TPTP Problem File: SLH0879^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Risk_Free_Lending/0000_Risk_Free_Lending/prob_00945_028745__5917380_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1354 ( 566 unt; 80 typ; 0 def)
% Number of atoms : 3688 (1183 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9845 ( 336 ~; 90 |; 116 &;7733 @)
% ( 0 <=>;1570 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 316 ( 316 >; 0 *; 0 +; 0 <<)
% Number of symbols : 76 ( 73 usr; 11 con; 0-3 aty)
% Number of variables : 3219 ( 189 ^;2916 !; 114 ?;3219 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:57:26.284
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $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__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (73)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
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__Complex__Ocomplex,type,
one_one_complex: complex ).
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__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
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_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
uminus1482373934393186551omplex: complex > complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Risk____Free____Lending__Oaccount,type,
uminus3377898441596595772ccount: risk_Free_account > risk_Free_account ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
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_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_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
semiri8010041392384452111omplex: nat > complex ).
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__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
neg_nu6511756317524482435omplex: complex > complex ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
neg_nu6075765906172075777c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
neg_nu8557863876264182079omplex: complex > complex ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_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__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_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
real_V1022390504157884413omplex: complex > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
real_V7735802525324610683m_real: real > real ).
thf(sy_c_Real__Vector__Spaces_Orepresentation_001t__Complex__Ocomplex,type,
real_V4712799919980427094omplex: set_complex > complex > complex > real ).
thf(sy_c_Real__Vector__Spaces_Orepresentation_001t__Real__Oreal,type,
real_V2383402355066202452n_real: set_real > real > real > real ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
real_V2046097035970521341omplex: real > complex > complex ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
real_V1485227260804924795R_real: real > real > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
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_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
topolo6517432010174082258omplex: ( nat > complex ) > $o ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
topolo4055970368930404560y_real: ( nat > real ) > $o ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
powr_real: real > real > real ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
% Relevant facts (1268)
thf(fact_0_Rep__account__zero,axiom,
( ( risk_F170160801229183585ccount @ zero_z1425366712893667068ccount )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% Rep_account_zero
thf(fact_1_zero__account__alt__def,axiom,
( ( risk_Free_just_cash @ zero_zero_real )
= zero_z1425366712893667068ccount ) ).
% zero_account_alt_def
thf(fact_2_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_3_zero__reorient,axiom,
! [X: risk_Free_account] :
( ( zero_z1425366712893667068ccount = X )
= ( X = zero_z1425366712893667068ccount ) ) ).
% zero_reorient
thf(fact_4_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_5_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_6_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_7_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_8_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_9_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_10_representation__zero,axiom,
! [Basis: set_complex] :
( ( real_V4712799919980427094omplex @ Basis @ zero_zero_complex )
= ( ^ [B: complex] : zero_zero_real ) ) ).
% representation_zero
thf(fact_11_representation__zero,axiom,
! [Basis: set_real] :
( ( real_V2383402355066202452n_real @ Basis @ zero_zero_real )
= ( ^ [B: real] : zero_zero_real ) ) ).
% representation_zero
thf(fact_12_norm__zero,axiom,
( ( real_V7735802525324610683m_real @ zero_zero_real )
= zero_zero_real ) ).
% norm_zero
thf(fact_13_norm__zero,axiom,
( ( real_V1022390504157884413omplex @ zero_zero_complex )
= zero_zero_real ) ).
% norm_zero
thf(fact_14_norm__eq__zero,axiom,
! [X: real] :
( ( ( real_V7735802525324610683m_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_15_norm__eq__zero,axiom,
! [X: complex] :
( ( ( real_V1022390504157884413omplex @ X )
= zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_eq_zero
thf(fact_16_scaleR__eq__0__iff,axiom,
! [A: real,X: complex] :
( ( ( real_V2046097035970521341omplex @ A @ X )
= zero_zero_complex )
= ( ( A = zero_zero_real )
| ( X = zero_zero_complex ) ) ) ).
% scaleR_eq_0_iff
thf(fact_17_scaleR__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( real_V1485227260804924795R_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% scaleR_eq_0_iff
thf(fact_18_scaleR__zero__left,axiom,
! [X: complex] :
( ( real_V2046097035970521341omplex @ zero_zero_real @ X )
= zero_zero_complex ) ).
% scaleR_zero_left
thf(fact_19_scaleR__zero__left,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% scaleR_zero_left
thf(fact_20_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_21_scaleR__cancel__right,axiom,
! [A: real,X: complex,B2: real] :
( ( ( real_V2046097035970521341omplex @ A @ X )
= ( real_V2046097035970521341omplex @ B2 @ X ) )
= ( ( A = B2 )
| ( X = zero_zero_complex ) ) ) ).
% scaleR_cancel_right
thf(fact_22_scaleR__cancel__right,axiom,
! [A: real,X: real,B2: real] :
( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ B2 @ X ) )
= ( ( A = B2 )
| ( X = zero_zero_real ) ) ) ).
% scaleR_cancel_right
thf(fact_23_scaleR__zero__right,axiom,
! [A: real] :
( ( real_V2046097035970521341omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% scaleR_zero_right
thf(fact_24_scaleR__zero__right,axiom,
! [A: real] :
( ( real_V1485227260804924795R_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% scaleR_zero_right
thf(fact_25_scaleR__cancel__left,axiom,
! [A: real,X: real,Y: real] :
( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ A @ Y ) )
= ( ( X = Y )
| ( A = zero_zero_real ) ) ) ).
% scaleR_cancel_left
thf(fact_26_Rep__account__just__cash,axiom,
! [C: real] :
( ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ C ) )
= ( ^ [N2: nat] : ( if_real @ ( N2 = zero_zero_nat ) @ C @ zero_zero_real ) ) ) ).
% Rep_account_just_cash
thf(fact_27_norm__le__zero__iff,axiom,
! [X: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_le_zero_iff
thf(fact_28_norm__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_le_zero_iff
thf(fact_29_Rep__account__inject,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ( risk_F170160801229183585ccount @ X )
= ( risk_F170160801229183585ccount @ Y ) )
= ( X = Y ) ) ).
% Rep_account_inject
thf(fact_30_norm__ge__zero,axiom,
! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% norm_ge_zero
thf(fact_31_norm__ge__zero,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) ) ).
% norm_ge_zero
thf(fact_32_scaleR__left__mono,axiom,
! [X: real,Y: real,A: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ) ).
% scaleR_left_mono
thf(fact_33_scaleR__right__mono,axiom,
! [A: real,B2: real,X: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B2 @ X ) ) ) ) ).
% scaleR_right_mono
thf(fact_34_just__cash__embed,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A2: real,B: real] :
( ( risk_Free_just_cash @ A2 )
= ( risk_Free_just_cash @ B ) ) ) ) ).
% just_cash_embed
thf(fact_35_scaleR__left__commute,axiom,
! [A: real,B2: real,X: real] :
( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B2 @ X ) )
= ( real_V1485227260804924795R_real @ B2 @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ).
% scaleR_left_commute
thf(fact_36_scaleR__left__mono__neg,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B2 ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_37_scaleR__right__mono__neg,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B2 @ C ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_38_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_39_scaleR__nonpos__nonpos,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B2 ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_40_scaleR__nonpos__nonneg,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_41_scaleR__nonneg__nonpos,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_42_scaleR__nonneg__nonneg,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_43_split__scaleR__pos__le,axiom,
! [A: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B2 ) ) ) ).
% split_scaleR_pos_le
thf(fact_44_split__scaleR__neg__le,axiom,
! [A: real,X: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ X @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ X ) ) )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ).
% split_scaleR_neg_le
thf(fact_45_scaleR__mono_H,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B2 @ D ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_46_scaleR__mono,axiom,
! [A: real,B2: real,X: real,Y: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B2 @ Y ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_47_scaleR__right__imp__eq,axiom,
! [X: complex,A: real,B2: real] :
( ( X != zero_zero_complex )
=> ( ( ( real_V2046097035970521341omplex @ A @ X )
= ( real_V2046097035970521341omplex @ B2 @ X ) )
=> ( A = B2 ) ) ) ).
% scaleR_right_imp_eq
thf(fact_48_scaleR__right__imp__eq,axiom,
! [X: real,A: real,B2: real] :
( ( X != zero_zero_real )
=> ( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ B2 @ X ) )
=> ( A = B2 ) ) ) ).
% scaleR_right_imp_eq
thf(fact_49_scaleR__left__imp__eq,axiom,
! [A: real,X: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ A @ Y ) )
=> ( X = Y ) ) ) ).
% scaleR_left_imp_eq
thf(fact_50_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_51_representation__ne__zero,axiom,
! [Basis: set_real,V: real,B2: real] :
( ( ( real_V2383402355066202452n_real @ Basis @ V @ B2 )
!= zero_zero_real )
=> ( member_real @ B2 @ Basis ) ) ).
% representation_ne_zero
thf(fact_52_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_53_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_54_order__refl,axiom,
! [X: risk_Free_account] : ( ord_le4245800335709223507ccount @ X @ X ) ).
% order_refl
thf(fact_55_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_56_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_57_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_58_dual__order_Orefl,axiom,
! [A: risk_Free_account] : ( ord_le4245800335709223507ccount @ A @ A ) ).
% dual_order.refl
thf(fact_59_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_60_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_61_just__cash__valid__transfer,axiom,
! [C: real,T: real] :
( ( risk_F1023690899723030139ansfer @ ( risk_Free_just_cash @ C ) @ ( risk_Free_just_cash @ T ) )
= ( ( ord_less_eq_real @ zero_zero_real @ T )
& ( ord_less_eq_real @ T @ C ) ) ) ).
% just_cash_valid_transfer
thf(fact_62_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_63_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_64_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_65_cash__reserve__def,axiom,
( risk_F1914734008469130493eserve
= ( ^ [Alpha2: risk_Free_account] : ( risk_F170160801229183585ccount @ Alpha2 @ zero_zero_nat ) ) ) ).
% cash_reserve_def
thf(fact_66_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A3: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_68_scaleR__left__le__one__le,axiom,
! [X: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ X ) ) ) ).
% scaleR_left_le_one_le
thf(fact_69_zero__le__scaleR__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( A = zero_zero_real ) ) ) ).
% zero_le_scaleR_iff
thf(fact_70_scaleR__le__0__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( A = zero_zero_real ) ) ) ).
% scaleR_le_0_iff
thf(fact_71_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_72_zero__less__norm__iff,axiom,
! [X: complex] :
( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
= ( X != zero_zero_complex ) ) ).
% zero_less_norm_iff
thf(fact_73_zero__less__norm__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
= ( X != zero_zero_real ) ) ).
% zero_less_norm_iff
thf(fact_74_complete__real,axiom,
! [S: set_real] :
( ? [X3: real] : ( member_real @ X3 @ S )
=> ( ? [Z2: real] :
! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z2 ) )
=> ? [Y3: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Y3 ) )
& ! [Z2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z2 ) )
=> ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ) ) ).
% complete_real
thf(fact_75_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_76_scaleR__one,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ one_one_real @ X )
= X ) ).
% scaleR_one
thf(fact_77_norm__one,axiom,
( ( real_V1022390504157884413omplex @ one_one_complex )
= one_one_real ) ).
% norm_one
thf(fact_78_norm__one,axiom,
( ( real_V7735802525324610683m_real @ one_one_real )
= one_one_real ) ).
% norm_one
thf(fact_79_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_80_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_81_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_82_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_83_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_84_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_85_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_86_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_87_order__less__imp__not__less,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ~ ( ord_le2131251472502387783ccount @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_88_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_89_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_90_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_91_order__less__imp__not__eq2,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_92_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_93_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_94_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_95_order__less__imp__not__eq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_96_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_97_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_98_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_99_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_100_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_101_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_102_order__less__imp__triv,axiom,
! [X: risk_Free_account,Y: risk_Free_account,P: $o] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_103_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_104_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_105_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_106_order__less__not__sym,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ~ ( ord_le2131251472502387783ccount @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_107_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_108_order__less__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_109_order__less__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_110_order__less__subst2,axiom,
! [A: real,B2: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_111_order__less__subst2,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_112_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_113_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_114_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_115_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_116_order__less__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_117_order__less__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_118_order__less__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_119_order__less__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_120_order__less__subst1,axiom,
! [A: real,F: risk_Free_account > real,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_121_order__less__subst1,axiom,
! [A: real,F: int > real,B2: int,C: int] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_122_order__less__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_123_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_124_order__less__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_125_order__less__subst1,axiom,
! [A: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_126_order__less__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B2: real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_127_order__less__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B2: nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_128_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_129_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_130_order__less__irrefl,axiom,
! [X: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ X @ X ) ).
% order_less_irrefl
thf(fact_131_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_132_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_133_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_134_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_135_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_136_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_137_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_138_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_139_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_140_ord__less__eq__subst,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_141_ord__less__eq__subst,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_142_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_143_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_144_ord__eq__less__subst,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_145_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_146_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_147_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_148_ord__eq__less__subst,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_149_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_150_ord__eq__less__subst,axiom,
! [A: real,F: risk_Free_account > real,B2: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_151_ord__eq__less__subst,axiom,
! [A: nat,F: risk_Free_account > nat,B2: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_152_order__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_153_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_154_order__less__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ Z3 )
=> ( ord_le2131251472502387783ccount @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_155_order__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_156_order__less__asym_H,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ~ ( ord_less_real @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_157_order__less__asym_H,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_158_order__less__asym_H,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ~ ( ord_le2131251472502387783ccount @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_159_order__less__asym_H,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ~ ( ord_less_int @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_160_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_161_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_162_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_163_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_164_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_165_order__less__asym,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ~ ( ord_le2131251472502387783ccount @ Y @ X ) ) ).
% order_less_asym
thf(fact_166_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_167_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_168_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_169_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_170_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_171_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_172_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_173_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_174_order_Ostrict__implies__not__eq,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_175_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_176_order_Ostrict__implies__not__eq,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_177_order_Ostrict__implies__not__eq,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_178_dual__order_Ostrict__trans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_179_dual__order_Ostrict__trans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_180_dual__order_Ostrict__trans,axiom,
! [B2: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A )
=> ( ( ord_le2131251472502387783ccount @ C @ B2 )
=> ( ord_le2131251472502387783ccount @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_181_dual__order_Ostrict__trans,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_182_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_183_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_184_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_185_order_Ostrict__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_186_order_Ostrict__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_187_order_Ostrict__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_188_order_Ostrict__trans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_189_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B2: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_190_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_191_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B2: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_192_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_193_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_194_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_195_dual__order_Oirrefl,axiom,
! [A: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A @ A ) ).
% dual_order.irrefl
thf(fact_196_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_197_dual__order_Oasym,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ~ ( ord_less_real @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_198_dual__order_Oasym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ~ ( ord_less_nat @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_199_dual__order_Oasym,axiom,
! [B2: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A )
=> ~ ( ord_le2131251472502387783ccount @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_200_dual__order_Oasym,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ~ ( ord_less_int @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_201_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_202_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_203_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_204_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_205_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_206_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_207_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X4 )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_208_ord__less__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_209_ord__less__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_210_ord__less__eq__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_211_ord__less__eq__trans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_212_ord__eq__less__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_213_ord__eq__less__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_214_ord__eq__less__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( A = B2 )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_215_ord__eq__less__trans,axiom,
! [A: int,B2: int,C: int] :
( ( A = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_216_order_Oasym,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ~ ( ord_less_real @ B2 @ A ) ) ).
% order.asym
thf(fact_217_order_Oasym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order.asym
thf(fact_218_order_Oasym,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ~ ( ord_le2131251472502387783ccount @ B2 @ A ) ) ).
% order.asym
thf(fact_219_order_Oasym,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ~ ( ord_less_int @ B2 @ A ) ) ).
% order.asym
thf(fact_220_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_221_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_222_less__imp__neq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_223_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_224_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z4: real] :
( ( ord_less_real @ X @ Z4 )
& ( ord_less_real @ Z4 @ Y ) ) ) ).
% dense
thf(fact_225_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_226_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_227_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_228_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_229_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_230_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_231_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_232_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_233_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_234_order__le__imp__less__or__eq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_235_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_236_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_237_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_238_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_239_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_240_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_241_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_242_order__less__le__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_243_order__less__le__subst2,axiom,
! [A: int,B2: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_244_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_245_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_246_order__less__le__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_247_order__less__le__subst2,axiom,
! [A: int,B2: int,F: int > risk_Free_account,C: risk_Free_account] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_248_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_249_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_250_order__less__le__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_251_order__less__le__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B2: real,C: real] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_252_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_253_order__less__le__subst1,axiom,
! [A: int,F: real > int,B2: real,C: real] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_254_order__less__le__subst1,axiom,
! [A: real,F: risk_Free_account > real,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_255_order__less__le__subst1,axiom,
! [A: risk_Free_account,F: risk_Free_account > risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_256_order__less__le__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_257_order__less__le__subst1,axiom,
! [A: int,F: risk_Free_account > int,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_258_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_259_order__less__le__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B2: nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_260_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_261_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_262_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_263_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_264_order__le__less__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_265_order__le__less__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_266_order__le__less__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_267_order__le__less__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > int,C: int] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_268_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_269_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_270_order__le__less__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_271_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_272_order__le__less__subst1,axiom,
! [A: real,F: risk_Free_account > real,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_273_order__le__less__subst1,axiom,
! [A: real,F: int > real,B2: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_274_order__le__less__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B2: real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_275_order__le__less__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B2: nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_276_order__le__less__subst1,axiom,
! [A: risk_Free_account,F: risk_Free_account > risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B2 ) )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_277_order__le__less__subst1,axiom,
! [A: risk_Free_account,F: int > risk_Free_account,B2: int,C: int] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_278_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_279_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_280_order__less__le__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_281_order__less__le__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ Z3 )
=> ( ord_le2131251472502387783ccount @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_282_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_283_order__less__le__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_284_order__le__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_285_order__le__less__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ Z3 )
=> ( ord_le2131251472502387783ccount @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_286_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_287_order__le__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_288_order__neq__le__trans,axiom,
! [A: real,B2: real] :
( ( A != B2 )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_289_order__neq__le__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( A != B2 )
=> ( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ord_le2131251472502387783ccount @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_290_order__neq__le__trans,axiom,
! [A: nat,B2: nat] :
( ( A != B2 )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_291_order__neq__le__trans,axiom,
! [A: int,B2: int] :
( ( A != B2 )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_292_order__le__neq__trans,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_293_order__le__neq__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_le2131251472502387783ccount @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_294_order__le__neq__trans,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_295_order__le__neq__trans,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_296_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_297_order__less__imp__le,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_298_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_299_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_300_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_301_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_302_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_303_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_304_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_305_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_306_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_307_order__less__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X2: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_308_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_309_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_310_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_311_order__le__less,axiom,
( ord_le4245800335709223507ccount
= ( ^ [X2: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_312_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_313_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_int @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_314_dual__order_Ostrict__implies__order,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ( ord_less_eq_real @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_315_dual__order_Ostrict__implies__order,axiom,
! [B2: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A )
=> ( ord_le4245800335709223507ccount @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_316_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_317_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ( ord_less_eq_int @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_318_order_Ostrict__implies__order,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_319_order_Ostrict__implies__order,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ord_le4245800335709223507ccount @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_320_order_Ostrict__implies__order,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_321_order_Ostrict__implies__order,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_eq_int @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_322_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
& ~ ( ord_less_eq_real @ A2 @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_323_dual__order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B: risk_Free_account,A2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A2 )
& ~ ( ord_le4245800335709223507ccount @ A2 @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_324_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_325_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
& ~ ( ord_less_eq_int @ A2 @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_326_dual__order_Ostrict__trans2,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_327_dual__order_Ostrict__trans2,axiom,
! [B2: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A )
=> ( ( ord_le4245800335709223507ccount @ C @ B2 )
=> ( ord_le2131251472502387783ccount @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_328_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_329_dual__order_Ostrict__trans2,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_330_dual__order_Ostrict__trans1,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_331_dual__order_Ostrict__trans1,axiom,
! [B2: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A )
=> ( ( ord_le2131251472502387783ccount @ C @ B2 )
=> ( ord_le2131251472502387783ccount @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_332_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_333_dual__order_Ostrict__trans1,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_334_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
& ( A2 != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_335_dual__order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B: risk_Free_account,A2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A2 )
& ( A2 != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_336_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
& ( A2 != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_337_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
& ( A2 != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_338_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
| ( A2 = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_339_dual__order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [B: risk_Free_account,A2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A2 )
| ( A2 = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_340_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
| ( A2 = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_341_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
| ( A2 = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_342_dense__le__bounded,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_343_dense__ge__bounded,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_344_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
& ~ ( ord_less_eq_real @ B @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_345_order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B )
& ~ ( ord_le4245800335709223507ccount @ B @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_346_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
& ~ ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_347_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
& ~ ( ord_less_eq_int @ B @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_348_order_Ostrict__trans2,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_349_order_Ostrict__trans2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_350_order_Ostrict__trans2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_351_order_Ostrict__trans2,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_352_order_Ostrict__trans1,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_353_order_Ostrict__trans1,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ B2 @ C )
=> ( ord_le2131251472502387783ccount @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_354_order_Ostrict__trans1,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_355_order_Ostrict__trans1,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_356_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
& ( A2 != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_357_order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B )
& ( A2 != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_358_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
& ( A2 != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_359_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
& ( A2 != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_360_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
| ( A2 = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_361_order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A2 @ B )
| ( A2 = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_362_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
| ( A2 = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_363_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
| ( A2 = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_364_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_365_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_366_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_367_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_368_less__le__not__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X2: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y5 )
& ~ ( ord_le4245800335709223507ccount @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_369_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_370_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_371_dense__le,axiom,
! [Y: real,Z3: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_eq_real @ X4 @ Z3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_le
thf(fact_372_dense__ge,axiom,
! [Z3: real,Y: real] :
( ! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( ord_less_eq_real @ Y @ X4 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_ge
thf(fact_373_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_374_antisym__conv2,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ~ ( ord_le2131251472502387783ccount @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_375_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_376_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_377_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_378_antisym__conv1,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ~ ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_379_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_380_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_381_nless__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_real @ A @ B2 ) )
= ( ~ ( ord_less_eq_real @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_382_nless__le,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ~ ( ord_le2131251472502387783ccount @ A @ B2 ) )
= ( ~ ( ord_le4245800335709223507ccount @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_383_nless__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_384_nless__le,axiom,
! [A: int,B2: int] :
( ( ~ ( ord_less_int @ A @ B2 ) )
= ( ~ ( ord_less_eq_int @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_385_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_386_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_387_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_388_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_389_leD,axiom,
! [Y: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y @ X )
=> ~ ( ord_le2131251472502387783ccount @ X @ Y ) ) ).
% leD
thf(fact_390_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_391_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_392_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_393_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_394_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_395_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_396_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_397_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_398_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_399_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_400_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_401_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_402_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_403_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_404_valid__transfer__alt__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha2: risk_Free_account,Tau: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ Tau )
& ( ord_le4245800335709223507ccount @ Tau @ Alpha2 ) ) ) ) ).
% valid_transfer_alt_def
thf(fact_405_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_406_norm__not__less__zero,axiom,
! [X: complex] :
~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% norm_not_less_zero
thf(fact_407_norm__not__less__zero,axiom,
! [X: real] :
~ ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real ) ).
% norm_not_less_zero
thf(fact_408_just__cash__order__embed,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B: real] : ( ord_le4245800335709223507ccount @ ( risk_Free_just_cash @ A2 ) @ ( risk_Free_just_cash @ B ) ) ) ) ).
% just_cash_order_embed
thf(fact_409_strictly__solvent__alt__def,axiom,
( risk_F1636578016437888323olvent
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount ) ) ).
% strictly_solvent_alt_def
thf(fact_410_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_411_order__antisym__conv,axiom,
! [Y: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y @ X )
=> ( ( ord_le4245800335709223507ccount @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_412_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_413_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_414_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_415_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_416_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_417_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_418_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_419_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_420_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_421_ord__le__eq__subst,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_422_ord__le__eq__subst,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_423_ord__le__eq__subst,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_424_ord__le__eq__subst,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > int,C: int] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_425_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_426_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_427_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_428_ord__eq__le__subst,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_429_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_430_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_431_ord__eq__le__subst,axiom,
! [A: real,F: risk_Free_account > real,B2: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_432_ord__eq__le__subst,axiom,
! [A: risk_Free_account,F: risk_Free_account > risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_433_ord__eq__le__subst,axiom,
! [A: nat,F: risk_Free_account > nat,B2: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_434_ord__eq__le__subst,axiom,
! [A: int,F: risk_Free_account > int,B2: risk_Free_account,C: risk_Free_account] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_435_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_436_ord__eq__le__subst,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_437_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_438_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_439_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_440_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_441_order__eq__refl,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( X = Y )
=> ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% order_eq_refl
thf(fact_442_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_443_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_444_order__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_445_order__subst2,axiom,
! [A: real,B2: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_446_order__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_447_order__subst2,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_448_order__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_449_order__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_450_order__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_451_order__subst2,axiom,
! [A: risk_Free_account,B2: risk_Free_account,F: risk_Free_account > int,C: int] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_452_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_453_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_454_order__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_455_order__subst1,axiom,
! [A: real,F: risk_Free_account > real,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_456_order__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_457_order__subst1,axiom,
! [A: real,F: int > real,B2: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_458_order__subst1,axiom,
! [A: risk_Free_account,F: real > risk_Free_account,B2: real,C: real] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_459_order__subst1,axiom,
! [A: risk_Free_account,F: risk_Free_account > risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_460_order__subst1,axiom,
! [A: risk_Free_account,F: nat > risk_Free_account,B2: nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_461_order__subst1,axiom,
! [A: risk_Free_account,F: int > risk_Free_account,B2: int,C: int] :
( ( ord_le4245800335709223507ccount @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_462_order__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_463_order__subst1,axiom,
! [A: nat,F: risk_Free_account > nat,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ! [X4: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_464_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
& ( ord_less_eq_real @ B @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_465_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: risk_Free_account,Z: risk_Free_account] : ( Y2 = Z ) )
= ( ^ [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A2 @ B )
& ( ord_le4245800335709223507ccount @ B @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_466_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
& ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_467_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
& ( ord_less_eq_int @ B @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_468_antisym,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_469_antisym,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_470_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_471_antisym,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_472_dual__order_Otrans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_473_dual__order_Otrans,axiom,
! [B2: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A )
=> ( ( ord_le4245800335709223507ccount @ C @ B2 )
=> ( ord_le4245800335709223507ccount @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_474_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_475_dual__order_Otrans,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_476_dual__order_Oantisym,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_477_dual__order_Oantisym,axiom,
! [B2: risk_Free_account,A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A )
=> ( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_478_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_479_dual__order_Oantisym,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_480_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A2: real,B: real] :
( ( ord_less_eq_real @ B @ A2 )
& ( ord_less_eq_real @ A2 @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_481_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: risk_Free_account,Z: risk_Free_account] : ( Y2 = Z ) )
= ( ^ [A2: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A2 )
& ( ord_le4245800335709223507ccount @ A2 @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_482_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A2: nat,B: nat] :
( ( ord_less_eq_nat @ B @ A2 )
& ( ord_less_eq_nat @ A2 @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_483_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A2: int,B: int] :
( ( ord_less_eq_int @ B @ A2 )
& ( ord_less_eq_int @ A2 @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_484_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B2: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_485_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_486_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B2: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_487_order__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_488_order__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ Z3 )
=> ( ord_le4245800335709223507ccount @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_489_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_490_order__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_491_order_Otrans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_492_order_Otrans,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ord_le4245800335709223507ccount @ A @ C ) ) ) ).
% order.trans
thf(fact_493_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_494_order_Otrans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_495_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_496_order__antisym,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_497_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_498_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_499_ord__le__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_500_ord__le__eq__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le4245800335709223507ccount @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_501_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_502_ord__le__eq__trans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_503_ord__eq__le__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_504_ord__eq__le__trans,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( A = B2 )
=> ( ( ord_le4245800335709223507ccount @ B2 @ C )
=> ( ord_le4245800335709223507ccount @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_505_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_506_ord__eq__le__trans,axiom,
! [A: int,B2: int,C: int] :
( ( A = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_507_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_508_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: risk_Free_account,Z: risk_Free_account] : ( Y2 = Z ) )
= ( ^ [X2: risk_Free_account,Y5: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y5 )
& ( ord_le4245800335709223507ccount @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_509_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_510_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_511_le__cases3,axiom,
! [X: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_512_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_513_le__cases3,axiom,
! [X: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_514_nle__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_515_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_516_nle__le,axiom,
! [A: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_517_scaleR__le__cancel__left__pos,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B2 ) )
= ( ord_less_eq_real @ A @ B2 ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_518_scaleR__le__cancel__left__neg,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B2 ) )
= ( ord_less_eq_real @ B2 @ A ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_519_scaleR__le__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B2 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_520_only__strictly__solvent__accounts__can__transfer,axiom,
! [Alpha: risk_Free_account,Tau2: risk_Free_account] :
( ( risk_F1023690899723030139ansfer @ Alpha @ Tau2 )
=> ( risk_F1636578016437888323olvent @ Alpha ) ) ).
% only_strictly_solvent_accounts_can_transfer
thf(fact_521_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_522_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_523_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_524_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_525_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_526_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_527_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_528_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_529_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_530_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_531_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_532_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_533_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_534_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_535_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_536_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_537_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_538_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_539_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_540_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_541_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_542_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_543_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_544_complete__interval,axiom,
! [A: real,B2: real,P: real > $o] :
( ( ord_less_real @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B2 )
& ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ C2 ) )
=> ( P @ X3 ) )
& ! [D3: real] :
( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_real @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_545_complete__interval,axiom,
! [A: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B2 )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ C2 ) )
=> ( P @ X3 ) )
& ! [D3: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_546_complete__interval,axiom,
! [A: int,B2: int,P: int > $o] :
( ( ord_less_int @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B2 )
& ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ C2 ) )
=> ( P @ X3 ) )
& ! [D3: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_int @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_547_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_548_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_549_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_550_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_551_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X )
= ( ln_ln_real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_552_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_553_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_554_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_555_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_556_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= zero_zero_real )
= ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_557_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_558_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_559_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_560_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_561_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_562_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_563_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_564_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_565_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_566_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_567_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_568_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_569_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_570_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_571_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_572_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_573_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_574_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_575_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_576_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_577_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_578_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_579_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_580_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_581_ln__bound,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_bound
thf(fact_582_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_583_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_584_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_gt_zero
thf(fact_585_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_ge_zero
thf(fact_586_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_587_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_588_ex__gt__or__lt,axiom,
! [A: real] :
? [B3: real] :
( ( ord_less_real @ A @ B3 )
| ( ord_less_real @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_589_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_590_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_591_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_592_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_593_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_594_greater__than__shortest__period__zero,axiom,
! [Alpha: risk_Free_account,M2: nat] :
( ( ord_less_nat @ ( risk_F4612863212915232279period @ Alpha ) @ M2 )
=> ( ( risk_F170160801229183585ccount @ Alpha @ M2 )
= zero_zero_real ) ) ).
% greater_than_shortest_period_zero
thf(fact_595_minf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ~ ( ord_less_eq_real @ T @ X3 ) ) ).
% minf(8)
thf(fact_596_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_597_minf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ~ ( ord_less_eq_int @ T @ X3 ) ) ).
% minf(8)
thf(fact_598_minf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ord_less_eq_real @ X3 @ T ) ) ).
% minf(6)
thf(fact_599_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_600_minf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ord_less_eq_int @ X3 @ T ) ) ).
% minf(6)
thf(fact_601_pinf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ord_less_eq_real @ T @ X3 ) ) ).
% pinf(8)
thf(fact_602_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_603_pinf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ord_less_eq_int @ T @ X3 ) ) ).
% pinf(8)
thf(fact_604_pinf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ T ) ) ).
% pinf(6)
thf(fact_605_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_606_pinf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ T ) ) ).
% pinf(6)
thf(fact_607_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
= ( ord_less_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_608_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_609_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_610_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_611_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_612_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
= one_one_complex ) ).
% dbl_inc_simps(2)
thf(fact_613_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_614_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_615_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_616_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_617_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_618_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_619_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_620_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_621_verit__la__disequality,axiom,
! [A: real,B2: real] :
( ( A = B2 )
| ~ ( ord_less_eq_real @ A @ B2 )
| ~ ( ord_less_eq_real @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_622_verit__la__disequality,axiom,
! [A: nat,B2: nat] :
( ( A = B2 )
| ~ ( ord_less_eq_nat @ A @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_623_verit__la__disequality,axiom,
! [A: int,B2: int] :
( ( A = B2 )
| ~ ( ord_less_eq_int @ A @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_624_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_625_verit__comp__simplify1_I2_J,axiom,
! [A: risk_Free_account] : ( ord_le4245800335709223507ccount @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_626_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_627_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_628_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_629_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_630_verit__comp__simplify1_I1_J,axiom,
! [A: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_631_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_632_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_633_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_634_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_635_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_636_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_637_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_638_pinf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_639_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_640_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_641_pinf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_642_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_643_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_644_pinf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ~ ( ord_less_real @ X3 @ T ) ) ).
% pinf(5)
thf(fact_645_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ~ ( ord_less_nat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_646_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ~ ( ord_less_int @ X3 @ T ) ) ).
% pinf(5)
thf(fact_647_pinf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ord_less_real @ T @ X3 ) ) ).
% pinf(7)
thf(fact_648_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ord_less_nat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_649_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ord_less_int @ T @ X3 ) ) ).
% pinf(7)
thf(fact_650_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_651_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_652_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_653_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_654_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_655_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_656_minf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_657_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_658_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_659_minf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_660_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_661_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_662_minf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ord_less_real @ X3 @ T ) ) ).
% minf(5)
thf(fact_663_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_664_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ord_less_int @ X3 @ T ) ) ).
% minf(5)
thf(fact_665_minf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ~ ( ord_less_real @ T @ X3 ) ) ).
% minf(7)
thf(fact_666_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_667_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ~ ( ord_less_int @ T @ X3 ) ) ).
% minf(7)
thf(fact_668_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_669_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% ln_le_minus_one
thf(fact_670_cos__coeff__0,axiom,
( ( cos_coeff @ zero_zero_nat )
= one_one_real ) ).
% cos_coeff_0
thf(fact_671_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_672_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_673_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_674_zero__le__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_675_log__le__zero__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_676_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_677_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_678_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_679_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_680_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_681_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A @ A )
= zero_z1425366712893667068ccount ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_682_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_683_diff__zero,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_zero
thf(fact_684_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_685_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_686_diff__zero,axiom,
! [A: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% diff_zero
thf(fact_687_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_688_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_689_diff__0__right,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_0_right
thf(fact_690_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_691_diff__0__right,axiom,
! [A: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A @ zero_z1425366712893667068ccount )
= A ) ).
% diff_0_right
thf(fact_692_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_693_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_694_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_695_diff__self,axiom,
! [A: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A @ A )
= zero_z1425366712893667068ccount ) ).
% diff_self
thf(fact_696_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_697_norm__of__nat,axiom,
! [N: nat] :
( ( real_V1022390504157884413omplex @ ( semiri8010041392384452111omplex @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% norm_of_nat
thf(fact_698_norm__of__nat,axiom,
! [N: nat] :
( ( real_V7735802525324610683m_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% norm_of_nat
thf(fact_699_diff__ge__0__iff__ge,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
= ( ord_less_eq_real @ B2 @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_700_diff__ge__0__iff__ge,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( minus_4846202936726426316ccount @ A @ B2 ) )
= ( ord_le4245800335709223507ccount @ B2 @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_701_diff__ge__0__iff__ge,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
= ( ord_less_eq_int @ B2 @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_702_diff__gt__0__iff__gt,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
= ( ord_less_real @ B2 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_703_diff__gt__0__iff__gt,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( minus_4846202936726426316ccount @ A @ B2 ) )
= ( ord_le2131251472502387783ccount @ B2 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_704_diff__gt__0__iff__gt,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
= ( ord_less_int @ B2 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_705_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_706_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_707_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_708_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_709_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri8010041392384452111omplex @ M2 )
= zero_zero_complex )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_710_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_711_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_712_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_713_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_714_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_715_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_716_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_717_of__nat__0,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_nat_0
thf(fact_718_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_719_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_720_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_721_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_722_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_723_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_724_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_725_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_726_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_727_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_728_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_729_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_730_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_731_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_732_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_733_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_734_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_735_log__one,axiom,
! [A: real] :
( ( log @ A @ one_one_real )
= zero_zero_real ) ).
% log_one
thf(fact_736_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_737_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_738_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_739_zero__less__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_740_log__less__zero__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_741_one__less__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
= ( ord_less_real @ A @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_742_log__less__one__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
= ( ord_less_real @ X @ A ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_743_log__less__cancel__iff,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_744_log__eq__one,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ A )
= one_one_real ) ) ) ).
% log_eq_one
thf(fact_745_log__le__cancel__iff,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_746_log__le__one__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ A ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_747_one__le__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
= ( ord_less_eq_real @ A @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_748_diff__right__commute,axiom,
! [A: real,C: real,B2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 )
= ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_749_diff__right__commute,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_750_diff__right__commute,axiom,
! [A: risk_Free_account,C: risk_Free_account,B2: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( minus_4846202936726426316ccount @ A @ C ) @ B2 )
= ( minus_4846202936726426316ccount @ ( minus_4846202936726426316ccount @ A @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_751_diff__right__commute,axiom,
! [A: int,C: int,B2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 )
= ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_752_diff__eq__diff__eq,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B2 )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B2 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_753_diff__eq__diff__eq,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ( minus_4846202936726426316ccount @ A @ B2 )
= ( minus_4846202936726426316ccount @ C @ D ) )
=> ( ( A = B2 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_754_diff__eq__diff__eq,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B2 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_755_scaleR__left__diff__distrib,axiom,
! [A: real,B2: real,X: real] :
( ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B2 ) @ X )
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B2 @ X ) ) ) ).
% scaleR_left_diff_distrib
thf(fact_756_scaleR__left_Odiff,axiom,
! [X: real,Y: real,Xa: real] :
( ( real_V1485227260804924795R_real @ ( minus_minus_real @ X @ Y ) @ Xa )
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) @ ( real_V1485227260804924795R_real @ Y @ Xa ) ) ) ).
% scaleR_left.diff
thf(fact_757_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_758_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_759_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_760_norm__triangle__ineq2,axiom,
! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).
% norm_triangle_ineq2
thf(fact_761_norm__triangle__ineq2,axiom,
! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% norm_triangle_ineq2
thf(fact_762_diff__mono,axiom,
! [A: real,B2: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% diff_mono
thf(fact_763_diff__mono,axiom,
! [A: risk_Free_account,B2: risk_Free_account,D: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ( ord_le4245800335709223507ccount @ D @ C )
=> ( ord_le4245800335709223507ccount @ ( minus_4846202936726426316ccount @ A @ C ) @ ( minus_4846202936726426316ccount @ B2 @ D ) ) ) ) ).
% diff_mono
thf(fact_764_diff__mono,axiom,
! [A: int,B2: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_mono
thf(fact_765_diff__left__mono,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_766_diff__left__mono,axiom,
! [B2: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B2 @ A )
=> ( ord_le4245800335709223507ccount @ ( minus_4846202936726426316ccount @ C @ A ) @ ( minus_4846202936726426316ccount @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_767_diff__left__mono,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_768_diff__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_769_diff__right__mono,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ord_le4245800335709223507ccount @ ( minus_4846202936726426316ccount @ A @ C ) @ ( minus_4846202936726426316ccount @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_770_diff__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_771_diff__eq__diff__less__eq,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B2 )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B2 )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_772_diff__eq__diff__less__eq,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ( minus_4846202936726426316ccount @ A @ B2 )
= ( minus_4846202936726426316ccount @ C @ D ) )
=> ( ( ord_le4245800335709223507ccount @ A @ B2 )
= ( ord_le4245800335709223507ccount @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_773_diff__eq__diff__less__eq,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B2 )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_774_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: complex,Z: complex] : ( Y2 = Z ) )
= ( ^ [A2: complex,B: complex] :
( ( minus_minus_complex @ A2 @ B )
= zero_zero_complex ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_775_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A2: real,B: real] :
( ( minus_minus_real @ A2 @ B )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_776_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: risk_Free_account,Z: risk_Free_account] : ( Y2 = Z ) )
= ( ^ [A2: risk_Free_account,B: risk_Free_account] :
( ( minus_4846202936726426316ccount @ A2 @ B )
= zero_z1425366712893667068ccount ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_777_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A2: int,B: int] :
( ( minus_minus_int @ A2 @ B )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_778_diff__strict__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_779_diff__strict__right__mono,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ord_le2131251472502387783ccount @ ( minus_4846202936726426316ccount @ A @ C ) @ ( minus_4846202936726426316ccount @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_780_diff__strict__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_781_diff__strict__left__mono,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_782_diff__strict__left__mono,axiom,
! [B2: risk_Free_account,A: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B2 @ A )
=> ( ord_le2131251472502387783ccount @ ( minus_4846202936726426316ccount @ C @ A ) @ ( minus_4846202936726426316ccount @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_783_diff__strict__left__mono,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_784_diff__eq__diff__less,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B2 )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B2 )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_785_diff__eq__diff__less,axiom,
! [A: risk_Free_account,B2: risk_Free_account,C: risk_Free_account,D: risk_Free_account] :
( ( ( minus_4846202936726426316ccount @ A @ B2 )
= ( minus_4846202936726426316ccount @ C @ D ) )
=> ( ( ord_le2131251472502387783ccount @ A @ B2 )
= ( ord_le2131251472502387783ccount @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_786_diff__eq__diff__less,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B2 )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_787_diff__strict__mono,axiom,
! [A: real,B2: real,D: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_788_diff__strict__mono,axiom,
! [A: risk_Free_account,B2: risk_Free_account,D: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ( ord_le2131251472502387783ccount @ D @ C )
=> ( ord_le2131251472502387783ccount @ ( minus_4846202936726426316ccount @ A @ C ) @ ( minus_4846202936726426316ccount @ B2 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_789_diff__strict__mono,axiom,
! [A: int,B2: int,D: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_790_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_791_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A2: nat,B: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_792_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_793_norm__minus__commute,axiom,
! [A: complex,B2: complex] :
( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) )
= ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ A ) ) ) ).
% norm_minus_commute
thf(fact_794_norm__minus__commute,axiom,
! [A: real,B2: real] :
( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) )
= ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% norm_minus_commute
thf(fact_795_scaleR__right__diff__distrib,axiom,
! [A: real,X: real,Y: real] :
( ( real_V1485227260804924795R_real @ A @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).
% scaleR_right_diff_distrib
thf(fact_796_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_797_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_798_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_799_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_800_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_801_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_802_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_803_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_804_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_805_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_806_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_807_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_808_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_809_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_810_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_811_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_812_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_813_le__iff__diff__le__0,axiom,
( ord_le4245800335709223507ccount
= ( ^ [A2: risk_Free_account,B: risk_Free_account] : ( ord_le4245800335709223507ccount @ ( minus_4846202936726426316ccount @ A2 @ B ) @ zero_z1425366712893667068ccount ) ) ) ).
% le_iff_diff_le_0
thf(fact_814_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_815_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A2: real,B: real] : ( ord_less_real @ ( minus_minus_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_816_less__iff__diff__less__0,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A2: risk_Free_account,B: risk_Free_account] : ( ord_le2131251472502387783ccount @ ( minus_4846202936726426316ccount @ A2 @ B ) @ zero_z1425366712893667068ccount ) ) ) ).
% less_iff_diff_less_0
thf(fact_817_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A2: int,B: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_818_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= ( minus_minus_real @ X @ one_one_real ) )
=> ( X = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_819_Bolzano,axiom,
! [A: real,B2: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ! [A4: real,B3: real,C2: real] :
( ( P @ A4 @ B3 )
=> ( ( P @ B3 @ C2 )
=> ( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( P @ A4 @ C2 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A4: real,B3: real] :
( ( ( ord_less_eq_real @ A4 @ X4 )
& ( ord_less_eq_real @ X4 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A4 ) @ D3 ) )
=> ( P @ A4 @ B3 ) ) ) ) )
=> ( P @ A @ B2 ) ) ) ) ).
% Bolzano
thf(fact_820_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_821_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_822_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_823_real__arch__simple,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_824_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_825_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_826_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_827_just__cash__subtract,axiom,
! [A: real,B2: real] :
( ( minus_4846202936726426316ccount @ ( risk_Free_just_cash @ A ) @ ( risk_Free_just_cash @ B2 ) )
= ( risk_Free_just_cash @ ( minus_minus_real @ A @ B2 ) ) ) ).
% just_cash_subtract
thf(fact_828_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_829_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_830_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_831_zle__diff1__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z3 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_832_int__ops_I6_J,axiom,
! [A: nat,B2: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_833_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_834_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_835_int__if,axiom,
! [P: $o,A: nat,B2: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% int_if
thf(fact_836_nat__int__comparison_I1_J,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ A2 )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_837_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_838_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_839_int__diff__cases,axiom,
! [Z3: int] :
~ ! [M4: nat,N3: nat] :
( Z3
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_840_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_841_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_842_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_843_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_844_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_845_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_846_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_847_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_848_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_849_le__diff__iff_H,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_850_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_851_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_852_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_853_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_854_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_855_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_856_diff__less__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_857_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_858_valid__transfer__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha2: risk_Free_account,Tau: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Tau )
& ( risk_F1636578016437888323olvent @ ( minus_4846202936726426316ccount @ Alpha2 @ Tau ) ) ) ) ) ).
% valid_transfer_def
thf(fact_859_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_860_log__of__power__le,axiom,
! [M2: nat,B2: real,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B2 @ N ) )
=> ( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_eq_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_861_log__pow__cancel,axiom,
! [A: real,B2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ ( power_power_real @ A @ B2 ) )
= ( semiri5074537144036343181t_real @ B2 ) ) ) ) ).
% log_pow_cancel
thf(fact_862_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_863_log__of__power__less,axiom,
! [M2: nat,B2: real,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B2 @ N ) )
=> ( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_864_CauchyD,axiom,
! [X6: nat > complex,E2: real] :
( ( topolo6517432010174082258omplex @ X6 )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [M5: nat] :
! [M3: nat] :
( ( ord_less_eq_nat @ M5 @ M3 )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X6 @ M3 ) @ ( X6 @ N4 ) ) ) @ E2 ) ) ) ) ) ).
% CauchyD
thf(fact_865_CauchyD,axiom,
! [X6: nat > real,E2: real] :
( ( topolo4055970368930404560y_real @ X6 )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [M5: nat] :
! [M3: nat] :
( ( ord_less_eq_nat @ M5 @ M3 )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X6 @ M3 ) @ ( X6 @ N4 ) ) ) @ E2 ) ) ) ) ) ).
% CauchyD
thf(fact_866_verit__minus__simplify_I4_J,axiom,
! [B2: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B2 ) )
= B2 ) ).
% verit_minus_simplify(4)
thf(fact_867_verit__minus__simplify_I4_J,axiom,
! [B2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B2 ) )
= B2 ) ).
% verit_minus_simplify(4)
thf(fact_868_verit__minus__simplify_I4_J,axiom,
! [B2: risk_Free_account] :
( ( uminus3377898441596595772ccount @ ( uminus3377898441596595772ccount @ B2 ) )
= B2 ) ).
% verit_minus_simplify(4)
thf(fact_869_neg__equal__iff__equal,axiom,
! [A: real,B2: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B2 ) )
= ( A = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_870_neg__equal__iff__equal,axiom,
! [A: int,B2: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B2 ) )
= ( A = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_871_neg__equal__iff__equal,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A )
= ( uminus3377898441596595772ccount @ B2 ) )
= ( A = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_872_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_873_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_874_add_Oinverse__inverse,axiom,
! [A: risk_Free_account] :
( ( uminus3377898441596595772ccount @ ( uminus3377898441596595772ccount @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_875_arsinh__minus__real,axiom,
! [X: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arsinh_real @ X ) ) ) ).
% arsinh_minus_real
thf(fact_876_neg__le__iff__le,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B2 ) ) ).
% neg_le_iff_le
thf(fact_877_neg__le__iff__le,axiom,
! [B2: risk_Free_account,A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ B2 ) @ ( uminus3377898441596595772ccount @ A ) )
= ( ord_le4245800335709223507ccount @ A @ B2 ) ) ).
% neg_le_iff_le
thf(fact_878_neg__le__iff__le,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B2 ) ) ).
% neg_le_iff_le
thf(fact_879_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_880_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_881_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_882_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_883_neg__equal__0__iff__equal,axiom,
! [A: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_884_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_885_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_886_neg__equal__0__iff__equal,axiom,
! [A: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A )
= zero_z1425366712893667068ccount )
= ( A = zero_z1425366712893667068ccount ) ) ).
% neg_equal_0_iff_equal
thf(fact_887_neg__0__equal__iff__equal,axiom,
! [A: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A ) )
= ( zero_zero_complex = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_888_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_889_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_890_neg__0__equal__iff__equal,axiom,
! [A: risk_Free_account] :
( ( zero_z1425366712893667068ccount
= ( uminus3377898441596595772ccount @ A ) )
= ( zero_z1425366712893667068ccount = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_891_add_Oinverse__neutral,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% add.inverse_neutral
thf(fact_892_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_893_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_894_add_Oinverse__neutral,axiom,
( ( uminus3377898441596595772ccount @ zero_z1425366712893667068ccount )
= zero_z1425366712893667068ccount ) ).
% add.inverse_neutral
thf(fact_895_neg__less__iff__less,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B2 ) ) ).
% neg_less_iff_less
thf(fact_896_neg__less__iff__less,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B2 ) ) ).
% neg_less_iff_less
thf(fact_897_neg__less__iff__less,axiom,
! [B2: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ B2 ) @ ( uminus3377898441596595772ccount @ A ) )
= ( ord_le2131251472502387783ccount @ A @ B2 ) ) ).
% neg_less_iff_less
thf(fact_898_minus__diff__eq,axiom,
! [A: real,B2: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) )
= ( minus_minus_real @ B2 @ A ) ) ).
% minus_diff_eq
thf(fact_899_minus__diff__eq,axiom,
! [A: int,B2: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) )
= ( minus_minus_int @ B2 @ A ) ) ).
% minus_diff_eq
thf(fact_900_minus__diff__eq,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( uminus3377898441596595772ccount @ ( minus_4846202936726426316ccount @ A @ B2 ) )
= ( minus_4846202936726426316ccount @ B2 @ A ) ) ).
% minus_diff_eq
thf(fact_901_norm__minus__cancel,axiom,
! [X: complex] :
( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ X ) )
= ( real_V1022390504157884413omplex @ X ) ) ).
% norm_minus_cancel
thf(fact_902_norm__minus__cancel,axiom,
! [X: real] :
( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ X ) )
= ( real_V7735802525324610683m_real @ X ) ) ).
% norm_minus_cancel
thf(fact_903_scaleR__minus__right,axiom,
! [A: real,X: real] :
( ( real_V1485227260804924795R_real @ A @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ).
% scaleR_minus_right
thf(fact_904_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_905_neg__0__le__iff__le,axiom,
! [A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( uminus3377898441596595772ccount @ A ) )
= ( ord_le4245800335709223507ccount @ A @ zero_z1425366712893667068ccount ) ) ).
% neg_0_le_iff_le
thf(fact_906_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_907_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_908_neg__le__0__iff__le,axiom,
! [A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ A ) @ zero_z1425366712893667068ccount )
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ A ) ) ).
% neg_le_0_iff_le
thf(fact_909_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_910_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_911_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_912_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_913_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_914_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_915_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_916_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_917_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_918_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_919_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_920_neg__0__less__iff__less,axiom,
! [A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ ( uminus3377898441596595772ccount @ A ) )
= ( ord_le2131251472502387783ccount @ A @ zero_z1425366712893667068ccount ) ) ).
% neg_0_less_iff_less
thf(fact_921_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_922_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_923_neg__less__0__iff__less,axiom,
! [A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ A ) @ zero_z1425366712893667068ccount )
= ( ord_le2131251472502387783ccount @ zero_z1425366712893667068ccount @ A ) ) ).
% neg_less_0_iff_less
thf(fact_924_verit__minus__simplify_I3_J,axiom,
! [B2: complex] :
( ( minus_minus_complex @ zero_zero_complex @ B2 )
= ( uminus1482373934393186551omplex @ B2 ) ) ).
% verit_minus_simplify(3)
thf(fact_925_verit__minus__simplify_I3_J,axiom,
! [B2: real] :
( ( minus_minus_real @ zero_zero_real @ B2 )
= ( uminus_uminus_real @ B2 ) ) ).
% verit_minus_simplify(3)
thf(fact_926_verit__minus__simplify_I3_J,axiom,
! [B2: int] :
( ( minus_minus_int @ zero_zero_int @ B2 )
= ( uminus_uminus_int @ B2 ) ) ).
% verit_minus_simplify(3)
thf(fact_927_verit__minus__simplify_I3_J,axiom,
! [B2: risk_Free_account] :
( ( minus_4846202936726426316ccount @ zero_z1425366712893667068ccount @ B2 )
= ( uminus3377898441596595772ccount @ B2 ) ) ).
% verit_minus_simplify(3)
thf(fact_928_diff__0,axiom,
! [A: complex] :
( ( minus_minus_complex @ zero_zero_complex @ A )
= ( uminus1482373934393186551omplex @ A ) ) ).
% diff_0
thf(fact_929_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_930_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_931_diff__0,axiom,
! [A: risk_Free_account] :
( ( minus_4846202936726426316ccount @ zero_z1425366712893667068ccount @ A )
= ( uminus3377898441596595772ccount @ A ) ) ).
% diff_0
thf(fact_932_scaleR__minus__left,axiom,
! [A: real,X: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ A ) @ X )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ).
% scaleR_minus_left
thf(fact_933_scaleR__left_Ominus,axiom,
! [X: real,Xa: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ X ) @ Xa )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) ) ) ).
% scaleR_left.minus
thf(fact_934_scaleR__power,axiom,
! [X: real,Y: real,N: nat] :
( ( power_power_real @ ( real_V1485227260804924795R_real @ X @ Y ) @ N )
= ( real_V1485227260804924795R_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).
% scaleR_power
thf(fact_935_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_inc_simps(4)
thf(fact_936_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_937_diff__numeral__special_I12_J,axiom,
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% diff_numeral_special(12)
thf(fact_938_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_939_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_940_scaleR__minus1__left,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ one_one_real ) @ X )
= ( uminus_uminus_real @ X ) ) ).
% scaleR_minus1_left
thf(fact_941_norm__power__ineq,axiom,
! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% norm_power_ineq
thf(fact_942_norm__power__ineq,axiom,
! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% norm_power_ineq
thf(fact_943_verit__negate__coefficient_I3_J,axiom,
! [A: real,B2: real] :
( ( A = B2 )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B2 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_944_verit__negate__coefficient_I3_J,axiom,
! [A: int,B2: int] :
( ( A = B2 )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B2 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_945_verit__negate__coefficient_I3_J,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( A = B2 )
=> ( ( uminus3377898441596595772ccount @ A )
= ( uminus3377898441596595772ccount @ B2 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_946_minus__equation__iff,axiom,
! [A: real,B2: real] :
( ( ( uminus_uminus_real @ A )
= B2 )
= ( ( uminus_uminus_real @ B2 )
= A ) ) ).
% minus_equation_iff
thf(fact_947_minus__equation__iff,axiom,
! [A: int,B2: int] :
( ( ( uminus_uminus_int @ A )
= B2 )
= ( ( uminus_uminus_int @ B2 )
= A ) ) ).
% minus_equation_iff
thf(fact_948_minus__equation__iff,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ( uminus3377898441596595772ccount @ A )
= B2 )
= ( ( uminus3377898441596595772ccount @ B2 )
= A ) ) ).
% minus_equation_iff
thf(fact_949_equation__minus__iff,axiom,
! [A: real,B2: real] :
( ( A
= ( uminus_uminus_real @ B2 ) )
= ( B2
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_950_equation__minus__iff,axiom,
! [A: int,B2: int] :
( ( A
= ( uminus_uminus_int @ B2 ) )
= ( B2
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_951_equation__minus__iff,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( A
= ( uminus3377898441596595772ccount @ B2 ) )
= ( B2
= ( uminus3377898441596595772ccount @ A ) ) ) ).
% equation_minus_iff
thf(fact_952_norm__power,axiom,
! [X: complex,N: nat] :
( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) )
= ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% norm_power
thf(fact_953_norm__power,axiom,
! [X: real,N: nat] :
( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) )
= ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% norm_power
thf(fact_954_le__minus__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B2 ) )
= ( ord_less_eq_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_955_le__minus__iff,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ ( uminus3377898441596595772ccount @ B2 ) )
= ( ord_le4245800335709223507ccount @ B2 @ ( uminus3377898441596595772ccount @ A ) ) ) ).
% le_minus_iff
thf(fact_956_le__minus__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_957_minus__le__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% minus_le_iff
thf(fact_958_minus__le__iff,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ A ) @ B2 )
= ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ B2 ) @ A ) ) ).
% minus_le_iff
thf(fact_959_minus__le__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% minus_le_iff
thf(fact_960_le__imp__neg__le,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_961_le__imp__neg__le,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B2 )
=> ( ord_le4245800335709223507ccount @ ( uminus3377898441596595772ccount @ B2 ) @ ( uminus3377898441596595772ccount @ A ) ) ) ).
% le_imp_neg_le
thf(fact_962_le__imp__neg__le,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_963_minus__less__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 )
= ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% minus_less_iff
thf(fact_964_minus__less__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 )
= ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% minus_less_iff
thf(fact_965_minus__less__iff,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ A ) @ B2 )
= ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ B2 ) @ A ) ) ).
% minus_less_iff
thf(fact_966_less__minus__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B2 ) )
= ( ord_less_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_967_less__minus__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_968_less__minus__iff,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ ( uminus3377898441596595772ccount @ B2 ) )
= ( ord_le2131251472502387783ccount @ B2 @ ( uminus3377898441596595772ccount @ A ) ) ) ).
% less_minus_iff
thf(fact_969_verit__negate__coefficient_I2_J,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_970_verit__negate__coefficient_I2_J,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_971_verit__negate__coefficient_I2_J,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B2 )
=> ( ord_le2131251472502387783ccount @ ( uminus3377898441596595772ccount @ B2 ) @ ( uminus3377898441596595772ccount @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_972_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_973_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_974_minus__diff__minus,axiom,
! [A: real,B2: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% minus_diff_minus
thf(fact_975_minus__diff__minus,axiom,
! [A: int,B2: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) ) ) ).
% minus_diff_minus
thf(fact_976_minus__diff__minus,axiom,
! [A: risk_Free_account,B2: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( uminus3377898441596595772ccount @ A ) @ ( uminus3377898441596595772ccount @ B2 ) )
= ( uminus3377898441596595772ccount @ ( minus_4846202936726426316ccount @ A @ B2 ) ) ) ).
% minus_diff_minus
thf(fact_977_minus__diff__commute,axiom,
! [B2: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B2 ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% minus_diff_commute
thf(fact_978_minus__diff__commute,axiom,
! [B2: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% minus_diff_commute
thf(fact_979_minus__diff__commute,axiom,
! [B2: risk_Free_account,A: risk_Free_account] :
( ( minus_4846202936726426316ccount @ ( uminus3377898441596595772ccount @ B2 ) @ A )
= ( minus_4846202936726426316ccount @ ( uminus3377898441596595772ccount @ A ) @ B2 ) ) ).
% minus_diff_commute
thf(fact_980_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_981_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_982_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_983_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_984_zero__neq__neg__one,axiom,
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% zero_neq_neg_one
thf(fact_985_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_986_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_987_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_988_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_989_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_990_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_991_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_992_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_993_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_994_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_995_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_996_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_997_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_998_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_999_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_1000_power__eq__imp__eq__norm,axiom,
! [W2: complex,N: nat,Z3: complex] :
( ( ( power_power_complex @ W2 @ N )
= ( power_power_complex @ Z3 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V1022390504157884413omplex @ W2 )
= ( real_V1022390504157884413omplex @ Z3 ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_1001_power__eq__imp__eq__norm,axiom,
! [W2: real,N: nat,Z3: real] :
( ( ( power_power_real @ W2 @ N )
= ( power_power_real @ Z3 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V7735802525324610683m_real @ W2 )
= ( real_V7735802525324610683m_real @ Z3 ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_1002_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1003_power__eq__1__iff,axiom,
! [W2: complex,N: nat] :
( ( ( power_power_complex @ W2 @ N )
= one_one_complex )
=> ( ( ( real_V1022390504157884413omplex @ W2 )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_1004_power__eq__1__iff,axiom,
! [W2: real,N: nat] :
( ( ( power_power_real @ W2 @ N )
= one_one_real )
=> ( ( ( real_V7735802525324610683m_real @ W2 )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_1005_less__log__of__power,axiom,
! [B2: real,N: nat,M2: real] :
( ( ord_less_real @ ( power_power_real @ B2 @ N ) @ M2 )
=> ( ( ord_less_real @ one_one_real @ B2 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M2 ) ) ) ) ).
% less_log_of_power
thf(fact_1006_log__of__power__eq,axiom,
! [M2: nat,B2: real,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( power_power_real @ B2 @ N ) )
=> ( ( ord_less_real @ one_one_real @ B2 )
=> ( ( semiri5074537144036343181t_real @ N )
= ( log @ B2 @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ) ).
% log_of_power_eq
thf(fact_1007_le__log__of__power,axiom,
! [B2: real,N: nat,M2: real] :
( ( ord_less_eq_real @ ( power_power_real @ B2 @ N ) @ M2 )
=> ( ( ord_less_real @ one_one_real @ B2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M2 ) ) ) ) ).
% le_log_of_power
thf(fact_1008_Cauchy__iff,axiom,
( topolo6517432010174082258omplex
= ( ^ [X7: nat > complex] :
! [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
=> ? [M6: nat] :
! [M: nat] :
( ( ord_less_eq_nat @ M6 @ M )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X7 @ M ) @ ( X7 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_1009_Cauchy__iff,axiom,
( topolo4055970368930404560y_real
= ( ^ [X7: nat > real] :
! [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
=> ? [M6: nat] :
! [M: nat] :
( ( ord_less_eq_nat @ M6 @ M )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X7 @ M ) @ ( X7 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_1010_CauchyI,axiom,
! [X6: nat > complex] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [M7: nat] :
! [M4: nat] :
( ( ord_less_eq_nat @ M7 @ M4 )
=> ! [N3: nat] :
( ( ord_less_eq_nat @ M7 @ N3 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) ) @ E ) ) ) )
=> ( topolo6517432010174082258omplex @ X6 ) ) ).
% CauchyI
thf(fact_1011_CauchyI,axiom,
! [X6: nat > real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [M7: nat] :
! [M4: nat] :
( ( ord_less_eq_nat @ M7 @ M4 )
=> ! [N3: nat] :
( ( ord_less_eq_nat @ M7 @ N3 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) ) @ E ) ) ) )
=> ( topolo4055970368930404560y_real @ X6 ) ) ).
% CauchyI
thf(fact_1012_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1013_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1014_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1015_power__decreasing__iff,axiom,
! [B2: real,M2: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_real @ B2 @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B2 @ M2 ) @ ( power_power_real @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_1016_power__decreasing__iff,axiom,
! [B2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ B2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M2 ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_1017_power__decreasing__iff,axiom,
! [B2: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ B2 @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M2 ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_1018_power__mono__iff,axiom,
! [A: real,B2: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
= ( ord_less_eq_real @ A @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_1019_power__mono__iff,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_1020_power__mono__iff,axiom,
! [A: int,B2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_eq_int @ A @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_1021_power__increasing__iff,axiom,
! [B2: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_eq_real @ ( power_power_real @ B2 @ X ) @ ( power_power_real @ B2 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1022_power__increasing__iff,axiom,
! [B2: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1023_power__increasing__iff,axiom,
! [B2: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B2 )
=> ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1024_power__strict__decreasing__iff,axiom,
! [B2: real,M2: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_real @ B2 @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B2 @ M2 ) @ ( power_power_real @ B2 @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1025_power__strict__decreasing__iff,axiom,
! [B2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ B2 @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B2 @ M2 ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1026_power__strict__decreasing__iff,axiom,
! [B2: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ B2 @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B2 @ M2 ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1027_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_1028_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_1029_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_1030_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1031_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W2: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W2 ) )
= ( X
= ( power_power_nat @ B2 @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1032_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W2: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W2 ) )
= ( X
= ( power_power_nat @ B2 @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1033_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W2: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W2 ) )
= ( X
= ( power_power_nat @ B2 @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1034_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B2: nat,W2: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W2 )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B2 @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1035_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B2: nat,W2: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W2 )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B2 @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1036_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B2: nat,W2: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W2 )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B2 @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1037_of__nat__power,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M2 @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ N ) ) ).
% of_nat_power
thf(fact_1038_of__nat__power,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M2 @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M2 ) @ N ) ) ).
% of_nat_power
thf(fact_1039_of__nat__power,axiom,
! [M2: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M2 @ N ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M2 ) @ N ) ) ).
% of_nat_power
thf(fact_1040_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_1041_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_1042_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1043_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_1044_Rep__account__uminus,axiom,
! [Alpha: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( uminus3377898441596595772ccount @ Alpha ) )
= ( ^ [N2: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha @ N2 ) ) ) ) ).
% Rep_account_uminus
thf(fact_1045_just__cash__uminus,axiom,
! [A: real] :
( ( uminus3377898441596595772ccount @ ( risk_Free_just_cash @ A ) )
= ( risk_Free_just_cash @ ( uminus_uminus_real @ A ) ) ) ).
% just_cash_uminus
thf(fact_1046_power__inject__exp,axiom,
! [A: real,M2: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M2 )
= ( power_power_real @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_1047_power__inject__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M2 )
= ( power_power_nat @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_1048_power__inject__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M2 )
= ( power_power_int @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_1049_power__strict__increasing__iff,axiom,
! [B2: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_real @ ( power_power_real @ B2 @ X ) @ ( power_power_real @ B2 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1050_power__strict__increasing__iff,axiom,
! [B2: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B2 )
=> ( ( ord_less_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1051_power__strict__increasing__iff,axiom,
! [B2: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B2 )
=> ( ( ord_less_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1052_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1053_power__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( ( power_power_complex @ A @ N )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1054_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1055_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1056_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B2: nat,W2: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1057_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B2: nat,W2: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1058_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B2: nat,W2: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1059_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1060_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1061_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1062_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B2: nat,W2: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B2 @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_1063_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B2: nat,W2: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B2 @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_1064_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B2: nat,W2: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B2 @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_1065_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_1066_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_1067_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_1068_int__cases2,axiom,
! [Z3: int] :
( ! [N3: nat] :
( Z3
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_1069_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1070_shortest__period__uminus,axiom,
! [Alpha: risk_Free_account] :
( ( risk_F4612863212915232279period @ ( uminus3377898441596595772ccount @ Alpha ) )
= ( risk_F4612863212915232279period @ Alpha ) ) ).
% shortest_period_uminus
thf(fact_1071_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1072_not__int__zless__negative,axiom,
! [N: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_1073_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1074_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1075_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1076_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1077_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1078_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_1079_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_1080_power__not__zero,axiom,
! [A: complex,N: nat] :
( ( A != zero_zero_complex )
=> ( ( power_power_complex @ A @ N )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_1081_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_1082_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_1083_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1084_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1085_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1086_power__mono,axiom,
! [A: real,B2: real,N: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_1087_power__mono,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_1088_power__mono,axiom,
! [A: int,B2: int,N: nat] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_1089_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1090_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1091_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1092_one__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% one_le_power
thf(fact_1093_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_1094_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_1095_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_1096_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_1097_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_1098_power__less__imp__less__base,axiom,
! [A: real,N: nat,B2: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% power_less_imp_less_base
thf(fact_1099_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B2: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% power_less_imp_less_base
thf(fact_1100_power__less__imp__less__base,axiom,
! [A: int,N: nat,B2: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% power_less_imp_less_base
thf(fact_1101_power__le__one,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_1102_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1103_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_1104_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_1105_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% power_0_left
thf(fact_1106_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_1107_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1108_power__increasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1109_power__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1110_power__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1111_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_1112_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_1113_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_1114_power__less__imp__less__exp,axiom,
! [A: real,M2: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1115_power__less__imp__less__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1116_power__less__imp__less__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1117_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_1118_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ).
% zero_power
thf(fact_1119_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_1120_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_1121_power__decreasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1122_power__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1123_power__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1124_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1125_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1126_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1127_power__le__imp__le__exp,axiom,
! [A: real,M2: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1128_power__le__imp__le__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1129_power__le__imp__le__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1130_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B2: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B2 @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B2 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1131_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B2: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B2 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B2 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1132_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B2: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B2 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B2 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1133_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B2 @ N ) )
= ( A = B2 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1134_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B2 @ N ) )
= ( A = B2 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1135_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B2: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B2 @ N ) )
= ( A = B2 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1136_self__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1137_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1138_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1139_one__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_1140_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_1141_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_1142_power__strict__mono,axiom,
! [A: real,B2: real,N: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1143_power__strict__mono,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1144_power__strict__mono,axiom,
! [A: int,B2: int,N: nat] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1145_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1146_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ( power_power_real @ X4 @ N )
= A )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N )
= A ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1147_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_dec_simps(2)
thf(fact_1148_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_1149_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_1150_complex__mod__minus__le__complex__mod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_1151_norm__power__diff,axiom,
! [Z3: complex,W2: complex,M2: nat] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z3 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W2 ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z3 @ M2 ) @ ( power_power_complex @ W2 @ M2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z3 @ W2 ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_1152_norm__power__diff,axiom,
! [Z3: real,W2: real,M2: nat] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z3 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W2 ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z3 @ M2 ) @ ( power_power_real @ W2 @ M2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z3 @ W2 ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_1153_mult__cancel__right,axiom,
! [A: complex,C: complex,B2: complex] :
( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B2 @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_1154_mult__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_1155_mult__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B2 @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_1156_mult__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_1157_mult__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_1158_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_1159_real__scaleR__def,axiom,
real_V1485227260804924795R_real = times_times_real ).
% real_scaleR_def
thf(fact_1160_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1161_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N3: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1162_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 )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1163_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% ln_realpow
thf(fact_1164_log__nat__power,axiom,
! [X: real,B2: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ B2 @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X ) ) ) ) ).
% log_nat_power
thf(fact_1165_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1166_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1167_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1168_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1169_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1170_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1171_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1172_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1173_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1174_int__ops_I7_J,axiom,
! [A: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_1175_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1176_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1177_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1178_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1179_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1180_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1181_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1182_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1183_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_1184_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_1185_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_1186_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1187_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_1188_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_1189_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1190_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K2: int] :
( ( P1 @ X4 )
= ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( P @ X4 )
= ( P1 @ X4 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1191_plusinfinity,axiom,
! [D: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K2: int] :
( ( P4 @ X4 )
= ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1192_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_1193_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1194_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1195_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1196_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1197_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1198_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1199_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1200_log__minus__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( B2 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( minus_minus_real @ ( log @ B2 @ X ) @ Y )
= ( log @ B2 @ ( times_times_real @ X @ ( powr_real @ B2 @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_1201_powr__gt__zero,axiom,
! [X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
= ( X != zero_zero_real ) ) ).
% powr_gt_zero
thf(fact_1202_powr__nonneg__iff,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% powr_nonneg_iff
thf(fact_1203_powr__less__cancel__iff,axiom,
! [X: real,A: real,B2: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B2 ) )
= ( ord_less_real @ A @ B2 ) ) ) ).
% powr_less_cancel_iff
thf(fact_1204_powr__eq__one__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( powr_real @ A @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ) ).
% powr_eq_one_iff
thf(fact_1205_powr__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ one_one_real )
= X ) ) ).
% powr_one
thf(fact_1206_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr_real @ X @ one_one_real )
= X )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% powr_one_gt_zero_iff
thf(fact_1207_powr__le__cancel__iff,axiom,
! [X: real,A: real,B2: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B2 ) )
= ( ord_less_eq_real @ A @ B2 ) ) ) ).
% powr_le_cancel_iff
thf(fact_1208_powr__log__cancel,axiom,
! [A: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ A @ ( log @ A @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_1209_log__powr__cancel,axiom,
! [A: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ ( powr_real @ A @ Y ) )
= Y ) ) ) ).
% log_powr_cancel
thf(fact_1210_powr__powr,axiom,
! [X: real,A: real,B2: real] :
( ( powr_real @ ( powr_real @ X @ A ) @ B2 )
= ( powr_real @ X @ ( times_times_real @ A @ B2 ) ) ) ).
% powr_powr
thf(fact_1211_powr__less__mono2__neg,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_1212_powr__non__neg,axiom,
! [A: real,X: real] :
~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% powr_non_neg
thf(fact_1213_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).
% powr_ge_pzero
thf(fact_1214_powr__mono2,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_mono2
thf(fact_1215_powr__less__mono,axiom,
! [A: real,B2: real,X: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B2 ) ) ) ) ).
% powr_less_mono
thf(fact_1216_powr__less__cancel,axiom,
! [X: real,A: real,B2: real] :
( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B2 ) )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% powr_less_cancel
thf(fact_1217_powr__mono,axiom,
! [A: real,B2: real,X: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B2 ) ) ) ) ).
% powr_mono
thf(fact_1218_powr__powr__swap,axiom,
! [X: real,A: real,B2: real] :
( ( powr_real @ ( powr_real @ X @ A ) @ B2 )
= ( powr_real @ ( powr_real @ X @ B2 ) @ A ) ) ).
% powr_powr_swap
thf(fact_1219_powr__mono2_H,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% powr_mono2'
thf(fact_1220_powr__less__mono2,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_less_mono2
thf(fact_1221_gr__one__powr,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_1222_powr__inj,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ( powr_real @ A @ X )
= ( powr_real @ A @ Y ) )
= ( X = Y ) ) ) ) ).
% powr_inj
thf(fact_1223_powr__le1,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% powr_le1
thf(fact_1224_powr__mono__both,axiom,
! [A: real,B2: real,X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B2 ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_1225_ge__one__powr__ge__zero,axiom,
! [X: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_1226_powr__mult,axiom,
! [X: real,Y: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
= ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_mult
thf(fact_1227_ln__powr,axiom,
! [X: real,Y: real] :
( ( X != zero_zero_real )
=> ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
= ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).
% ln_powr
thf(fact_1228_log__powr,axiom,
! [X: real,B2: real,Y: real] :
( ( X != zero_zero_real )
=> ( ( log @ B2 @ ( powr_real @ X @ Y ) )
= ( times_times_real @ Y @ ( log @ B2 @ X ) ) ) ) ).
% log_powr
thf(fact_1229_powr__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
= ( power_power_real @ X @ N ) ) ) ).
% powr_realpow
thf(fact_1230_powr__less__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( powr_real @ B2 @ Y ) @ X )
= ( ord_less_real @ Y @ ( log @ B2 @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_1231_less__powr__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( powr_real @ B2 @ Y ) )
= ( ord_less_real @ ( log @ B2 @ X ) @ Y ) ) ) ) ).
% less_powr_iff
thf(fact_1232_log__less__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ B2 @ X ) @ Y )
= ( ord_less_real @ X @ ( powr_real @ B2 @ Y ) ) ) ) ) ).
% log_less_iff
thf(fact_1233_less__log__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ ( log @ B2 @ X ) )
= ( ord_less_real @ ( powr_real @ B2 @ Y ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_1234_powr__le__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ B2 @ Y ) @ X )
= ( ord_less_eq_real @ Y @ ( log @ B2 @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_1235_le__powr__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ ( powr_real @ B2 @ Y ) )
= ( ord_less_eq_real @ ( log @ B2 @ X ) @ Y ) ) ) ) ).
% le_powr_iff
thf(fact_1236_log__le__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ B2 @ X ) @ Y )
= ( ord_less_eq_real @ X @ ( powr_real @ B2 @ Y ) ) ) ) ) ).
% log_le_iff
thf(fact_1237_le__log__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ ( log @ B2 @ X ) )
= ( ord_less_eq_real @ ( powr_real @ B2 @ Y ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_1238_ln__powr__bound2,axiom,
! [X: real,A: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_1239_log__base__root,axiom,
! [N: nat,B2: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( log @ ( root @ N @ B2 ) @ X )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_1240_minus__log__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( B2 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( minus_minus_real @ Y @ ( log @ B2 @ X ) )
= ( log @ B2 @ ( divide_divide_real @ ( powr_real @ B2 @ Y ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_1241_add__log__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( B2 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( plus_plus_real @ Y @ ( log @ B2 @ X ) )
= ( log @ B2 @ ( times_times_real @ ( powr_real @ B2 @ Y ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_1242_real__root__zero,axiom,
! [N: nat] :
( ( root @ N @ zero_zero_real )
= zero_zero_real ) ).
% real_root_zero
thf(fact_1243_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X = A ) ) ).
% real_add_minus_iff
thf(fact_1244_root__0,axiom,
! [X: real] :
( ( root @ zero_zero_nat @ X )
= zero_zero_real ) ).
% root_0
thf(fact_1245_real__root__eq__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_1246_real__root__eq__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_1247_real__root__less__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_1248_real__root__le__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_1249_real__root__one,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_1250_real__root__eq__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= one_one_real )
= ( X = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_1251_real__root__gt__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_1252_real__root__lt__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_1253_real__root__le__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_1254_real__root__ge__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_1255_real__root__gt__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_1256_real__root__lt__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_1257_real__root__ge__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_1258_real__root__le__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_1259_real__root__pow__pos2,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_1260_real__root__pos__pos__le,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_1261_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X2: real,Y5: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y5 ) ) ) ) ).
% minus_real_def
thf(fact_1262_log__def,axiom,
( log
= ( ^ [A2: real,X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ A2 ) ) ) ) ).
% log_def
thf(fact_1263_real__root__less__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_1264_real__root__le__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_1265_real__root__power,axiom,
! [N: nat,X: real,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( power_power_real @ X @ K ) )
= ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_1266_ln__root,axiom,
! [N: nat,B2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ln_ln_real @ ( root @ N @ B2 ) )
= ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% ln_root
thf(fact_1267_log__root,axiom,
! [N: nat,A: real,B2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( log @ B2 @ ( root @ N @ A ) )
= ( divide_divide_real @ ( log @ B2 @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_root
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( risk_F2906766666041932210_value @ zero_z1425366712893667068ccount )
= zero_zero_real ) ).
%------------------------------------------------------------------------------