groups6591440286371151544t_real
@ ^ [A2: nat] : ( abs_abs_real @ ( F @ A2 ) )
@ A4 ) ) ).
% abs_sum_abs
thf(fact_444_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_445_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_446_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_447_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_448_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_449_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_450_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_451_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_452_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_453_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_454_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_455_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_456_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_457_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_458_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_459_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_460_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_461_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_462_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_463_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_464_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_465_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_466_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_467_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_468_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_469_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_470_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_471_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_472_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_473_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_474_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_475_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_476_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_477_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_478_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_479_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_480_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_481_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_482_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_483_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_484_abs__le__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_485_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_486_abs__le__self__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% abs_le_self_iff
thf(fact_487_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_488_abs__of__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_489_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_490_zero__less__abs__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
= ( A != zero_zero_real ) ) ).
% zero_less_abs_iff
thf(fact_491_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_492_sum__abs,axiom,
! [F: nat > real,A4: set_nat] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
@ A4 ) ) ).
% sum_abs
thf(fact_493_abs__of__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_pos
thf(fact_494_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_495_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_496_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_497_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_498_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_499_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_500_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_501_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_502_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_503_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_504_abs__not__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_505_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_506_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_507_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_508_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_509_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_510_nat__neq__iff,axiom,
! [M3: nat,N2: nat] :
( ( M3 != N2 )
= ( ( ord_less_nat @ M3 @ N2 )
| ( ord_less_nat @ N2 @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_511_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_512_less__not__refl2,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ N2 @ M3 )
=> ( M3 != N2 ) ) ).
% less_not_refl2
thf(fact_513_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_514_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_515_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_516_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_517_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_518_UNIV__eq__I,axiom,
! [A4: set_real] :
( ! [X3: real] : ( member_real @ X3 @ A4 )
=> ( top_top_set_real = A4 ) ) ).
% UNIV_eq_I
thf(fact_519_UNIV__eq__I,axiom,
! [A4: set_nat_real] :
( ! [X3: nat > real] : ( member_nat_real @ X3 @ A4 )
=> ( top_top_set_nat_real = A4 ) ) ).
% UNIV_eq_I
thf(fact_520_UNIV__eq__I,axiom,
! [A4: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A4 )
=> ( top_top_set_nat = A4 ) ) ).
% UNIV_eq_I
thf(fact_521_UNIV__witness,axiom,
? [X3: real] : ( member_real @ X3 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_522_UNIV__witness,axiom,
? [X3: nat > real] : ( member_nat_real @ X3 @ top_top_set_nat_real ) ).
% UNIV_witness
thf(fact_523_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_524_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_525_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_526_verit__comp__simplify1_I1_J,axiom,
! [A: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_527_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_528_lt__ex,axiom,
! [X: real] :
? [Y: real] : ( ord_less_real @ Y @ X ) ).
% lt_ex
thf(fact_529_lt__ex,axiom,
! [X: int] :
? [Y: int] : ( ord_less_int @ Y @ X ) ).
% lt_ex
thf(fact_530_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_531_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_532_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_533_dense,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ? [Z5: real] :
( ( ord_less_real @ X @ Z5 )
& ( ord_less_real @ Z5 @ Y2 ) ) ) ).
% dense
thf(fact_534_less__imp__neq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_535_less__imp__neq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_536_less__imp__neq,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_537_less__imp__neq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_538_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_539_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_540_order_Oasym,axiom,
! [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
=> ~ ( ord_le2131251472502387783ccount @ B @ A ) ) ).
% order.asym
thf(fact_541_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_542_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_543_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_544_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_545_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_546_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_547_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_548_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_549_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_550_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_551_antisym__conv3,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_nat @ Y2 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_552_antisym__conv3,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_real @ Y2 @ X )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_553_antisym__conv3,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_int @ Y2 @ X )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_554_linorder__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_555_linorder__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_556_linorder__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_557_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_558_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_559_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_560_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_561_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_562_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_563_dual__order_Oirrefl,axiom,
! [A: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A @ A ) ).
% dual_order.irrefl
thf(fact_564_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_565_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_566_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_567_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_568_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_569_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_570_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_571_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_572_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_573_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_574_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
= ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_575_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_576_not__less__iff__gr__or__eq,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_577_not__less__iff__gr__or__eq,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_578_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_579_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_580_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_581_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_582_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_583_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_584_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_585_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_586_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_587_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_588_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_589_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_590_linorder__neqE,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_591_linorder__neqE,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_592_linorder__neqE,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_593_order__less__asym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_594_order__less__asym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_595_order__less__asym,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y2 )
=> ~ ( ord_le2131251472502387783ccount @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_596_order__less__asym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_597_linorder__neq__iff,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
= ( ( ord_less_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_598_linorder__neq__iff,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
= ( ( ord_less_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_599_linorder__neq__iff,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
= ( ( ord_less_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_600_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_601_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_602_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_603_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_604_order__less__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_605_order__less__trans,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_606_order__less__trans,axiom,
! [X: risk_Free_account,Y2: risk_Free_account,Z2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y2 )
=> ( ( ord_le2131251472502387783ccount @ Y2 @ Z2 )
=> ( ord_le2131251472502387783ccount @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_607_order__less__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_608_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_609_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_610_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,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_611_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_612_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_613_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_614_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,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_615_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_616_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,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_617_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,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_618_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_619_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_620_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,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_621_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_622_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_623_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_624_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,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_625_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_626_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,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_627_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,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_628_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_629_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_630_order__less__irrefl,axiom,
! [X: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ X @ X ) ).
% order_less_irrefl
thf(fact_631_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_632_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,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_633_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,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_634_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,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_635_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,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_636_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,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_637_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_638_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,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_639_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_640_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,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_641_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,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_642_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,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_643_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,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_644_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,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_645_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,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_646_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,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_647_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_648_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,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_le2131251472502387783ccount @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_649_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_650_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,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_651_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,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X3 @ Y )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_652_order__less__not__sym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_653_order__less__not__sym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_654_order__less__not__sym,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y2 )
=> ~ ( ord_le2131251472502387783ccount @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_655_order__less__not__sym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_656_order__less__imp__triv,axiom,
! [X: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_657_order__less__imp__triv,axiom,
! [X: real,Y2: real,P: $o] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_658_order__less__imp__triv,axiom,
! [X: risk_Free_account,Y2: risk_Free_account,P: $o] :
( ( ord_le2131251472502387783ccount @ X @ Y2 )
=> ( ( ord_le2131251472502387783ccount @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_659_order__less__imp__triv,axiom,
! [X: int,Y2: int,P: $o] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_660_linorder__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_661_linorder__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_662_linorder__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_663_order__less__imp__not__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_664_order__less__imp__not__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_665_order__less__imp__not__eq,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_666_order__less__imp__not__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_667_order__less__imp__not__eq2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_668_order__less__imp__not__eq2,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_669_order__less__imp__not__eq2,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_670_order__less__imp__not__eq2,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_671_order__less__imp__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_672_order__less__imp__not__less,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_673_order__less__imp__not__less,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y2 )
=> ~ ( ord_le2131251472502387783ccount @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_674_order__less__imp__not__less,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_675_atMost__eq__UNIV__iff,axiom,
! [X: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X )
= top_top_set_set_nat )
= ( X = top_top_set_nat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_676_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_677_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_678_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_679_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_680_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_681_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_682_add__mono__thms__linordered__field_I5_J,axiom,
! [I: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le2131251472502387783ccount @ I @ J )
& ( ord_le2131251472502387783ccount @ K @ L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_683_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_684_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_685_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_686_add__mono__thms__linordered__field_I2_J,axiom,
! [I: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( I = J )
& ( ord_le2131251472502387783ccount @ K @ L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_687_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_688_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_689_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_690_add__mono__thms__linordered__field_I1_J,axiom,
! [I: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le2131251472502387783ccount @ I @ J )
& ( K = L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_691_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_692_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_693_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_694_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_p1863581527469039996ccount @ I @ K )
= ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_695_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_696_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_697_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_698_group__cancel_Oadd1,axiom,
! [A4: risk_Free_account,K: risk_Free_account,A: risk_Free_account,B: risk_Free_account] :
( ( A4
= ( plus_p1863581527469039996ccount @ K @ A ) )
=> ( ( plus_p1863581527469039996ccount @ A4 @ B )
= ( plus_p1863581527469039996ccount @ K @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_699_group__cancel_Oadd1,axiom,
! [A4: nat,K: nat,A: nat,B: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_700_group__cancel_Oadd1,axiom,
! [A4: real,K: real,A: real,B: real] :
( ( A4
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A4 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_701_group__cancel_Oadd1,axiom,
! [A4: int,K: int,A: int,B: int] :
( ( A4
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A4 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_702_group__cancel_Oadd2,axiom,
! [B4: risk_Free_account,K: risk_Free_account,B: risk_Free_account,A: risk_Free_account] :
( ( B4
= ( plus_p1863581527469039996ccount @ K @ B ) )
=> ( ( plus_p1863581527469039996ccount @ A @ B4 )
= ( plus_p1863581527469039996ccount @ K @ ( plus_p1863581527469039996ccount @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_703_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_704_group__cancel_Oadd2,axiom,
! [B4: real,K: real,B: real,A: real] :
( ( B4
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B4 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_705_group__cancel_Oadd2,axiom,
! [B4: int,K: int,B: int,A: int] :
( ( B4
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B4 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_706_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_707_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_708_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_709_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_710_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_711_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_712_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_713_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_714_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_715_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_716_add_Ocommute,axiom,
( plus_p1863581527469039996ccount
= ( ^ [A2: risk_Free_account,B2: risk_Free_account] : ( plus_p1863581527469039996ccount @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_717_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_718_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_719_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_720_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_721_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_722_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_723_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_724_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_725_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_726_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_727_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_728_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_729_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_730_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_731_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_732_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_733_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_734_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_735_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_736_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_737_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_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_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_746_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_747_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_748_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_749_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_750_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_751_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_752_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_753_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_754_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_755_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_756_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_757_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_758_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_759_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_760_abs__triangle__ineq,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_761_abs__triangle__ineq,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_762_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_763_add__mono__thms__linordered__field_I3_J,axiom,
! [I: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le2131251472502387783ccount @ I @ J )
& ( ord_le4245800335709223507ccount @ K @ L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_764_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_765_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_766_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_767_add__mono__thms__linordered__field_I4_J,axiom,
! [I: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le4245800335709223507ccount @ I @ J )
& ( ord_le2131251472502387783ccount @ K @ L ) )
=> ( ord_le2131251472502387783ccount @ ( plus_p1863581527469039996ccount @ I @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_768_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_769_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_770_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X4: nat] : $true ) ) ).
% UNIV_def
thf(fact_771_dense__eq0__I,axiom,
! [X: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
=> ( X = zero_zero_real ) ) ).
% dense_eq0_I
thf(fact_772_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_773_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_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_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_787_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_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_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_796_abs__ge__self,axiom,
! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% abs_ge_self
thf(fact_797_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_798_abs__le__D1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% abs_le_D1
thf(fact_799_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_800_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_801_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_802_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_803_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_804_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le4245800335709223507ccount @ I @ J )
& ( K = L ) )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ I @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_805_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_806_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_807_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_808_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( I = J )
& ( ord_le4245800335709223507ccount @ K @ L ) )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ I @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_809_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_810_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_811_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_812_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: risk_Free_account,J: risk_Free_account,K: risk_Free_account,L: risk_Free_account] :
( ( ( ord_le4245800335709223507ccount @ I @ J )
& ( ord_le4245800335709223507ccount @ K @ L ) )
=> ( ord_le4245800335709223507ccount @ ( plus_p1863581527469039996ccount @ I @ K ) @ ( plus_p1863581527469039996ccount @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_813_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_814_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_815_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_816_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_817_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_818_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_819_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_820_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_821_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_822_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_823_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_824_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_825_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_826_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_827_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_828_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_829_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_830_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_831_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_832_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_833_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_834_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_835_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_836_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_837_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_838_add_Ogroup__left__neutral,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ zero_z1425366712893667068ccount @ A )
= A ) ).
% add.group_left_neutral
thf(fact_839_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_840_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_841_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_842_add_Ocomm__neutral,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% add.comm_neutral
thf(fact_843_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_844_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_845_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_846_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_847_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_848_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_849_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_850_verit__sum__simplify,axiom,
! [A: risk_Free_account] :
( ( plus_p1863581527469039996ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% verit_sum_simplify
thf(fact_851_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_852_verit__comp__simplify1_I3_J,axiom,
! [B6: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
= ( ord_less_real @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_853_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_854_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_855_leD,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_real @ X @ Y2 ) ) ).
% leD
thf(fact_856_leD,axiom,
! [Y2: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y2 @ X )
=> ~ ( ord_le2131251472502387783ccount @ X @ Y2 ) ) ).
% leD
thf(fact_857_leD,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_nat @ X @ Y2 ) ) ).
% leD
thf(fact_858_leD,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_int @ X @ Y2 ) ) ).
% leD
thf(fact_859_leI,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% leI
thf(fact_860_leI,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% leI
thf(fact_861_leI,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% leI
thf(fact_862_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_863_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_864_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_865_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_866_antisym__conv1,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_867_antisym__conv1,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ~ ( ord_le2131251472502387783ccount @ X @ Y2 )
=> ( ( ord_le4245800335709223507ccount @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_868_antisym__conv1,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_869_antisym__conv1,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_870_antisym__conv2,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_871_antisym__conv2,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y2 )
=> ( ( ~ ( ord_le2131251472502387783ccount @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_872_antisym__conv2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_873_antisym__conv2,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_874_dense__ge,axiom,
! [Z2: real,Y2: real] :
( ! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ord_less_eq_real @ Y2 @ X3 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_ge
thf(fact_875_dense__le,axiom,
! [Y2: real,Z2: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_le
thf(fact_876_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_877_less__le__not__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X4: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y4 )
& ~ ( ord_le4245800335709223507ccount @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_878_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_879_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_880_not__le__imp__less,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_eq_real @ Y2 @ X )
=> ( ord_less_real @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_881_not__le__imp__less,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X )
=> ( ord_less_nat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_882_not__le__imp__less,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_eq_int @ Y2 @ X )
=> ( ord_less_int @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_883_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_884_order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [A2: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_885_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_886_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_887_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_888_order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A2: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_889_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_890_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_891_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_892_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_893_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_894_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_895_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_896_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_897_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_898_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_899_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_900_order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A2: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B2 )
& ~ ( ord_le4245800335709223507ccount @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_901_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_902_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_903_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y2: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y2 @ W ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_904_dense__le__bounded,axiom,
! [X: real,Y2: real,Z2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y2 )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_905_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_906_dual__order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [B2: risk_Free_account,A2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_907_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_908_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_909_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_910_dual__order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B2: risk_Free_account,A2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_911_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_912_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_913_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_914_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_915_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_916_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_917_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_918_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_919_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_920_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_921_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ~ ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_922_dual__order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B2: risk_Free_account,A2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A2 )
& ~ ( ord_le4245800335709223507ccount @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_923_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_924_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_925_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_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_934_order__le__less,axiom,
( ord_le4245800335709223507ccount
= ( ^ [X4: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_935_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_936_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_937_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_938_order__less__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X4: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_939_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_940_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_941_linorder__not__le,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X @ Y2 ) )
= ( ord_less_real @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_942_linorder__not__le,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y2 ) )
= ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_943_linorder__not__le,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X @ Y2 ) )
= ( ord_less_int @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_944_linorder__not__less,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_945_linorder__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_946_linorder__not__less,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_947_order__less__imp__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_948_order__less__imp__le,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y2 )
=> ( ord_le4245800335709223507ccount @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_949_order__less__imp__le,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_950_order__less__imp__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_951_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_952_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_953_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_954_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_955_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_956_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_957_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_958_gr__implies__not0,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_959_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_960_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_961_le__neq__implies__less,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( M3 != N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_962_less__or__eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_963_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N: nat] :
( ( ord_less_nat @ M5 @ N )
| ( M5 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_964_less__imp__le__nat,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_965_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
& ( M5 != N ) ) ) ) ).
% nat_less_le
thf(fact_966_additive__strictly__solvent,axiom,
! [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha_12 )
=> ( ( risk_F1636578016437888323olvent @ Alpha_22 )
=> ( risk_F1636578016437888323olvent @ ( plus_p1863581527469039996ccount @ Alpha_12 @ Alpha_22 ) ) ) ) ).
% additive_strictly_solvent
thf(fact_967_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_968_Rep__account__inverse,axiom,
! [X: risk_Free_account] :
( ( risk_F5458100604530014700ccount @ ( risk_F170160801229183585ccount @ X ) )
= X ) ).
% Rep_account_inverse
thf(fact_969_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
@ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
@ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_970_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M3: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N2 @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_971_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_972_add__is__0,axiom,
! [M3: nat,N2: nat] :
( ( ( plus_plus_nat @ M3 @ N2 )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_973_nat__add__left__cancel__less,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_974_nat__add__left__cancel__le,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_975_add__gr__0,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_976_Rep__account__plus,axiom,
! [Alpha_12: risk_Free_account,Alpha_22: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( plus_p1863581527469039996ccount @ Alpha_12 @ Alpha_22 ) )
= ( ^ [N: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_12 @ N ) @ ( risk_F170160801229183585ccount @ Alpha_22 @ N ) ) ) ) ).
% Rep_account_plus
thf(fact_977_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_978_add__eq__self__zero,axiom,
! [M3: nat,N2: nat] :
( ( ( plus_plus_nat @ M3 @ N2 )
= M3 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_979_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_980_less__add__eq__less,axiom,
! [K: nat,L: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_981_trans__less__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_982_trans__less__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_983_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_984_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_985_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_986_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_987_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_988_add__leE,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M3 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_989_le__add1,axiom,
! [N2: nat,M3: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M3 ) ) ).
% le_add1
thf(fact_990_le__add2,axiom,
! [N2: nat,M3: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M3 @ N2 ) ) ).
% le_add2
thf(fact_991_add__leD1,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% add_leD1
thf(fact_992_add__leD2,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_993_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_994_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_995_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_996_trans__le__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_997_trans__le__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_998_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N: nat] :
? [K4: nat] :
( N
= ( plus_plus_nat @ M5 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_999_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1000_nat__add__max__left,axiom,
! [M3: nat,N2: nat,Q2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M3 @ N2 ) @ Q2 )
= ( ord_max_nat @ ( plus_plus_nat @ M3 @ Q2 ) @ ( plus_plus_nat @ N2 @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_1001_nat__add__max__right,axiom,
! [M3: nat,N2: nat,Q2: nat] :
( ( plus_plus_nat @ M3 @ ( ord_max_nat @ N2 @ Q2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M3 @ N2 ) @ ( plus_plus_nat @ M3 @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_1002_less__account__def,axiom,
( ord_le2131251472502387783ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Alpha_1 @ Alpha_2 )
& ~ ( ord_le4245800335709223507ccount @ Alpha_2 @ Alpha_1 ) ) ) ) ).
% less_account_def
thf(fact_1003_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1004_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K: nat] :
( ! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K ) @ ( F @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1005_plus__account__def,axiom,
( plus_p1863581527469039996ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N: nat] : ( plus_plus_real @ ( risk_F170160801229183585ccount @ Alpha_1 @ N ) @ ( risk_F170160801229183585ccount @ Alpha_2 @ N ) ) ) ) ) ).
% plus_account_def
thf(fact_1006_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
= ( P @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1007_Abs__account__inverse,axiom,
! [Y2: nat > real] :
( ( member_nat_real @ Y2 @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ( risk_F170160801229183585ccount @ ( risk_F5458100604530014700ccount @ Y2 ) )
= Y2 ) ) ).
% Abs_account_inverse
thf(fact_1008_Rep__account__induct,axiom,
! [Y2: nat > real,P: ( nat > real ) > $o] :
( ( member_nat_real @ Y2 @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ! [X3: risk_Free_account] : ( P @ ( risk_F170160801229183585ccount @ X3 ) )
=> ( P @ Y2 ) ) ) ).
% Rep_account_induct
thf(fact_1009_Rep__account__cases,axiom,
! [Y2: nat > real] :
( ( member_nat_real @ Y2 @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ~ ! [X3: risk_Free_account] :
( Y2
!= ( risk_F170160801229183585ccount @ X3 ) ) ) ).
% Rep_account_cases
thf(fact_1010_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_1011_Abs__account__inject,axiom,
! [X: nat > real,Y2: nat > real] :
( ( member_nat_real @ X @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ( member_nat_real @ Y2 @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ( ( risk_F5458100604530014700ccount @ X )
= ( risk_F5458100604530014700ccount @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% Abs_account_inject
thf(fact_1012_Abs__account__induct,axiom,
! [P: risk_Free_account > $o,X: risk_Free_account] :
( ! [Y: nat > real] :
( ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( P @ ( risk_F5458100604530014700ccount @ Y ) ) )
=> ( P @ X ) ) ).
% Abs_account_induct
thf(fact_1013_Abs__account__cases,axiom,
! [X: risk_Free_account] :
~ ! [Y: nat > real] :
( ( X
= ( risk_F5458100604530014700ccount @ Y ) )
=> ~ ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) ) ) ).
% Abs_account_cases
thf(fact_1014_type__definition__account,axiom,
type_d8982087200295354172t_real @ risk_F170160801229183585ccount @ risk_F5458100604530014700ccount @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) ).
% type_definition_account
thf(fact_1015_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
@ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1016_num__of__nat__plus__distrib,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( num_of_nat @ ( plus_plus_nat @ M3 @ N2 ) )
= ( plus_plus_num @ ( num_of_nat @ M3 ) @ ( num_of_nat @ N2 ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_1017_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1018_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1019_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1020_Suc__less__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_less_eq
thf(fact_1021_Suc__mono,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1022_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1023_Suc__le__mono,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% Suc_le_mono
thf(fact_1024_add__Suc__right,axiom,
! [M3: nat,N2: nat] :
( ( plus_plus_nat @ M3 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M3 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1025_max__Suc__Suc,axiom,
! [M3: nat,N2: nat] :
( ( ord_max_nat @ ( suc @ M3 ) @ ( suc @ N2 ) )
= ( suc @ ( ord_max_nat @ M3 @ N2 ) ) ) ).
% max_Suc_Suc
thf(fact_1026_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1027_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1028_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1029_add__Suc,axiom,
! [M3: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M3 @ N2 ) ) ) ).
% add_Suc
thf(fact_1030_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1031_add__Suc__shift,axiom,
! [M3: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N2 )
= ( plus_plus_nat @ M3 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1032_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1033_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1034_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_1035_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1036_int__Suc,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1037_int__plus,axiom,
! [N2: nat,M3: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M3 ) ) ) ).
% int_plus
thf(fact_1038_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_1039_Suc__inject,axiom,
! [X: nat,Y2: nat] :
( ( ( suc @ X )
= ( suc @ Y2 ) )
=> ( X = Y2 ) ) ).
% Suc_inject
thf(fact_1040_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1041_transitive__stepwise__le,axiom,
! [M3: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y: nat,Z5: nat] :
( ( R @ X3 @ Y )
=> ( ( R @ Y @ Z5 )
=> ( R @ X3 @ Z5 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M3 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1042_nat__induct__at__least,axiom,
! [M3: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( P @ M3 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1043_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1044_not__less__eq__eq,axiom,
! [M3: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_1045_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1046_le__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M3 @ N2 )
| ( M3
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1047_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M2: nat] :
( M6
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_1048_le__SucI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1049_le__SucE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N2 )
=> ( M3
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1050_Suc__leD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% Suc_leD
thf(fact_1051_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M2: nat] :
( N2
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_1052_Zero__not__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_not_Suc
thf(fact_1053_Zero__neq__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_neq_Suc
thf(fact_1054_Suc__neq__Zero,axiom,
! [M3: nat] :
( ( suc @ M3 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1055_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1056_diff__induct,axiom,
! [P: nat > nat > $o,M3: nat,N2: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X3: nat,Y: nat] :
( ( P @ X3 @ Y )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y ) ) )
=> ( P @ M3 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_1057_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_1058_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1059_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1060_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1061_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1062_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1063_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1064_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1065_zle__int,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% zle_int
thf(fact_1066_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1067_Suc__lessD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_lessD
thf(fact_1068_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1069_Suc__lessI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ( ( suc @ M3 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M3 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1070_less__SucE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M3 @ N2 )
=> ( M3 = N2 ) ) ) ).
% less_SucE
thf(fact_1071_less__SucI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ M3 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1072_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ N2 )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1073_less__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1074_not__less__eq,axiom,
! [M3: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M3 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M3 ) ) ) ).
% not_less_eq
thf(fact_1075_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_1076_Suc__less__eq2,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M3 )
= ( ? [M7: nat] :
( ( M3
= ( suc @ M7 ) )
& ( ord_less_nat @ N2 @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1077_less__antisym,axiom,
! [N2: nat,M3: nat] :
( ~ ( ord_less_nat @ N2 @ M3 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
=> ( M3 = N2 ) ) ) ).
% less_antisym
thf(fact_1078_Suc__less__SucD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_1079_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1080_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I2 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1081_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1082_not__less__less__Suc__eq,axiom,
! [N2: nat,M3: nat] :
( ~ ( ord_less_nat @ N2 @ M3 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
= ( N2 = M3 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1083_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1084_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1085_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_1086_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1087_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N: nat,M5: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).
% nat_less_real_le
thf(fact_1088_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1089_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_1090_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_1091_less__Suc__eq__0__disj,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ( M3 = zero_zero_nat )
| ? [J2: nat] :
( ( M3
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1092_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M2: nat] :
( N2
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_1093_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1094_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1095_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1096_Suc__leI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 ) ) ).
% Suc_leI
thf(fact_1097_Suc__le__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_le_eq
thf(fact_1098_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1099_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1100_Suc__le__lessD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1101_le__less__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
= ( N2 = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_1102_less__Suc__eq__le,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1103_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1104_le__imp__less__Suc,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_nat @ M3 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1105_one__is__add,axiom,
! [M3: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M3 @ N2 ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1106_add__is__1,axiom,
! [M3: nat,N2: nat] :
( ( ( plus_plus_nat @ M3 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1107_less__imp__Suc__add,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1108_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N: nat] :
? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M5 @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1109_less__add__Suc2,axiom,
! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M3 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1110_less__add__Suc1,axiom,
! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M3 ) ) ) ).
% less_add_Suc1
thf(fact_1111_less__natE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ~ ! [Q3: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M3 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1112_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N2 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1113_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1114_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1115_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1116_Suc__diff__diff,axiom,
! [M3: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M3 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1117_diff__Suc__Suc,axiom,
! [M3: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M3 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1118_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1119_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1120_zero__less__diff,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M3 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% zero_less_diff
thf(fact_1121_diff__is__0__eq,axiom,
! [M3: nat,N2: nat] :
( ( ( minus_minus_nat @ M3 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1122_diff__is__0__eq_H,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( minus_minus_nat @ M3 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1123_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1124_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1125_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1126_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_1127_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1128_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1129_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1130_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1131_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1132_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1133_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1134_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1135_Nat_Odiff__cancel,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1136_diff__cancel2,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ).
% diff_cancel2
thf(fact_1137_diff__add__inverse,axiom,
! [N2: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M3 ) @ N2 )
= M3 ) ).
% diff_add_inverse
thf(fact_1138_diff__add__inverse2,axiom,
! [M3: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N2 ) @ N2 )
= M3 ) ).
% diff_add_inverse2
thf(fact_1139_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_1140_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A2 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1141_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1142_diff__Suc__eq__diff__pred,axiom,
! [M3: nat,N2: nat] :
( ( minus_minus_nat @ M3 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1143_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1144_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_1145_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_1146_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1147_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1148_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1149_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1150_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1151_diff__less__mono2,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_1152_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1153_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_1154_diffs0__imp__equal,axiom,
! [M3: nat,N2: nat] :
( ( ( minus_minus_nat @ M3 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M3 )
= zero_zero_nat )
=> ( M3 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1155_eq__diff__iff,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M3 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1156_le__diff__iff,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1157_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1158_diff__le__mono,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1159_diff__le__self,axiom,
! [M3: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N2 ) @ M3 ) ).
% diff_le_self
thf(fact_1160_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_1161_diff__le__mono2,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1162_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1163_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1164_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1165_diff__less,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N2 ) @ M3 ) ) ) ).
% diff_less
thf(fact_1166_Suc__diff__Suc,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ N2 @ M3 )
=> ( ( suc @ ( minus_minus_nat @ M3 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1167_diff__less__Suc,axiom,
! [M3: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M3 @ N2 ) @ ( suc @ M3 ) ) ).
% diff_less_Suc
thf(fact_1168_Suc__diff__le,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_nat @ N2 @ M3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M3 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1169_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_1170_less__diff__iff,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M3 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1171_diff__add__0,axiom,
! [N2: nat,M3: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1172_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1173_add__diff__inverse__nat,axiom,
! [M3: nat,N2: nat] :
( ~ ( ord_less_nat @ M3 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M3 @ N2 ) )
= M3 ) ) ).
% add_diff_inverse_nat
thf(fact_1174_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1175_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1176_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1177_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1178_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1179_nat__minus__add__max,axiom,
! [N2: nat,M3: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M3 ) @ M3 )
= ( ord_max_nat @ N2 @ M3 ) ) ).
% nat_minus_add_max
thf(fact_1180_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1181_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N2 )
= ( minus_minus_nat @ M3 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1182_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1183_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N ) ) ) ) ) ).
% add_eq_if
thf(fact_1184_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 ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_1185_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 ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1186_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1187_nat0__intermed__int__val,axiom,
! [N2: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1188_mult__is__0,axiom,
! [M3: nat,N2: nat] :
( ( ( times_times_nat @ M3 @ N2 )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1189_mult__0__right,axiom,
! [M3: nat] :
( ( times_times_nat @ M3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1190_mult__cancel1,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ( times_times_nat @ K @ M3 )
= ( times_times_nat @ K @ N2 ) )
= ( ( M3 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1191_mult__cancel2,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M3 @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M3 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1192_nat__mult__eq__1__iff,axiom,
! [M3: nat,N2: nat] :
( ( ( times_times_nat @ M3 @ N2 )
= one_one_nat )
= ( ( M3 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1193_nat__1__eq__mult__iff,axiom,
! [M3: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M3 @ N2 ) )
= ( ( M3 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1194_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_1195_one__eq__mult__iff,axiom,
! [M3: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M3 @ N2 ) )
= ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1196_mult__eq__1__iff,axiom,
! [M3: nat,N2: nat] :
( ( ( times_times_nat @ M3 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1197_nat__0__less__mult__iff,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M3 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1198_mult__less__cancel2,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M3 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1199_mult__Suc__right,axiom,
! [M3: nat,N2: nat] :
( ( times_times_nat @ M3 @ ( suc @ N2 ) )
= ( plus_plus_nat @ M3 @ ( times_times_nat @ M3 @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_1200_one__le__mult__iff,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M3 @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M3 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_1201_mult__le__cancel2,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1202_add__mult__distrib,axiom,
! [M3: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M3 @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1203_add__mult__distrib2,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M3 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1204_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1205_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1206_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1207_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1208_plusinfinity,axiom,
! [D: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K3: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z: int] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1209_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K3: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1210_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1211_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1212_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1213_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1214_le__cube,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ) ).
% le_cube
thf(fact_1215_le__square,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ).
% le_square
thf(fact_1216_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1217_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1218_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1219_diff__mult__distrib,axiom,
! [M3: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M3 @ N2 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1220_diff__mult__distrib2,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M3 @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1221_nat__mult__max__right,axiom,
! [M3: nat,N2: nat,Q2: nat] :
( ( times_times_nat @ M3 @ ( ord_max_nat @ N2 @ Q2 ) )
= ( ord_max_nat @ ( times_times_nat @ M3 @ N2 ) @ ( times_times_nat @ M3 @ Q2 ) ) ) ).
% nat_mult_max_right
thf(fact_1222_nat__mult__max__left,axiom,
! [M3: nat,N2: nat,Q2: nat] :
( ( times_times_nat @ ( ord_max_nat @ M3 @ N2 ) @ Q2 )
= ( ord_max_nat @ ( times_times_nat @ M3 @ Q2 ) @ ( times_times_nat @ N2 @ Q2 ) ) ) ).
% nat_mult_max_left
thf(fact_1223_Suc__mult__cancel1,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M3 )
= ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( M3 = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_1224_decr__lemma,axiom,
! [D: int,X: int,Z2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_1225_incr__lemma,axiom,
! [D: int,Z2: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z2 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).
% incr_lemma
thf(fact_1226_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1227_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1228_Suc__mult__less__cancel1,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M3 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1229_Suc__mult__le__cancel1,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M3 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1230_mult__Suc,axiom,
! [M3: nat,N2: nat] :
( ( times_times_nat @ ( suc @ M3 ) @ N2 )
= ( plus_plus_nat @ N2 @ ( times_times_nat @ M3 @ N2 ) ) ) ).
% mult_Suc
thf(fact_1231_mult__eq__self__implies__10,axiom,
! [M3: nat,N2: nat] :
( ( M3
= ( times_times_nat @ M3 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M3 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1232_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1233_valid__transfer__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha2: risk_Free_account,Tau2: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Tau2 )
& ( risk_F1636578016437888323olvent @ ( minus_4846202936726426316ccount @ Alpha2 @ Tau2 ) ) ) ) ) ).
% valid_transfer_def
thf(fact_1234_one__less__mult,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M3 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M3 @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1235_n__less__m__mult__n,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M3 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M3 @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1236_n__less__n__mult__m,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M3 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M3 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1237_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1238_pos__zmult__eq__1__iff,axiom,
! [M3: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M3 )
=> ( ( ( times_times_int @ M3 @ N2 )
= one_one_int )
= ( ( M3 = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1239_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_1240_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1241_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M5: nat,N: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N ) ) ) ) ) ).
% mult_eq_if
thf(fact_1242_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1243_nat__intermed__int__val,axiom,
! [M3: nat,N2: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M3 @ I2 )
& ( ord_less_nat @ I2 @ N2 ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_eq_int @ ( F @ M3 ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M3 @ I2 )
& ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1244_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y2: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y2 ) ) )
= ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
& ( ( ord_less_nat @ X @ Y2 )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1245_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1246_nat__ivt__aux,axiom,
! [N2: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1247_nat__mult__le__cancel__disj,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1248_nat__mult__less__cancel__disj,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M3 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1249_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ( times_times_nat @ K @ M3 )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M3 = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1250_nat__mult__less__cancel1,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1251_nat__mult__eq__cancel1,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M3 )
= ( times_times_nat @ K @ N2 ) )
= ( M3 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1252_nat__mult__le__cancel1,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1253_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M3 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U2 ) @ M3 )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1254_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M3 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N2 ) )
= ( M3
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U2 ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1255_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U2 ) @ M3 ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1256_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N2 ) )
= ( ord_less_eq_nat @ M3 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U2 ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1257_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U2 ) @ M3 ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1258_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N2 ) )
= ( minus_minus_nat @ M3 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U2 ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1259_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N2 ) )
= ( ord_less_nat @ M3 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U2 ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1260_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U2 ) @ M3 ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1261_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1262_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A3: real,B3: real,C3: real] :
( ( P @ A3 @ B3 )
=> ( ( P @ B3 @ C3 )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( P @ A3 @ C3 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A3: real,B3: real] :
( ( ( ord_less_eq_real @ A3 @ X3 )
& ( ord_less_eq_real @ X3 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D3 ) )
=> ( P @ A3 @ B3 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1263_lemma__interval,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D4 )
=> ( ( ord_less_eq_real @ A @ Y5 )
& ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% lemma_interval
thf(fact_1264_lemma__interval__lt,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D4 )
=> ( ( ord_less_real @ A @ Y5 )
& ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_1265_sin__bound__lemma,axiom,
! [X: real,Y2: real,U2: real,V: real] :
( ( X = Y2 )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ U2 ) @ V )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U2 ) @ Y2 ) ) @ V ) ) ) ).
% sin_bound_lemma
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $true @ X @ Y2 )
= 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,Y2: risk_Free_account] :
( ( if_Risk_Free_account @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Risk____Free____Lending__Oaccount_T,axiom,
! [X: risk_Free_account,Y2: risk_Free_account] :
( ( if_Risk_Free_account @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_real @ ( risk_F2906766666041932210_value @ alpha ) @ ( risk_F2906766666041932210_value @ beta ) ).
%------------------------------------------------------------------------------