TPTP Problem File: SLH0041^1.p
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- 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 : Median_Method/0000_Median/prob_00450_016793__14877808_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1358 ( 886 unt; 88 typ; 0 def)
% Number of atoms : 3052 (1533 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 10043 ( 277 ~; 59 |; 155 &;8772 @)
% ( 0 <=>; 780 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 5 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 166 ( 166 >; 0 *; 0 +; 0 <<)
% Number of symbols : 77 ( 74 usr; 17 con; 0-4 aty)
% Number of variables : 2455 ( 33 ^;2395 !; 27 ?;2455 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:46:33.425
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__b_J,type,
sigma_measure_b: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__a_J,type,
sigma_measure_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (74)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001tf__b,type,
borel_5459123734250506525orel_b: sigma_measure_b ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
one_on2969667320475766781nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nonnegative____Real__Oennreal,type,
plus_p1859984266308609217nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nonnegative____Real__Oennreal,type,
times_1893300245718287421nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
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__Num__Onum,type,
times_times_num: num > num > num ).
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__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_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001t__Nat__Onat_001tf__b,type,
indepe3245197900929106295_nat_b: sigma_measure_a > ( nat > sigma_measure_b ) > ( nat > a > b ) > set_nat > $o ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nonnegative____Real__Oennreal,type,
semiri6283507881447550617nnreal: nat > extend8495563244428889912nnreal ).
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_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nonnegative____Real__Oennreal,type,
numera4658534427948366547nnreal: num > extend8495563244428889912nnreal ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le7381754540660121996nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Extended____Nonnegative____Real__Oennreal,type,
power_6007165696250533058nnreal: extend8495563244428889912nnreal > nat > extend8495563244428889912nnreal ).
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_Probability__Measure_Oprob__space_001tf__a,type,
probab7247484486040049089pace_a: sigma_measure_a > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(sy_c_Sigma__Algebra_Omeasure_001tf__a,type,
sigma_measure_a2: sigma_measure_a > set_a > real ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__a,type,
sigma_space_a: sigma_measure_a > set_a ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_I,type,
i: set_b ).
thf(sy_v_M,type,
m: sigma_measure_a ).
thf(sy_v_X,type,
x: nat > a > b ).
thf(sy_v__092_060alpha_062,type,
alpha: real ).
thf(sy_v__092_060epsilon_062,type,
epsilon: real ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1266)
thf(fact_0_assms_I2_J,axiom,
ord_less_real @ zero_zero_real @ alpha ).
% assms(2)
thf(fact_1_assms_I3_J,axiom,
member_real @ epsilon @ ( set_or1633881224788618240n_real @ zero_zero_real @ one_one_real ) ).
% assms(3)
thf(fact_2_assms_I5_J,axiom,
ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ ( ln_ln_real @ epsilon ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ alpha @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( semiri5074537144036343181t_real @ n ) ).
% assms(5)
thf(fact_3_indep__vars__cong,axiom,
! [I: set_nat,J: set_nat,X: nat > a > b,Y: nat > a > b,M: nat > sigma_measure_b,N: nat > sigma_measure_b] :
( ( I = J )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( X @ I2 )
= ( Y @ I2 ) ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( M @ I2 )
= ( N @ I2 ) ) )
=> ( ( indepe3245197900929106295_nat_b @ m @ M @ X @ I )
= ( indepe3245197900929106295_nat_b @ m @ N @ Y @ J ) ) ) ) ) ).
% indep_vars_cong
thf(fact_4_calculation,axiom,
ord_less_real @ zero_zero_real @ ( divide_divide_real @ ( uminus_uminus_real @ ( ln_ln_real @ epsilon ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ alpha @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% calculation
thf(fact_5_power2__minus,axiom,
! [A: real] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_6_power2__minus,axiom,
! [A: int] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_7_numeral__power__le__of__nat__cancel__iff,axiom,
! [I3: num,N2: nat,X2: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I3 ) @ N2 ) @ ( semiri5074537144036343181t_real @ X2 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) @ X2 ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_8_numeral__power__le__of__nat__cancel__iff,axiom,
! [I3: num,N2: nat,X2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X2 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) @ X2 ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_9_numeral__power__le__of__nat__cancel__iff,axiom,
! [I3: num,N2: nat,X2: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I3 ) @ N2 ) @ ( semiri1314217659103216013at_int @ X2 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) @ X2 ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_10_of__nat__le__numeral__power__cancel__iff,axiom,
! [X2: nat,I3: num,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I3 ) @ N2 ) )
= ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_11_of__nat__le__numeral__power__cancel__iff,axiom,
! [X2: nat,I3: num,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) )
= ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_12_of__nat__le__numeral__power__cancel__iff,axiom,
! [X2: nat,I3: num,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I3 ) @ N2 ) )
= ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_13_divide__le__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_14_le__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_15_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X2: num,N2: nat,Y2: nat] :
( ( ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 )
= ( semiri5074537144036343181t_real @ Y2 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
= Y2 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_16_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X2: num,N2: nat,Y2: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
= ( semiri1314217659103216013at_int @ Y2 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
= Y2 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_17_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X2: num,N2: nat,Y2: nat] :
( ( ( power_6007165696250533058nnreal @ ( numera4658534427948366547nnreal @ X2 ) @ N2 )
= ( semiri6283507881447550617nnreal @ Y2 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
= Y2 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_18_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X2: num,N2: nat,Y2: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
= ( semiri1316708129612266289at_nat @ Y2 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
= Y2 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_19_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: nat,X2: num,N2: nat] :
( ( ( semiri5074537144036343181t_real @ Y2 )
= ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) )
= ( Y2
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_20_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: nat,X2: num,N2: nat] :
( ( ( semiri1314217659103216013at_int @ Y2 )
= ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) )
= ( Y2
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_21_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: nat,X2: num,N2: nat] :
( ( ( semiri6283507881447550617nnreal @ Y2 )
= ( power_6007165696250533058nnreal @ ( numera4658534427948366547nnreal @ X2 ) @ N2 ) )
= ( Y2
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_22_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: nat,X2: num,N2: nat] :
( ( ( semiri1316708129612266289at_nat @ Y2 )
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) )
= ( Y2
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_23_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_24_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_25_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_real @ N2 ) ) ).
% of_nat_numeral
thf(fact_26_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% of_nat_numeral
thf(fact_27_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri6283507881447550617nnreal @ ( numeral_numeral_nat @ N2 ) )
= ( numera4658534427948366547nnreal @ N2 ) ) ).
% of_nat_numeral
thf(fact_28_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ N2 ) ) ).
% of_nat_numeral
thf(fact_29_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_30_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_31_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_32_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_33_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_34_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_35_indep__vars__subset,axiom,
! [M: nat > sigma_measure_b,X: nat > a > b,I: set_nat,J: set_nat] :
( ( indepe3245197900929106295_nat_b @ m @ M @ X @ I )
=> ( ( ord_less_eq_set_nat @ J @ I )
=> ( indepe3245197900929106295_nat_b @ m @ M @ X @ J ) ) ) ).
% indep_vars_subset
thf(fact_36_numeral__eq__iff,axiom,
! [M2: num,N2: num] :
( ( ( numeral_numeral_real @ M2 )
= ( numeral_numeral_real @ N2 ) )
= ( M2 = N2 ) ) ).
% numeral_eq_iff
thf(fact_37_numeral__eq__iff,axiom,
! [M2: num,N2: num] :
( ( ( numeral_numeral_nat @ M2 )
= ( numeral_numeral_nat @ N2 ) )
= ( M2 = N2 ) ) ).
% numeral_eq_iff
thf(fact_38_numeral__eq__iff,axiom,
! [M2: num,N2: num] :
( ( ( numeral_numeral_int @ M2 )
= ( numeral_numeral_int @ N2 ) )
= ( M2 = N2 ) ) ).
% numeral_eq_iff
thf(fact_39_numeral__eq__iff,axiom,
! [M2: num,N2: num] :
( ( ( numera4658534427948366547nnreal @ M2 )
= ( numera4658534427948366547nnreal @ N2 ) )
= ( M2 = N2 ) ) ).
% numeral_eq_iff
thf(fact_40_measure__ge__1__iff,axiom,
! [A2: set_a] :
( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_a2 @ m @ A2 ) )
= ( ( sigma_measure_a2 @ m @ A2 )
= one_one_real ) ) ).
% measure_ge_1_iff
thf(fact_41_numeral__le__iff,axiom,
! [M2: num,N2: num] :
( ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ M2 ) @ ( numera4658534427948366547nnreal @ N2 ) )
= ( ord_less_eq_num @ M2 @ N2 ) ) ).
% numeral_le_iff
thf(fact_42_numeral__le__iff,axiom,
! [M2: num,N2: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N2 ) )
= ( ord_less_eq_num @ M2 @ N2 ) ) ).
% numeral_le_iff
thf(fact_43_numeral__le__iff,axiom,
! [M2: num,N2: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
= ( ord_less_eq_num @ M2 @ N2 ) ) ).
% numeral_le_iff
thf(fact_44_numeral__le__iff,axiom,
! [M2: num,N2: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
= ( ord_less_eq_num @ M2 @ N2 ) ) ).
% numeral_le_iff
thf(fact_45_numeral__less__iff,axiom,
! [M2: num,N2: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N2 ) )
= ( ord_less_num @ M2 @ N2 ) ) ).
% numeral_less_iff
thf(fact_46_numeral__less__iff,axiom,
! [M2: num,N2: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
= ( ord_less_num @ M2 @ N2 ) ) ).
% numeral_less_iff
thf(fact_47_numeral__less__iff,axiom,
! [M2: num,N2: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
= ( ord_less_num @ M2 @ N2 ) ) ).
% numeral_less_iff
thf(fact_48_numeral__less__iff,axiom,
! [M2: num,N2: num] :
( ( ord_le7381754540660121996nnreal @ ( numera4658534427948366547nnreal @ M2 ) @ ( numera4658534427948366547nnreal @ N2 ) )
= ( ord_less_num @ M2 @ N2 ) ) ).
% numeral_less_iff
thf(fact_49_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_50_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_51_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_52_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ V ) @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ W ) @ Z ) )
= ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_53_numeral__times__numeral,axiom,
! [M2: num,N2: num] :
( ( times_times_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N2 ) )
= ( numeral_numeral_real @ ( times_times_num @ M2 @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_54_numeral__times__numeral,axiom,
! [M2: num,N2: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( times_times_num @ M2 @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_55_numeral__times__numeral,axiom,
! [M2: num,N2: num] :
( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
= ( numeral_numeral_int @ ( times_times_num @ M2 @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_56_numeral__times__numeral,axiom,
! [M2: num,N2: num] :
( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ M2 ) @ ( numera4658534427948366547nnreal @ N2 ) )
= ( numera4658534427948366547nnreal @ ( times_times_num @ M2 @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_57_neg__numeral__eq__iff,axiom,
! [M2: num,N2: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( M2 = N2 ) ) ).
% neg_numeral_eq_iff
thf(fact_58_neg__numeral__eq__iff,axiom,
! [M2: num,N2: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( M2 = N2 ) ) ).
% neg_numeral_eq_iff
thf(fact_59_num__double,axiom,
! [N2: num] :
( ( times_times_num @ ( bit0 @ one ) @ N2 )
= ( bit0 @ N2 ) ) ).
% num_double
thf(fact_60_power__one,axiom,
! [N2: nat] :
( ( power_6007165696250533058nnreal @ one_on2969667320475766781nnreal @ N2 )
= one_on2969667320475766781nnreal ) ).
% power_one
thf(fact_61_power__one,axiom,
! [N2: nat] :
( ( power_power_real @ one_one_real @ N2 )
= one_one_real ) ).
% power_one
thf(fact_62_power__one,axiom,
! [N2: nat] :
( ( power_power_nat @ one_one_nat @ N2 )
= one_one_nat ) ).
% power_one
thf(fact_63_power__one,axiom,
! [N2: nat] :
( ( power_power_int @ one_one_int @ N2 )
= one_one_int ) ).
% power_one
thf(fact_64_power__mult__numeral,axiom,
! [A: real,M2: num,N2: num] :
( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M2 @ N2 ) ) ) ) ).
% power_mult_numeral
thf(fact_65_power__mult__numeral,axiom,
! [A: nat,M2: num,N2: num] :
( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M2 @ N2 ) ) ) ) ).
% power_mult_numeral
thf(fact_66_power__mult__numeral,axiom,
! [A: int,M2: num,N2: num] :
( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M2 @ N2 ) ) ) ) ).
% power_mult_numeral
thf(fact_67_prob__space__axioms,axiom,
probab7247484486040049089pace_a @ m ).
% prob_space_axioms
thf(fact_68_neg__numeral__le__iff,axiom,
! [M2: num,N2: num] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( ord_less_eq_num @ N2 @ M2 ) ) ).
% neg_numeral_le_iff
thf(fact_69_neg__numeral__le__iff,axiom,
! [M2: num,N2: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( ord_less_eq_num @ N2 @ M2 ) ) ).
% neg_numeral_le_iff
thf(fact_70_neg__numeral__less__iff,axiom,
! [M2: num,N2: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( ord_less_num @ N2 @ M2 ) ) ).
% neg_numeral_less_iff
thf(fact_71_neg__numeral__less__iff,axiom,
! [M2: num,N2: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( ord_less_num @ N2 @ M2 ) ) ).
% neg_numeral_less_iff
thf(fact_72_mult__minus1__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1_right
thf(fact_73_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_74_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_75_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_76_mult__neg__numeral__simps_I1_J,axiom,
! [M2: num,N2: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( numeral_numeral_real @ ( times_times_num @ M2 @ N2 ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_77_mult__neg__numeral__simps_I1_J,axiom,
! [M2: num,N2: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( numeral_numeral_int @ ( times_times_num @ M2 @ N2 ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_78_mult__neg__numeral__simps_I2_J,axiom,
! [M2: num,N2: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N2 ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M2 @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_79_mult__neg__numeral__simps_I2_J,axiom,
! [M2: num,N2: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_80_mult__neg__numeral__simps_I3_J,axiom,
! [M2: num,N2: num] :
( ( times_times_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M2 @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_81_mult__neg__numeral__simps_I3_J,axiom,
! [M2: num,N2: num] :
( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_82_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_real
= ( numeral_numeral_real @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_83_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_84_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_int
= ( numeral_numeral_int @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_85_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_on2969667320475766781nnreal
= ( numera4658534427948366547nnreal @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_86_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numeral_numeral_real @ N2 )
= one_one_real )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_87_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numeral_numeral_nat @ N2 )
= one_one_nat )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_88_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numeral_numeral_int @ N2 )
= one_one_int )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_89_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numera4658534427948366547nnreal @ N2 )
= one_on2969667320475766781nnreal )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_90_power__strict__increasing__iff,axiom,
! [B: real,X2: nat,Y2: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y2 ) )
= ( ord_less_nat @ X2 @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_91_power__strict__increasing__iff,axiom,
! [B: nat,X2: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_nat @ X2 @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_92_power__strict__increasing__iff,axiom,
! [B: int,X2: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_nat @ X2 @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_93_power__inject__exp,axiom,
! [A: real,M2: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M2 )
= ( power_power_real @ A @ N2 ) )
= ( M2 = N2 ) ) ) ).
% power_inject_exp
thf(fact_94_power__inject__exp,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M2 )
= ( power_power_nat @ A @ N2 ) )
= ( M2 = N2 ) ) ) ).
% power_inject_exp
thf(fact_95_power__inject__exp,axiom,
! [A: int,M2: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M2 )
= ( power_power_int @ A @ N2 ) )
= ( M2 = N2 ) ) ) ).
% power_inject_exp
thf(fact_96_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_97_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_98_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_99_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X2 )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_100_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X2 )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_101_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri6283507881447550617nnreal @ X2 )
= ( power_6007165696250533058nnreal @ ( semiri6283507881447550617nnreal @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_102_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X2 )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_103_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
= ( semiri5074537144036343181t_real @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_104_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_105_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_6007165696250533058nnreal @ ( semiri6283507881447550617nnreal @ B ) @ W )
= ( semiri6283507881447550617nnreal @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_106_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_107_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_108_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_109_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_111_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_112_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_113_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_115_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_116_of__nat__power,axiom,
! [M2: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M2 @ N2 ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M2 ) @ N2 ) ) ).
% of_nat_power
thf(fact_117_of__nat__power,axiom,
! [M2: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M2 @ N2 ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M2 ) @ N2 ) ) ).
% of_nat_power
thf(fact_118_of__nat__power,axiom,
! [M2: nat,N2: nat] :
( ( semiri6283507881447550617nnreal @ ( power_power_nat @ M2 @ N2 ) )
= ( power_6007165696250533058nnreal @ ( semiri6283507881447550617nnreal @ M2 ) @ N2 ) ) ).
% of_nat_power
thf(fact_119_of__nat__power,axiom,
! [M2: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M2 @ N2 ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ N2 ) ) ).
% of_nat_power
thf(fact_120_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ one_on2969667320475766781nnreal )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_121_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_122_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_123_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_124_power__strict__decreasing__iff,axiom,
! [B: real,M2: nat,N2: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_125_power__strict__decreasing__iff,axiom,
! [B: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_126_power__strict__decreasing__iff,axiom,
! [B: int,M2: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_127_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
= B ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_128_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
= A )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_129_one__less__numeral__iff,axiom,
! [N2: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
= ( ord_less_num @ one @ N2 ) ) ).
% one_less_numeral_iff
thf(fact_130_one__less__numeral__iff,axiom,
! [N2: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
= ( ord_less_num @ one @ N2 ) ) ).
% one_less_numeral_iff
thf(fact_131_one__less__numeral__iff,axiom,
! [N2: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
= ( ord_less_num @ one @ N2 ) ) ).
% one_less_numeral_iff
thf(fact_132_one__less__numeral__iff,axiom,
! [N2: num] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N2 ) )
= ( ord_less_num @ one @ N2 ) ) ).
% one_less_numeral_iff
thf(fact_133_less__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_134_divide__less__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_135_numeral__eq__neg__one__iff,axiom,
! [N2: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) )
= ( uminus_uminus_real @ one_one_real ) )
= ( N2 = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_136_numeral__eq__neg__one__iff,axiom,
! [N2: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N2 = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_137_neg__one__eq__numeral__iff,axiom,
! [N2: num] :
( ( ( uminus_uminus_real @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( N2 = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_138_neg__one__eq__numeral__iff,axiom,
! [N2: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( N2 = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_139_of__nat__zero__less__power__iff,axiom,
! [X2: nat,N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X2 ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_140_of__nat__zero__less__power__iff,axiom,
! [X2: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X2 ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_141_of__nat__zero__less__power__iff,axiom,
! [X2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_142_left__minus__one__mult__self,axiom,
! [N2: nat,A: real] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_143_left__minus__one__mult__self,axiom,
! [N2: nat,A: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_144_minus__one__mult__self,axiom,
! [N2: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) )
= one_one_real ) ).
% minus_one_mult_self
thf(fact_145_minus__one__mult__self,axiom,
! [N2: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_146_bounded__measure,axiom,
! [A2: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) ) ).
% bounded_measure
thf(fact_147_prob__space,axiom,
( ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) )
= one_one_real ) ).
% prob_space
thf(fact_148_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_149_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_150_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_151_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_152_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_153_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_154_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) )
= ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_155_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) )
= ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_156_eq__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= B ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_157_divide__eq__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= A )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_158_neg__numeral__less__neg__one__iff,axiom,
! [M2: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( M2 != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_159_neg__numeral__less__neg__one__iff,axiom,
! [M2: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M2 != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_160_less__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_161_divide__less__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_162_power__increasing__iff,axiom,
! [B: real,X2: nat,Y2: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_163_power__increasing__iff,axiom,
! [B: nat,X2: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_164_power__increasing__iff,axiom,
! [B: int,X2: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_165_zero__eq__power2,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% zero_eq_power2
thf(fact_166_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_167_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_168_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_169_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_170_power__decreasing__iff,axiom,
! [B: real,M2: nat,N2: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_171_power__decreasing__iff,axiom,
! [B: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_172_power__decreasing__iff,axiom,
! [B: int,M2: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_173_power2__eq__iff__nonneg,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X2 = Y2 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_174_power2__eq__iff__nonneg,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X2 = Y2 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_175_power2__eq__iff__nonneg,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X2 = Y2 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_176_power2__less__eq__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% power2_less_eq_zero_iff
thf(fact_177_power2__less__eq__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_178_zero__less__power2,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_real ) ) ).
% zero_less_power2
thf(fact_179_zero__less__power2,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_180_power__minus1__even,axiom,
! [N2: nat] :
( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= one_one_real ) ).
% power_minus1_even
thf(fact_181_power__minus1__even,axiom,
! [N2: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= one_one_int ) ).
% power_minus1_even
thf(fact_182_of__nat__less__numeral__power__cancel__iff,axiom,
! [X2: nat,I3: num,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I3 ) @ N2 ) )
= ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_183_of__nat__less__numeral__power__cancel__iff,axiom,
! [X2: nat,I3: num,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I3 ) @ N2 ) )
= ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_184_of__nat__less__numeral__power__cancel__iff,axiom,
! [X2: nat,I3: num,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) )
= ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_185_numeral__power__less__of__nat__cancel__iff,axiom,
! [I3: num,N2: nat,X2: nat] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I3 ) @ N2 ) @ ( semiri5074537144036343181t_real @ X2 ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) @ X2 ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_186_numeral__power__less__of__nat__cancel__iff,axiom,
! [I3: num,N2: nat,X2: nat] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I3 ) @ N2 ) @ ( semiri1314217659103216013at_int @ X2 ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) @ X2 ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_187_numeral__power__less__of__nat__cancel__iff,axiom,
! [I3: num,N2: nat,X2: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X2 ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I3 ) @ N2 ) @ X2 ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_188_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_189_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_190_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_191_less__numeral__extra_I4_J,axiom,
~ ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ one_on2969667320475766781nnreal ) ).
% less_numeral_extra(4)
thf(fact_192_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_193_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_194_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_195_less__numeral__extra_I3_J,axiom,
~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ) ).
% less_numeral_extra(3)
thf(fact_196_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_197_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_198_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_199_less__numeral__extra_I1_J,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% less_numeral_extra(1)
thf(fact_200_le__num__One__iff,axiom,
! [X2: num] :
( ( ord_less_eq_num @ X2 @ one )
= ( X2 = one ) ) ).
% le_num_One_iff
thf(fact_201_power__mult,axiom,
! [A: real,M2: nat,N2: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M2 @ N2 ) )
= ( power_power_real @ ( power_power_real @ A @ M2 ) @ N2 ) ) ).
% power_mult
thf(fact_202_power__mult,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M2 @ N2 ) )
= ( power_power_nat @ ( power_power_nat @ A @ M2 ) @ N2 ) ) ).
% power_mult
thf(fact_203_power__mult,axiom,
! [A: int,M2: nat,N2: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M2 @ N2 ) )
= ( power_power_int @ ( power_power_int @ A @ M2 ) @ N2 ) ) ).
% power_mult
thf(fact_204_power__strict__decreasing,axiom,
! [N2: nat,N3: nat,A: real] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_205_power__strict__decreasing,axiom,
! [N2: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_206_power__strict__decreasing,axiom,
! [N2: nat,N3: nat,A: int] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_207_power__strict__increasing,axiom,
! [N2: nat,N3: nat,A: real] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_208_power__strict__increasing,axiom,
! [N2: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_209_power__strict__increasing,axiom,
! [N2: nat,N3: nat,A: int] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_210_power__less__imp__less__exp,axiom,
! [A: real,M2: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_211_power__less__imp__less__exp,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_212_power__less__imp__less__exp,axiom,
! [A: int,M2: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_213_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_214_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_215_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_216_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_217_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_6007165696250533058nnreal @ zero_z7100319975126383169nnreal @ N2 )
= one_on2969667320475766781nnreal ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_6007165696250533058nnreal @ zero_z7100319975126383169nnreal @ N2 )
= zero_z7100319975126383169nnreal ) ) ) ).
% power_0_left
thf(fact_218_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= one_one_real ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_219_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= one_one_nat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_220_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= one_one_int ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_221_not__numeral__less__one,axiom,
! [N2: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_222_not__numeral__less__one,axiom,
! [N2: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_223_not__numeral__less__one,axiom,
! [N2: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_224_not__numeral__less__one,axiom,
! [N2: num] :
~ ( ord_le7381754540660121996nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ one_on2969667320475766781nnreal ) ).
% not_numeral_less_one
thf(fact_225_not__numeral__less__zero,axiom,
! [N2: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_226_not__numeral__less__zero,axiom,
! [N2: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_227_not__numeral__less__zero,axiom,
! [N2: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_228_not__numeral__less__zero,axiom,
! [N2: num] :
~ ( ord_le7381754540660121996nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ zero_z7100319975126383169nnreal ) ).
% not_numeral_less_zero
thf(fact_229_zero__less__numeral,axiom,
! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% zero_less_numeral
thf(fact_230_zero__less__numeral,axiom,
! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% zero_less_numeral
thf(fact_231_zero__less__numeral,axiom,
! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% zero_less_numeral
thf(fact_232_zero__less__numeral,axiom,
! [N2: num] : ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( numera4658534427948366547nnreal @ N2 ) ) ).
% zero_less_numeral
thf(fact_233_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_234_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_235_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_236_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_237_zero__less__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_238_zero__less__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_239_zero__less__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_240_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_241_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_242_power2__nat__le__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% power2_nat_le_imp_le
thf(fact_243_power2__nat__le__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% power2_nat_le_eq_le
thf(fact_244_self__le__ge2__pow,axiom,
! [K: nat,M2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M2 @ ( power_power_nat @ K @ M2 ) ) ) ).
% self_le_ge2_pow
thf(fact_245_power__strict__mono,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_246_power__strict__mono,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_247_power__strict__mono,axiom,
! [A: int,B: int,N2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_248_le__numeral__extra_I4_J,axiom,
ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ one_on2969667320475766781nnreal ).
% le_numeral_extra(4)
thf(fact_249_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_250_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_251_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_252_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_253_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_254_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_255_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_256_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_257_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N2 ) ) ).
% zero_neq_numeral
thf(fact_258_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N2 ) ) ).
% zero_neq_numeral
thf(fact_259_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N2 ) ) ).
% zero_neq_numeral
thf(fact_260_zero__neq__numeral,axiom,
! [N2: num] :
( zero_z7100319975126383169nnreal
!= ( numera4658534427948366547nnreal @ N2 ) ) ).
% zero_neq_numeral
thf(fact_261_power__not__zero,axiom,
! [A: real,N2: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N2 )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_262_power__not__zero,axiom,
! [A: nat,N2: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N2 )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_263_power__not__zero,axiom,
! [A: int,N2: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N2 )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_264_power__Suc__less,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_265_power__Suc__less,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_266_power__Suc__less,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_267_power__le__imp__le__exp,axiom,
! [A: real,M2: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_268_power__le__imp__le__exp,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_269_power__le__imp__le__exp,axiom,
! [A: int,M2: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_270_power__less__imp__less__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_271_power__less__imp__less__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_272_power__less__imp__less__base,axiom,
! [A: int,N2: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_273_not__neg__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_274_not__neg__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_275_not__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).
% not_one_less_neg_numeral
thf(fact_276_not__one__less__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% not_one_less_neg_numeral
thf(fact_277_not__numeral__less__neg__one,axiom,
! [M2: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_less_neg_one
thf(fact_278_not__numeral__less__neg__one,axiom,
! [M2: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_279_neg__one__less__numeral,axiom,
! [M2: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M2 ) ) ).
% neg_one_less_numeral
thf(fact_280_neg__one__less__numeral,axiom,
! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).
% neg_one_less_numeral
thf(fact_281_neg__numeral__less__one,axiom,
! [M2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real ) ).
% neg_numeral_less_one
thf(fact_282_neg__numeral__less__one,axiom,
! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_283_power__less__power__Suc,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_284_power__less__power__Suc,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_285_power__less__power__Suc,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_286_power__gt1__lemma,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_287_power__gt1__lemma,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_288_power__gt1__lemma,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_289_not__zero__less__neg__numeral,axiom,
! [N2: num] :
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_290_not__zero__less__neg__numeral,axiom,
! [N2: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_291_neg__numeral__less__zero,axiom,
! [N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).
% neg_numeral_less_zero
thf(fact_292_neg__numeral__less__zero,axiom,
! [N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_293_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_294_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_295_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_296_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_297_power__decreasing,axiom,
! [N2: nat,N3: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_298_power__decreasing,axiom,
! [N2: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_299_power__decreasing,axiom,
! [N2: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_300_power__le__one,axiom,
! [A: real,N2: 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 @ N2 ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_301_power__le__one,axiom,
! [A: nat,N2: 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 @ N2 ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_302_power__le__one,axiom,
! [A: int,N2: 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 @ N2 ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_303_power__even__eq,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_304_power__even__eq,axiom,
! [A: nat,N2: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_305_power__even__eq,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_306_zero__le__even__power_H,axiom,
! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% zero_le_even_power'
thf(fact_307_zero__le__even__power_H,axiom,
! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% zero_le_even_power'
thf(fact_308_not__numeral__less__neg__numeral,axiom,
! [M2: num,N2: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_309_not__numeral__less__neg__numeral,axiom,
! [M2: num,N2: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_310_neg__numeral__less__numeral,axiom,
! [M2: num,N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N2 ) ) ).
% neg_numeral_less_numeral
thf(fact_311_neg__numeral__less__numeral,axiom,
! [M2: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% neg_numeral_less_numeral
thf(fact_312_one__le__numeral,axiom,
! [N2: num] : ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N2 ) ) ).
% one_le_numeral
thf(fact_313_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).
% one_le_numeral
thf(fact_314_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% one_le_numeral
thf(fact_315_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).
% one_le_numeral
thf(fact_316_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_317_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_318_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_319_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_320_power__increasing,axiom,
! [N2: nat,N3: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_321_power__increasing,axiom,
! [N2: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_322_power__increasing,axiom,
! [N2: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_323_one__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_324_one__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_325_one__le__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_326_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ zero_z7100319975126383169nnreal ) ).
% not_numeral_le_zero
thf(fact_327_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_328_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_329_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_330_zero__le__numeral,axiom,
! [N2: num] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( numera4658534427948366547nnreal @ N2 ) ) ).
% zero_le_numeral
thf(fact_331_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% zero_le_numeral
thf(fact_332_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% zero_le_numeral
thf(fact_333_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% zero_le_numeral
thf(fact_334_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_335_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_336_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_337_numeral__One,axiom,
( ( numera4658534427948366547nnreal @ one )
= one_on2969667320475766781nnreal ) ).
% numeral_One
thf(fact_338_one__neq__neg__numeral,axiom,
! [N2: num] :
( one_one_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% one_neq_neg_numeral
thf(fact_339_one__neq__neg__numeral,axiom,
! [N2: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% one_neq_neg_numeral
thf(fact_340_numeral__neq__neg__one,axiom,
! [N2: num] :
( ( numeral_numeral_real @ N2 )
!= ( uminus_uminus_real @ one_one_real ) ) ).
% numeral_neq_neg_one
thf(fact_341_numeral__neq__neg__one,axiom,
! [N2: num] :
( ( numeral_numeral_int @ N2 )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_342_left__right__inverse__power,axiom,
! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal,N2: nat] :
( ( ( times_1893300245718287421nnreal @ X2 @ Y2 )
= one_on2969667320475766781nnreal )
=> ( ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ X2 @ N2 ) @ ( power_6007165696250533058nnreal @ Y2 @ N2 ) )
= one_on2969667320475766781nnreal ) ) ).
% left_right_inverse_power
thf(fact_343_left__right__inverse__power,axiom,
! [X2: real,Y2: real,N2: nat] :
( ( ( times_times_real @ X2 @ Y2 )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X2 @ N2 ) @ ( power_power_real @ Y2 @ N2 ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_344_left__right__inverse__power,axiom,
! [X2: int,Y2: int,N2: nat] :
( ( ( times_times_int @ X2 @ Y2 )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N2 ) @ ( power_power_int @ Y2 @ N2 ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_345_left__right__inverse__power,axiom,
! [X2: nat,Y2: nat,N2: nat] :
( ( ( times_times_nat @ X2 @ Y2 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N2 ) @ ( power_power_nat @ Y2 @ N2 ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_346_zero__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_347_zero__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_348_zero__le__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_349_power__mono,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_350_power__mono,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_351_power__mono,axiom,
! [A: int,B: int,N2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_352_zero__neq__neg__numeral,axiom,
! [N2: num] :
( zero_zero_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% zero_neq_neg_numeral
thf(fact_353_zero__neq__neg__numeral,axiom,
! [N2: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% zero_neq_neg_numeral
thf(fact_354_power__one__over,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N2 )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% power_one_over
thf(fact_355_less__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_356_divide__less__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_357_le__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_358_divide__le__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_359_less__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_360_divide__less__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_361_half__gt__zero__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% half_gt_zero_iff
thf(fact_362_half__gt__zero,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_363_power2__less__0,axiom,
! [A: real] :
~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% power2_less_0
thf(fact_364_power2__less__0,axiom,
! [A: int] :
~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% power2_less_0
thf(fact_365_not__one__le__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).
% not_one_le_neg_numeral
thf(fact_366_not__one__le__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% not_one_le_neg_numeral
thf(fact_367_not__numeral__le__neg__one,axiom,
! [M2: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_le_neg_one
thf(fact_368_not__numeral__le__neg__one,axiom,
! [M2: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_369_neg__numeral__le__neg__one,axiom,
! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% neg_numeral_le_neg_one
thf(fact_370_neg__numeral__le__neg__one,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_371_neg__one__le__numeral,axiom,
! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M2 ) ) ).
% neg_one_le_numeral
thf(fact_372_neg__one__le__numeral,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).
% neg_one_le_numeral
thf(fact_373_neg__numeral__le__one,axiom,
! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real ) ).
% neg_numeral_le_one
thf(fact_374_neg__numeral__le__one,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_375_uminus__numeral__One,axiom,
( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% uminus_numeral_One
thf(fact_376_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_377_not__zero__le__neg__numeral,axiom,
! [N2: num] :
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_378_not__zero__le__neg__numeral,axiom,
! [N2: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_379_neg__numeral__le__zero,axiom,
! [N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).
% neg_numeral_le_zero
thf(fact_380_neg__numeral__le__zero,axiom,
! [N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_381_power__minus,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ A @ N2 ) ) ) ).
% power_minus
thf(fact_382_power__minus,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ A @ N2 ) ) ) ).
% power_minus
thf(fact_383_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ( numeral_numeral_real @ W )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_384_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ( divide_divide_real @ B @ C )
= ( numeral_numeral_real @ W ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_385_le__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_386_divide__le__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_387_power2__less__imp__less,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ X2 @ Y2 ) ) ) ).
% power2_less_imp_less
thf(fact_388_power2__less__imp__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ) ).
% power2_less_imp_less
thf(fact_389_power2__less__imp__less,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ord_less_int @ X2 @ Y2 ) ) ) ).
% power2_less_imp_less
thf(fact_390_numeral__neq__neg__numeral,axiom,
! [M2: num,N2: num] :
( ( numeral_numeral_real @ M2 )
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_391_numeral__neq__neg__numeral,axiom,
! [M2: num,N2: num] :
( ( numeral_numeral_int @ M2 )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_392_neg__numeral__neq__numeral,axiom,
! [M2: num,N2: num] :
( ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) )
!= ( numeral_numeral_real @ N2 ) ) ).
% neg_numeral_neq_numeral
thf(fact_393_neg__numeral__neq__numeral,axiom,
! [M2: num,N2: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
!= ( numeral_numeral_int @ N2 ) ) ).
% neg_numeral_neq_numeral
thf(fact_394_power__commuting__commutes,axiom,
! [X2: real,Y2: real,N2: nat] :
( ( ( times_times_real @ X2 @ Y2 )
= ( times_times_real @ Y2 @ X2 ) )
=> ( ( times_times_real @ ( power_power_real @ X2 @ N2 ) @ Y2 )
= ( times_times_real @ Y2 @ ( power_power_real @ X2 @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_395_power__commuting__commutes,axiom,
! [X2: int,Y2: int,N2: nat] :
( ( ( times_times_int @ X2 @ Y2 )
= ( times_times_int @ Y2 @ X2 ) )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N2 ) @ Y2 )
= ( times_times_int @ Y2 @ ( power_power_int @ X2 @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_396_power__commuting__commutes,axiom,
! [X2: nat,Y2: nat,N2: nat] :
( ( ( times_times_nat @ X2 @ Y2 )
= ( times_times_nat @ Y2 @ X2 ) )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N2 ) @ Y2 )
= ( times_times_nat @ Y2 @ ( power_power_nat @ X2 @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_397_power__mult__distrib,axiom,
! [A: real,B: real,N2: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N2 )
= ( times_times_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_398_power__mult__distrib,axiom,
! [A: int,B: int,N2: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N2 )
= ( times_times_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_399_power__mult__distrib,axiom,
! [A: nat,B: nat,N2: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N2 )
= ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_400_power__commutes,axiom,
! [A: real,N2: nat] :
( ( times_times_real @ ( power_power_real @ A @ N2 ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_401_power__commutes,axiom,
! [A: int,N2: nat] :
( ( times_times_int @ ( power_power_int @ A @ N2 ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_402_power__commutes,axiom,
! [A: nat,N2: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_403_power__divide,axiom,
! [A: real,B: real,N2: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N2 )
= ( divide_divide_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% power_divide
thf(fact_404_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_405_divide__eq__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( ( ( divide_divide_real @ B @ C )
= ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_406_one__power2,axiom,
( ( power_6007165696250533058nnreal @ one_on2969667320475766781nnreal @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_on2969667320475766781nnreal ) ).
% one_power2
thf(fact_407_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real ) ).
% one_power2
thf(fact_408_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_409_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_410_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_411_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_412_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_413_power2__eq__1__iff,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real )
= ( ( A = one_one_real )
| ( A
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% power2_eq_1_iff
thf(fact_414_power2__eq__1__iff,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int )
= ( ( A = one_one_int )
| ( A
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% power2_eq_1_iff
thf(fact_415_power2__le__imp__le,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ) ).
% power2_le_imp_le
thf(fact_416_power2__le__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).
% power2_le_imp_le
thf(fact_417_power2__le__imp__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ) ).
% power2_le_imp_le
thf(fact_418_power2__eq__imp__eq,axiom,
! [X2: real,Y2: real] :
( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( X2 = Y2 ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_419_power2__eq__imp__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( X2 = Y2 ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_420_power2__eq__imp__eq,axiom,
! [X2: int,Y2: int] :
( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( X2 = Y2 ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_421_zero__le__power2,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_422_zero__le__power2,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_423_not__numeral__le__neg__numeral,axiom,
! [M2: num,N2: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_424_not__numeral__le__neg__numeral,axiom,
! [M2: num,N2: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_425_neg__numeral__le__numeral,axiom,
! [M2: num,N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N2 ) ) ).
% neg_numeral_le_numeral
thf(fact_426_neg__numeral__le__numeral,axiom,
! [M2: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% neg_numeral_le_numeral
thf(fact_427_mult__numeral__1__right,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_428_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_429_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_430_mult__numeral__1__right,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A @ ( numera4658534427948366547nnreal @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_431_mult__numeral__1,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_432_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_433_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_434_mult__numeral__1,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_435_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_436_square__le__1,axiom,
! [X2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
=> ( ( ord_less_eq_real @ X2 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% square_le_1
thf(fact_437_square__le__1,axiom,
! [X2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X2 )
=> ( ( ord_less_eq_int @ X2 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% square_le_1
thf(fact_438_mult__1s__ring__1_I1_J,axiom,
! [B: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
= ( uminus_uminus_real @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_439_mult__1s__ring__1_I1_J,axiom,
! [B: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_440_mult__1s__ring__1_I2_J,axiom,
! [B: real] :
( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
= ( uminus_uminus_real @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_441_mult__1s__ring__1_I2_J,axiom,
! [B: int] :
( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_442_power__minus__Bit0,axiom,
! [X2: real,K: num] :
( ( power_power_real @ ( uminus_uminus_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_443_power__minus__Bit0,axiom,
! [X2: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_444_power2__eq__square,axiom,
! [A: real] :
( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ A @ A ) ) ).
% power2_eq_square
thf(fact_445_power2__eq__square,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A @ A ) ) ).
% power2_eq_square
thf(fact_446_power2__eq__square,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A @ A ) ) ).
% power2_eq_square
thf(fact_447_power4__eq__xxxx,axiom,
! [X2: real] :
( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_real @ ( times_times_real @ ( times_times_real @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_448_power4__eq__xxxx,axiom,
! [X2: int] :
( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_449_power4__eq__xxxx,axiom,
! [X2: nat] :
( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_450_power2__eq__iff,axiom,
! [X2: real,Y2: real] :
( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X2 = Y2 )
| ( X2
= ( uminus_uminus_real @ Y2 ) ) ) ) ).
% power2_eq_iff
thf(fact_451_power2__eq__iff,axiom,
! [X2: int,Y2: int] :
( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X2 = Y2 )
| ( X2
= ( uminus_uminus_int @ Y2 ) ) ) ) ).
% power2_eq_iff
thf(fact_452_minus__power__mult__self,axiom,
! [A: real,N2: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) )
= ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% minus_power_mult_self
thf(fact_453_minus__power__mult__self,axiom,
! [A: int,N2: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% minus_power_mult_self
thf(fact_454_sum__power2__eq__zero__iff,axiom,
! [X2: real,Y2: real] :
( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_455_sum__power2__eq__zero__iff,axiom,
! [X2: int,Y2: int] :
( ( ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_456_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_457_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_458_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_459_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_460_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_461_one__add__one,axiom,
( ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ one_on2969667320475766781nnreal )
= ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_462_minus__1__div__2__eq,axiom,
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_2_eq
thf(fact_463_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_464_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_465_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_466_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_467_subprob__measure__le__1,axiom,
! [X: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ X ) @ one_one_real ) ).
% subprob_measure_le_1
thf(fact_468_prob__le__1,axiom,
! [A2: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ one_one_real ) ).
% prob_le_1
thf(fact_469_numeral__le__real__of__nat__iff,axiom,
! [N2: num,M2: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M2 ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M2 ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_470_ln__le__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
= ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_471_ln__ge__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
= ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% ln_ge_zero_iff
thf(fact_472_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_473_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_474_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_475_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_476_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_477_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_478_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_479_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_480_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_481_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_482_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_483_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_484_numeral__plus__numeral,axiom,
! [M2: num,N2: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N2 ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_485_numeral__plus__numeral,axiom,
! [M2: num,N2: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_486_numeral__plus__numeral,axiom,
! [M2: num,N2: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_487_numeral__plus__numeral,axiom,
! [M2: num,N2: num] :
( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ M2 ) @ ( numera4658534427948366547nnreal @ N2 ) )
= ( numera4658534427948366547nnreal @ ( plus_plus_num @ M2 @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_488_add__numeral__left,axiom,
! [V: num,W: num,Z: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_489_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_490_add__numeral__left,axiom,
! [V: num,W: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_491_add__numeral__left,axiom,
! [V: num,W: num,Z: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ V ) @ ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ W ) @ Z ) )
= ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_492_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_493_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_494_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_495_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_496_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_497_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_498_real__add__minus__iff,axiom,
! [X2: real,A: real] :
( ( ( plus_plus_real @ X2 @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X2 = A ) ) ).
% real_add_minus_iff
thf(fact_499_sum__squares__eq__zero__iff,axiom,
! [X2: real,Y2: real] :
( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_500_sum__squares__eq__zero__iff,axiom,
! [X2: int,Y2: int] :
( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_501_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_502_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_503_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_504_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_505_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_506_distrib__left__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_507_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_508_distrib__left__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_509_distrib__left__numeral,axiom,
! [V: num,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ V ) @ ( plus_p1859984266308609217nnreal @ B @ C ) )
= ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ V ) @ B ) @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_510_distrib__right__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_511_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_512_distrib__right__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_513_distrib__right__numeral,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,V: num] :
( ( times_1893300245718287421nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ ( numera4658534427948366547nnreal @ V ) )
= ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ A @ ( numera4658534427948366547nnreal @ V ) ) @ ( times_1893300245718287421nnreal @ B @ ( numera4658534427948366547nnreal @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_514_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_515_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_516_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_517_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_518_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_519_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_520_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_521_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_522_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N2: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N2 ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_523_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N2: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_524_divide__minus1,axiom,
! [X2: real] :
( ( divide_divide_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X2 ) ) ).
% divide_minus1
thf(fact_525_power__add__numeral,axiom,
! [A: real,M2: num,N2: num] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) ) ).
% power_add_numeral
thf(fact_526_power__add__numeral,axiom,
! [A: int,M2: num,N2: num] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) ) ).
% power_add_numeral
thf(fact_527_power__add__numeral,axiom,
! [A: nat,M2: num,N2: num] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) ) ).
% power_add_numeral
thf(fact_528_power__add__numeral2,axiom,
! [A: real,M2: num,N2: num,B: real] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
= ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_529_power__add__numeral2,axiom,
! [A: int,M2: num,N2: num,B: int] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
= ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_530_power__add__numeral2,axiom,
! [A: nat,M2: num,N2: num,B: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
= ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_531_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_532_ln__inj__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ( ln_ln_real @ X2 )
= ( ln_ln_real @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ).
% ln_inj_iff
thf(fact_533_ln__less__cancel__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) )
= ( ord_less_real @ X2 @ Y2 ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_534_nat__zero__less__power__iff,axiom,
! [X2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_535_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_536_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_537_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_538_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_539_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_540_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_541_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_542_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_543_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_544_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_545_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_546_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_547_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_548_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_549_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_550_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_551_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_552_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ one_on2969667320475766781nnreal )
= ( numera4658534427948366547nnreal @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_553_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_554_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_555_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_556_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N2 ) )
= ( numera4658534427948366547nnreal @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_557_power__eq__0__iff,axiom,
! [A: real,N2: nat] :
( ( ( power_power_real @ A @ N2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_558_power__eq__0__iff,axiom,
! [A: nat,N2: nat] :
( ( ( power_power_nat @ A @ N2 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_559_power__eq__0__iff,axiom,
! [A: int,N2: nat] :
( ( ( power_power_int @ A @ N2 )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_560_ln__le__cancel__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) )
= ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_561_ln__eq__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ( ln_ln_real @ X2 )
= zero_zero_real )
= ( X2 = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_562_ln__gt__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
= ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% ln_gt_zero_iff
thf(fact_563_ln__less__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
= ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_564_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_565_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_566_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_567_power__mono__iff,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_568_power__mono__iff,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_569_power__mono__iff,axiom,
! [A: int,B: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_570_numeral__less__real__of__nat__iff,axiom,
! [W: num,N2: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N2 ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_571_real__of__nat__less__numeral__iff,axiom,
! [N2: nat,W: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W ) )
= ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_572_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2 != zero_zero_int )
=> ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_573_nat__induct2,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct2
thf(fact_574_add__One__commute,axiom,
! [N2: num] :
( ( plus_plus_num @ one @ N2 )
= ( plus_plus_num @ N2 @ one ) ) ).
% add_One_commute
thf(fact_575_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_576_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_577_minus__1__div__exp__eq__int,axiom,
! [N2: nat] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_exp_eq_int
thf(fact_578_not__exp__less__eq__0__int,axiom,
! [N2: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_579_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_580_add__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_581_power__add,axiom,
! [A: real,M2: nat,N2: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M2 @ N2 ) )
= ( times_times_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N2 ) ) ) ).
% power_add
thf(fact_582_power__add,axiom,
! [A: int,M2: nat,N2: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M2 @ N2 ) )
= ( times_times_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) ) ) ).
% power_add
thf(fact_583_power__add,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M2 @ N2 ) )
= ( times_times_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% power_add
thf(fact_584_ex__power__ivl1,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ? [N4: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N4 ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_585_ex__power__ivl2,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ? [N4: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N4 ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_586_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_587_nat__power__less__imp__less,axiom,
! [I3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I3 )
=> ( ( ord_less_nat @ ( power_power_nat @ I3 @ M2 ) @ ( power_power_nat @ I3 @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_588_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_589_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X5 ) ).
% linordered_field_no_lb
thf(fact_590_complete__real,axiom,
! [S: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S )
=> ( ? [Z2: real] :
! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z2 ) )
=> ? [Y3: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S )
=> ( ord_less_eq_real @ X5 @ 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_591_exp__add__not__zero__imp__left,axiom,
! [M2: nat,N2: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
thf(fact_592_exp__add__not__zero__imp__left,axiom,
! [M2: nat,N2: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
thf(fact_593_exp__add__not__zero__imp__right,axiom,
! [M2: nat,N2: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
thf(fact_594_exp__add__not__zero__imp__right,axiom,
! [M2: nat,N2: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
thf(fact_595_div__exp__eq,axiom,
! [A: int,M2: nat,N2: nat] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ).
% div_exp_eq
thf(fact_596_div__exp__eq,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ).
% div_exp_eq
thf(fact_597_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N5: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N5 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% nat_less_real_le
thf(fact_598_real__archimedian__rdiv__eq__0,axiom,
! [X2: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( 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 ) @ X2 ) @ C ) )
=> ( X2 = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_599_field__le__epsilon,axiom,
! [X2: real,Y2: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y2 @ E ) ) )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% field_le_epsilon
thf(fact_600_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_601_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_602_add__frac__eq,axiom,
! [Y2: real,Z: real,X2: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_603_add__frac__num,axiom,
! [Y2: real,X2: real,Z: real] :
( ( Y2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y2 ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y2 ) ) @ Y2 ) ) ) ).
% add_frac_num
thf(fact_604_add__num__frac,axiom,
! [Y2: real,Z: real,X2: real] :
( ( Y2 != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X2 @ Y2 ) )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y2 ) ) @ Y2 ) ) ) ).
% add_num_frac
thf(fact_605_add__divide__eq__iff,axiom,
! [Z: real,X2: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X2 @ ( divide_divide_real @ Y2 @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_606_divide__add__eq__iff,axiom,
! [Z: real,X2: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Z ) @ Y2 )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_607_gt__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_608_less__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_609_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_610_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_611_real__0__le__add__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y2 ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y2 ) ) ).
% real_0_le_add_iff
thf(fact_612_real__add__le__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
= ( ord_less_eq_real @ Y2 @ ( uminus_uminus_real @ X2 ) ) ) ).
% real_add_le_0_iff
thf(fact_613_real__0__less__add__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y2 ) )
= ( ord_less_real @ ( uminus_uminus_real @ X2 ) @ Y2 ) ) ).
% real_0_less_add_iff
thf(fact_614_real__add__less__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
= ( ord_less_real @ Y2 @ ( uminus_uminus_real @ X2 ) ) ) ).
% real_add_less_0_iff
thf(fact_615_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_616_times__divide__times__eq,axiom,
! [X2: real,Y2: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y2 @ W ) ) ) ).
% times_divide_times_eq
thf(fact_617_divide__divide__times__eq,axiom,
! [X2: real,Y2: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ W ) @ ( times_times_real @ Y2 @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_618_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_619_self__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% self_le_power
thf(fact_620_self__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% self_le_power
thf(fact_621_self__le__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% self_le_power
thf(fact_622_div__mult2__numeral__eq,axiom,
! [A: int,K: num,L: num] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_623_div__mult2__numeral__eq,axiom,
! [A: nat,K: num,L: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_624_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_625_minus__divide__right,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_divide_right
thf(fact_626_minus__divide__divide,axiom,
! [A: real,B: real] :
( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ).
% minus_divide_divide
thf(fact_627_minus__divide__left,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_628_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M2: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_629_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M2: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M2 @ N2 ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_630_add__divide__eq__if__simps_I3_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_631_minus__divide__add__eq__iff,axiom,
! [Z: real,X2: real,Y2: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y2 )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_632_field__sum__of__halves,axiom,
! [X2: real] :
( ( plus_plus_real @ ( divide_divide_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= X2 ) ).
% field_sum_of_halves
thf(fact_633_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N5: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N5 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_634_ln__add__one__self__le__self,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% ln_add_one_self_le_self
thf(fact_635_ln__mult,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ln_ln_real @ ( times_times_real @ X2 @ Y2 ) )
= ( plus_plus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) ) ) ) ) ).
% ln_mult
thf(fact_636_field__less__half__sum,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ X2 @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% field_less_half_sum
thf(fact_637_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X2 ) @ one_one_real ) ) ).
% one_plus_numeral_commute
thf(fact_638_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_639_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X2 ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_640_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ X2 ) )
= ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ X2 ) @ one_on2969667320475766781nnreal ) ) ).
% one_plus_numeral_commute
thf(fact_641_numeral__Bit0,axiom,
! [N2: num] :
( ( numeral_numeral_real @ ( bit0 @ N2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_642_numeral__Bit0,axiom,
! [N2: num] :
( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_643_numeral__Bit0,axiom,
! [N2: num] :
( ( numeral_numeral_int @ ( bit0 @ N2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_644_numeral__Bit0,axiom,
! [N2: num] :
( ( numera4658534427948366547nnreal @ ( bit0 @ N2 ) )
= ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ ( numera4658534427948366547nnreal @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_645_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= zero_zero_real ) ) ).
% zero_power
thf(fact_646_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_647_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= zero_zero_int ) ) ).
% zero_power
thf(fact_648_inverse__of__nat__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( N2 != zero_zero_nat )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_649_ln__add__one__self__le__self2,axiom,
! [X2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% ln_add_one_self_le_self2
thf(fact_650_power__0,axiom,
! [A: extend8495563244428889912nnreal] :
( ( power_6007165696250533058nnreal @ A @ zero_zero_nat )
= one_on2969667320475766781nnreal ) ).
% power_0
thf(fact_651_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_652_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_653_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_654_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_655_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_656_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_657_divide__nonneg__nonneg,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_658_divide__nonneg__nonpos,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_659_divide__nonpos__nonneg,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_660_divide__nonpos__nonpos,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_661_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_662_divide__neg__neg,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ zero_zero_real )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).
% divide_neg_neg
thf(fact_663_divide__neg__pos,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_664_divide__pos__neg,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_665_divide__pos__pos,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).
% divide_pos_pos
thf(fact_666_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_667_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_668_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_669_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_670_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_671_frac__eq__eq,axiom,
! [Y2: real,Z: real,X2: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X2 @ Y2 )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X2 @ Z )
= ( times_times_real @ W @ Y2 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_672_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_673_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_674_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_675_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_676_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_677_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_678_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_679_nonzero__minus__divide__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_680_nonzero__minus__divide__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_681_div__mult2__eq_H,axiom,
! [A: int,M2: nat,N2: nat] :
( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% div_mult2_eq'
thf(fact_682_div__mult2__eq_H,axiom,
! [A: nat,M2: nat,N2: nat] :
( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% div_mult2_eq'
thf(fact_683_real__arch__pow,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ? [N4: nat] : ( ord_less_real @ Y2 @ ( power_power_real @ X2 @ N4 ) ) ) ).
% real_arch_pow
thf(fact_684_real__minus__mult__self__le,axiom,
! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X2 @ X2 ) ) ).
% real_minus_mult_self_le
thf(fact_685_real__of__nat__div4,axiom,
! [N2: nat,X2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% real_of_nat_div4
thf(fact_686_ln__less__self,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% ln_less_self
thf(fact_687_Bernoulli__inequality,axiom,
! [X2: real,N2: nat] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N2 ) ) ) ).
% Bernoulli_inequality
thf(fact_688_sum__squares__le__zero__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) @ zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_689_sum__squares__le__zero__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) @ zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_690_sum__squares__gt__zero__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) )
= ( ( X2 != zero_zero_real )
| ( Y2 != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_691_sum__squares__gt__zero__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) )
= ( ( X2 != zero_zero_int )
| ( Y2 != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_692_power__eq__iff__eq__base,axiom,
! [N2: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N2 )
= ( power_power_real @ B @ N2 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_693_power__eq__iff__eq__base,axiom,
! [N2: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N2 )
= ( power_power_nat @ B @ N2 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_694_power__eq__iff__eq__base,axiom,
! [N2: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N2 )
= ( power_power_int @ B @ N2 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_695_power__eq__imp__eq__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ( power_power_real @ A @ N2 )
= ( power_power_real @ B @ N2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_696_power__eq__imp__eq__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ( power_power_nat @ A @ N2 )
= ( power_power_nat @ B @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_697_power__eq__imp__eq__base,axiom,
! [A: int,N2: nat,B: int] :
( ( ( power_power_int @ A @ N2 )
= ( power_power_int @ B @ N2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_698_frac__le,axiom,
! [Y2: real,X2: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_699_frac__less,axiom,
! [X2: real,Y2: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_700_frac__less2,axiom,
! [X2: real,Y2: real,W: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_701_divide__le__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_702_divide__nonneg__neg,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_703_divide__nonneg__pos,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).
% divide_nonneg_pos
thf(fact_704_divide__nonpos__neg,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).
% divide_nonpos_neg
thf(fact_705_divide__nonpos__pos,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_706_divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_707_less__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_708_neg__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_709_neg__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_710_pos__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_711_pos__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_712_mult__imp__div__pos__less,axiom,
! [Y2: real,X2: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X2 @ ( times_times_real @ Z @ Y2 ) )
=> ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_713_mult__imp__less__div__pos,axiom,
! [Y2: real,Z: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( times_times_real @ Z @ Y2 ) @ X2 )
=> ( ord_less_real @ Z @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_714_divide__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_715_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_716_divide__less__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_717_less__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_718_eq__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= ( uminus_uminus_real @ B ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_719_minus__divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
= A )
= ( ( ( C != zero_zero_real )
=> ( ( uminus_uminus_real @ B )
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_720_nonzero__neg__divide__eq__eq,axiom,
! [B: real,A: real,C: real] :
( ( B != zero_zero_real )
=> ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= C )
= ( ( uminus_uminus_real @ A )
= ( times_times_real @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_721_nonzero__neg__divide__eq__eq2,axiom,
! [B: real,C: real,A: real] :
( ( B != zero_zero_real )
=> ( ( C
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
= ( ( times_times_real @ C @ B )
= ( uminus_uminus_real @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_722_divide__eq__minus__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B != zero_zero_real )
& ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_723_numeral__Bit0__div__2,axiom,
! [N2: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N2 ) ) ).
% numeral_Bit0_div_2
thf(fact_724_numeral__Bit0__div__2,axiom,
! [N2: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N2 ) ) ).
% numeral_Bit0_div_2
thf(fact_725_real__arch__pow__inv,axiom,
! [Y2: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N4 ) @ Y2 ) ) ) ).
% real_arch_pow_inv
thf(fact_726_reals__Archimedean3,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ! [Y5: real] :
? [N4: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X2 ) ) ) ).
% reals_Archimedean3
thf(fact_727_ln__bound,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% ln_bound
thf(fact_728_ln__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq_real @ one_one_real @ X2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% ln_ge_zero
thf(fact_729_one__less__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_730_one__less__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_731_one__less__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_732_ln__gt__zero,axiom,
! [X2: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% ln_gt_zero
thf(fact_733_ln__less__zero,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_734_ln__gt__zero__imp__gt__one,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_735_field__le__mult__one__interval,axiom,
! [X2: real,Y2: real] :
( ! [Z3: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ Z3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z3 @ X2 ) @ Y2 ) ) )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% field_le_mult_one_interval
thf(fact_736_divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_737_le__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_738_divide__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_739_neg__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_740_neg__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_741_pos__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_742_pos__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_743_mult__imp__div__pos__le,axiom,
! [Y2: real,X2: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ X2 @ ( times_times_real @ Z @ Y2 ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_744_mult__imp__le__div__pos,axiom,
! [Y2: real,Z: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y2 ) @ X2 )
=> ( ord_less_eq_real @ Z @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_745_divide__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_746_divide__le__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_le_eq_1
thf(fact_747_le__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ A ) ) ) ) ).
% le_divide_eq_1
thf(fact_748_less__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_749_minus__divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_750_neg__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_751_neg__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_752_pos__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_753_pos__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_754_mult__2,axiom,
! [Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2
thf(fact_755_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_756_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_757_mult__2,axiom,
! [Z: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) @ Z )
= ( plus_p1859984266308609217nnreal @ Z @ Z ) ) ).
% mult_2
thf(fact_758_mult__2__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2_right
thf(fact_759_mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_760_mult__2__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2_right
thf(fact_761_mult__2__right,axiom,
! [Z: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ Z @ ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) )
= ( plus_p1859984266308609217nnreal @ Z @ Z ) ) ).
% mult_2_right
thf(fact_762_left__add__twice,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_763_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_764_left__add__twice,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_765_left__add__twice,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ A @ ( plus_p1859984266308609217nnreal @ A @ B ) )
= ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_766_ln__ge__zero__imp__ge__one,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_767_less__exp,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% less_exp
thf(fact_768_le__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_769_minus__divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_770_neg__le__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_771_neg__minus__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_772_pos__le__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_773_pos__minus__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_774_of__nat__less__two__power,axiom,
! [N2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).
% of_nat_less_two_power
thf(fact_775_of__nat__less__two__power,axiom,
! [N2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% of_nat_less_two_power
thf(fact_776_two__realpow__ge__one,axiom,
! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).
% two_realpow_ge_one
thf(fact_777_realpow__square__minus__le,axiom,
! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% realpow_square_minus_le
thf(fact_778_ln__realpow,axiom,
! [X2: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ln_ln_real @ ( power_power_real @ X2 @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X2 ) ) ) ) ).
% ln_realpow
thf(fact_779_sum__power2__ge__zero,axiom,
! [X2: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_780_sum__power2__ge__zero,axiom,
! [X2: int,Y2: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_781_sum__power2__le__zero__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_782_sum__power2__le__zero__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_783_not__sum__power2__lt__zero,axiom,
! [X2: real,Y2: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% not_sum_power2_lt_zero
thf(fact_784_not__sum__power2__lt__zero,axiom,
! [X2: int,Y2: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_785_sum__power2__gt__zero__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X2 != zero_zero_real )
| ( Y2 != zero_zero_real ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_786_sum__power2__gt__zero__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X2 != zero_zero_int )
| ( Y2 != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_787_power2__sum,axiom,
! [X2: real,Y2: real] :
( ( power_power_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_788_power2__sum,axiom,
! [X2: nat,Y2: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_789_power2__sum,axiom,
! [X2: int,Y2: int] :
( ( power_power_int @ ( plus_plus_int @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_790_power2__sum,axiom,
! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( power_6007165696250533058nnreal @ ( plus_p1859984266308609217nnreal @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_p1859984266308609217nnreal @ ( plus_p1859984266308609217nnreal @ ( power_6007165696250533058nnreal @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_6007165696250533058nnreal @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_791_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_792_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_793_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( semiri6283507881447550617nnreal @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_794_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_795_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y2: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).
% semiring_norm(171)
thf(fact_796_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y2: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).
% semiring_norm(171)
thf(fact_797_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y2: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(170)
thf(fact_798_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(170)
thf(fact_799_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y2: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y2 ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(169)
thf(fact_800_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y2: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y2 ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(169)
thf(fact_801_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_802_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_803_arith__geo__mean,axiom,
! [U: real,X2: real,Y2: real] :
( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ X2 @ Y2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_804_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ M2 ) @ zero_z7100319975126383169nnreal )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_805_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_806_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_807_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_808_semiring__norm_I87_J,axiom,
! [M2: num,N2: num] :
( ( ( bit0 @ M2 )
= ( bit0 @ N2 ) )
= ( M2 = N2 ) ) ).
% semiring_norm(87)
thf(fact_809_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_810_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_811_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri6283507881447550617nnreal @ M2 )
= ( semiri6283507881447550617nnreal @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_812_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_813_semiring__norm_I83_J,axiom,
! [N2: num] :
( one
!= ( bit0 @ N2 ) ) ).
% semiring_norm(83)
thf(fact_814_semiring__norm_I85_J,axiom,
! [M2: num] :
( ( bit0 @ M2 )
!= one ) ).
% semiring_norm(85)
thf(fact_815_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_816_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_817_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_818_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_819_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_820_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_821_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_822_real__divide__square__eq,axiom,
! [R: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
= ( divide_divide_real @ A @ R ) ) ).
% real_divide_square_eq
thf(fact_823_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_824_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_825_mult__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_826_mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_827_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_828_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_829_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_830_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_831_semiring__norm_I6_J,axiom,
! [M2: num,N2: num] :
( ( plus_plus_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( plus_plus_num @ M2 @ N2 ) ) ) ).
% semiring_norm(6)
thf(fact_832_semiring__norm_I13_J,axiom,
! [M2: num,N2: num] :
( ( times_times_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M2 @ N2 ) ) ) ) ).
% semiring_norm(13)
thf(fact_833_semiring__norm_I11_J,axiom,
! [M2: num] :
( ( times_times_num @ M2 @ one )
= M2 ) ).
% semiring_norm(11)
thf(fact_834_semiring__norm_I12_J,axiom,
! [N2: num] :
( ( times_times_num @ one @ N2 )
= N2 ) ).
% semiring_norm(12)
thf(fact_835_semiring__norm_I78_J,axiom,
! [M2: num,N2: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
= ( ord_less_num @ M2 @ N2 ) ) ).
% semiring_norm(78)
thf(fact_836_semiring__norm_I75_J,axiom,
! [M2: num] :
~ ( ord_less_num @ M2 @ one ) ).
% semiring_norm(75)
thf(fact_837_semiring__norm_I71_J,axiom,
! [M2: num,N2: num] :
( ( ord_less_eq_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
= ( ord_less_eq_num @ M2 @ N2 ) ) ).
% semiring_norm(71)
thf(fact_838_semiring__norm_I68_J,axiom,
! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).
% semiring_norm(68)
thf(fact_839_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_840_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_841_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri6283507881447550617nnreal @ M2 )
= zero_z7100319975126383169nnreal )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_842_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_843_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_844_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_845_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_z7100319975126383169nnreal
= ( semiri6283507881447550617nnreal @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_846_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_847_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_848_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_849_of__nat__0,axiom,
( ( semiri6283507881447550617nnreal @ zero_zero_nat )
= zero_z7100319975126383169nnreal ) ).
% of_nat_0
thf(fact_850_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_851_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ M2 ) @ ( semiri6283507881447550617nnreal @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_852_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_853_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_854_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_855_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_856_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_857_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_le7381754540660121996nnreal @ ( semiri6283507881447550617nnreal @ M2 ) @ ( semiri6283507881447550617nnreal @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_858_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_859_not__real__square__gt__zero,axiom,
! [X2: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
= ( X2 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_860_of__nat__add,axiom,
! [M2: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_add
thf(fact_861_of__nat__add,axiom,
! [M2: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_add
thf(fact_862_of__nat__add,axiom,
! [M2: nat,N2: nat] :
( ( semiri6283507881447550617nnreal @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_p1859984266308609217nnreal @ ( semiri6283507881447550617nnreal @ M2 ) @ ( semiri6283507881447550617nnreal @ N2 ) ) ) ).
% of_nat_add
thf(fact_863_of__nat__add,axiom,
! [M2: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_add
thf(fact_864_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_mult
thf(fact_865_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_mult
thf(fact_866_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri6283507881447550617nnreal @ ( times_times_nat @ M2 @ N2 ) )
= ( times_1893300245718287421nnreal @ ( semiri6283507881447550617nnreal @ M2 ) @ ( semiri6283507881447550617nnreal @ N2 ) ) ) ).
% of_nat_mult
thf(fact_867_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_868_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri5074537144036343181t_real @ N2 )
= one_one_real )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_869_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_870_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri6283507881447550617nnreal @ N2 )
= one_on2969667320475766781nnreal )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_871_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_872_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_873_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_874_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_on2969667320475766781nnreal
= ( semiri6283507881447550617nnreal @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_875_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_876_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_877_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_878_of__nat__1,axiom,
( ( semiri6283507881447550617nnreal @ one_one_nat )
= one_on2969667320475766781nnreal ) ).
% of_nat_1
thf(fact_879_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_880_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_881_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_882_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_883_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_884_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_885_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_886_semiring__norm_I76_J,axiom,
! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).
% semiring_norm(76)
thf(fact_887_semiring__norm_I69_J,axiom,
! [M2: num] :
~ ( ord_less_eq_num @ ( bit0 @ M2 ) @ one ) ).
% semiring_norm(69)
thf(fact_888_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y2: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(167)
thf(fact_889_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).
% semiring_norm(167)
thf(fact_890_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_891_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_892_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_893_le__trans,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_894_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_895_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_896_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_897_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_898_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_899_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_900_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_901_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_902_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_903_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_904_add__le__mono,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_905_add__le__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_906_trans__le__add1,axiom,
! [I3: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_907_trans__le__add2,axiom,
! [I3: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_908_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N5: nat] :
? [K3: nat] :
( N5
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_909_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_910_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_911_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_912_trans__less__add2,axiom,
! [I3: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_913_trans__less__add1,axiom,
! [I3: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_914_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_915_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_916_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_917_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_918_add__less__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_919_not__add__less2,axiom,
! [J2: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_920_not__add__less1,axiom,
! [I3: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ I3 ) ).
% not_add_less1
thf(fact_921_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_922_add__less__mono,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_923_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_924_add__lessD1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K )
=> ( ord_less_nat @ I3 @ K ) ) ).
% add_lessD1
thf(fact_925_left__add__mult__distrib,axiom,
! [I3: nat,U: nat,J2: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I3 @ J2 ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_926_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_927_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_928_add__mult__distrib,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_929_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_930_mult__of__nat__commute,axiom,
! [X2: nat,Y2: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y2 )
= ( times_times_real @ Y2 @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_931_mult__of__nat__commute,axiom,
! [X2: nat,Y2: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y2 )
= ( times_times_int @ Y2 @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_932_mult__of__nat__commute,axiom,
! [X2: nat,Y2: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( semiri6283507881447550617nnreal @ X2 ) @ Y2 )
= ( times_1893300245718287421nnreal @ Y2 @ ( semiri6283507881447550617nnreal @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_933_mult__of__nat__commute,axiom,
! [X2: nat,Y2: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y2 )
= ( times_times_nat @ Y2 @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_934_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_935_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_936_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_937_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_938_less__imp__add__positive,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I3 @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_939_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_940_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_941_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_942_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_943_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_944_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_945_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_946_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
& ( M3 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_947_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_948_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N5: nat] :
( ( ord_less_nat @ M3 @ N5 )
| ( M3 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_949_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_950_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_951_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_952_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_953_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_954_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_955_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_956_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_957_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_958_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_959_mult__le__mono,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_960_mult__le__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_961_mult__le__mono2,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_962_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( semiri6283507881447550617nnreal @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_963_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_964_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_965_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_966_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_967_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_968_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_le7381754540660121996nnreal @ ( semiri6283507881447550617nnreal @ M2 ) @ zero_z7100319975126383169nnreal ) ).
% of_nat_less_0_iff
thf(fact_969_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_970_numeral__times__minus__swap,axiom,
! [W: num,X2: real] :
( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X2 ) )
= ( times_times_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_971_numeral__times__minus__swap,axiom,
! [W: num,X2: int] :
( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X2 ) )
= ( times_times_int @ X2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_972_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ I3 ) @ ( semiri6283507881447550617nnreal @ J2 ) ) ) ).
% of_nat_mono
thf(fact_973_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I3 ) @ ( semiri5074537144036343181t_real @ J2 ) ) ) ).
% of_nat_mono
thf(fact_974_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).
% of_nat_mono
thf(fact_975_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I3 ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).
% of_nat_mono
thf(fact_976_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_977_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_978_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_le7381754540660121996nnreal @ ( semiri6283507881447550617nnreal @ M2 ) @ ( semiri6283507881447550617nnreal @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_979_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_980_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_981_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_982_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_le7381754540660121996nnreal @ ( semiri6283507881447550617nnreal @ M2 ) @ ( semiri6283507881447550617nnreal @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_983_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_984_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_985_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_986_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_987_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_988_mult__less__mono2,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_989_mult__less__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_990_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_991_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_992_realpow__pos__nth__unique,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ( power_power_real @ X4 @ N2 )
= A )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N2 )
= A ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_993_realpow__pos__nth,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N2 )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_994_four__x__squared,axiom,
! [X2: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% four_x_squared
thf(fact_995_L2__set__mult__ineq__lemma,axiom,
! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_996_sum__squares__bound,axiom,
! [X2: real,Y2: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_997_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_998_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_999_add__self__div__2,axiom,
! [M2: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M2 ) ).
% add_self_div_2
thf(fact_1000_div__mult__self1__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M2 ) @ N2 )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1001_div__mult__self__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N2 ) @ N2 )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1002_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_1003_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1004_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1005_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1006_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1007_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1008_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1009_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_1010_div__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1011_zdiv__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_1012_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1013_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1014_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1015_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1016_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1017_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1018_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1019_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1020_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1021_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1022_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_1023_div__le__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ).
% div_le_dividend
thf(fact_1024_div__le__mono,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_1025_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1026_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1027_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1028_div__mult2__eq,axiom,
! [M2: nat,N2: nat,Q2: nat] :
( ( divide_divide_nat @ M2 @ ( times_times_nat @ N2 @ Q2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M2 @ N2 ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_1029_zdiv__int,axiom,
! [M2: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zdiv_int
thf(fact_1030_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1031_times__div__less__eq__dividend,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1032_div__times__less__eq__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1033_less__mult__imp__div__less,axiom,
! [M2: nat,I3: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I3 @ N2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ I3 ) ) ).
% less_mult_imp_div_less
thf(fact_1034_int__div__less__self,axiom,
! [X2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X2 )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X2 @ K ) @ X2 ) ) ) ).
% int_div_less_self
thf(fact_1035_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1036_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1037_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1038_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I3: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I3 @ K ) )
= ( ord_less_eq_int @ K @ I3 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1039_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1040_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1041_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1042_zdiv__mono2__neg,axiom,
! [A: int,B2: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1043_zdiv__mono1__neg,axiom,
! [A: int,A3: int,B: int] :
( ( ord_less_eq_int @ A @ A3 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1044_zdiv__eq__0__iff,axiom,
! [I3: int,K: int] :
( ( ( divide_divide_int @ I3 @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I3 )
& ( ord_less_int @ I3 @ K ) )
| ( ( ord_less_eq_int @ I3 @ zero_zero_int )
& ( ord_less_int @ K @ I3 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1045_zdiv__mono2,axiom,
! [A: int,B2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1046_zdiv__mono1,axiom,
! [A: int,A3: int,B: int] :
( ( ord_less_eq_int @ A @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A3 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1047_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_1048_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_1049_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_1050_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_1051_div__greater__zero__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1052_div__le__mono2,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1053_div__less__dividend,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1054_div__eq__dividend__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N2 )
= M2 )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1055_div__less__iff__less__mult,axiom,
! [Q2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q2 ) @ N2 )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1056_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_1057_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1058_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N2 @ Q2 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q2 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1059_split__div,axiom,
! [P: nat > $o,M2: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N2 != zero_zero_nat )
=> ! [I5: nat,J4: nat] :
( ( ( ord_less_nat @ J4 @ N2 )
& ( M2
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I5 ) @ J4 ) ) )
=> ( P @ I5 ) ) ) ) ) ).
% split_div
thf(fact_1060_dividend__less__div__times,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1061_dividend__less__times__div,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1062_int__div__pos__eq,axiom,
! [A: int,B: int,Q2: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ R @ B )
=> ( ( divide_divide_int @ A @ B )
= Q2 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1063_int__div__neg__eq,axiom,
! [A: int,B: int,Q2: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ( ord_less_int @ B @ R )
=> ( ( divide_divide_int @ A @ B )
= Q2 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1064_split__zdiv,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( divide_divide_int @ N2 @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I5: int,J4: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J4 )
& ( ord_less_int @ J4 @ K )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I5 ) @ J4 ) ) )
=> ( P @ I5 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I5: int,J4: int] :
( ( ( ord_less_int @ K @ J4 )
& ( ord_less_eq_int @ J4 @ zero_zero_int )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I5 ) @ J4 ) ) )
=> ( P @ I5 ) ) ) ) ) ).
% split_zdiv
thf(fact_1065_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1066_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1067_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1068_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1069_nat__neq__4k1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ M2 )
!= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% nat_neq_4k1
thf(fact_1070_log__induct,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
=> ( ( P @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( P @ N4 ) ) )
=> ( P @ N2 ) ) ) ) ).
% log_induct
thf(fact_1071_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_1072_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_1073_int__eq__iff__numeral,axiom,
! [M2: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( numeral_numeral_int @ V ) )
= ( M2
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_1074_negative__eq__positive,axiom,
! [N2: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N2 = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1075_negative__zle,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_1076_int__int__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M2 = N2 ) ) ).
% int_int_eq
thf(fact_1077_int__cases2,axiom,
! [Z: int] :
( ! [N4: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% int_cases2
thf(fact_1078_zle__int,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% zle_int
thf(fact_1079_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1080_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N4: nat] :
( K
!= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% nonneg_int_cases
thf(fact_1081_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N4: nat] :
( K
= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1082_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1083_zadd__int__left,axiom,
! [M2: nat,N2: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N2 ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1084_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1085_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1086_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1087_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1088_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1089_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z4: int] :
? [N5: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N5 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1090_not__int__zless__negative,axiom,
! [N2: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_1091_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1092_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1093_int__cases4,axiom,
! [M2: int] :
( ! [N4: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% int_cases4
thf(fact_1094_int__zle__neg,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N2 = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1095_negative__zle__0,axiom,
! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1096_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N4: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1097_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1098_zmult__zless__mono2,axiom,
! [I3: int,J2: int,K: int] :
( ( ord_less_int @ I3 @ J2 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I3 ) @ ( times_times_int @ K @ J2 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1099_int__ge__induct,axiom,
! [K: int,I3: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_ge_induct
thf(fact_1100_int__gr__induct,axiom,
! [K: int,I3: int,P: int > $o] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_gr_induct
thf(fact_1101_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1102_zmult__eq__1__iff,axiom,
! [M2: int,N2: int] :
( ( ( times_times_int @ M2 @ N2 )
= one_one_int )
= ( ( ( M2 = one_one_int )
& ( N2 = one_one_int ) )
| ( ( M2
= ( uminus_uminus_int @ one_one_int ) )
& ( N2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1103_pos__zmult__eq__1__iff__lemma,axiom,
! [M2: int,N2: int] :
( ( ( times_times_int @ M2 @ N2 )
= one_one_int )
=> ( ( M2 = one_one_int )
| ( M2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1104_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( K
= ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1105_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N4: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% pos_int_cases
thf(fact_1106_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N4: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
=> ~ ! [N4: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% int_cases3
thf(fact_1107_zmult__zless__mono2__lemma,axiom,
! [I3: int,J2: int,K: nat] :
( ( ord_less_int @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I3 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J2 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1108_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1109_pos__zmult__eq__1__iff,axiom,
! [M2: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N2 )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1110_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1111_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1112_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1113_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N4: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% neg_int_cases
thf(fact_1114_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1115_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_1116_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_1117_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1118_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_1119_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_1120_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_1121_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_1122_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1123_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1124_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1125_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1126_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1127_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1128_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_1129_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_1130_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_1131_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_1132_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1133_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1134_not__gr__zero,axiom,
! [N2: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N2 ) )
= ( N2 = zero_z7100319975126383169nnreal ) ) ).
% not_gr_zero
thf(fact_1135_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_1136_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1137_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1138_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_1139_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1140_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1141_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_1142_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1143_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1144_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1145_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1146_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1147_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1148_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1149_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1150_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1151_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1152_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1153_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1154_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1155_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1156_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_1157_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1158_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_1159_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y2 ) )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1160_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1161_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_1162_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1163_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_1164_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_1165_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1166_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_1167_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_1168_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1169_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_1170_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_1171_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1172_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_1173_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_1174_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_1175_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_1176_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1177_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_1178_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_1179_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_1180_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_1181_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_1182_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_1183_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_1184_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_1185_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_1186_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_1187_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_1188_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_1189_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_1190_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1191_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1192_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1193_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1194_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1195_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1196_mult_Oright__neutral,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A @ one_on2969667320475766781nnreal )
= A ) ).
% mult.right_neutral
thf(fact_1197_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_1198_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1199_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1200_mult__1,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal @ A )
= A ) ).
% mult_1
thf(fact_1201_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_1202_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1203_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1204_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_1205_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_1206_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_1207_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_1208_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_1209_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1210_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_1211_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_1212_mult__minus__left,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_1213_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_1214_minus__mult__minus,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A @ B ) ) ).
% minus_mult_minus
thf(fact_1215_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_1216_mult__minus__right,axiom,
! [A: real,B: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_1217_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_1218_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_1219_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_1220_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_1221_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_1222_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_1223_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_1224_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_1225_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_1226_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1227_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_1228_linear__plus__1__le__power,axiom,
! [X2: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X2 @ one_one_real ) @ N2 ) ) ) ).
% linear_plus_1_le_power
thf(fact_1229_reals__power__lt__ex,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ one_one_real @ Y2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_real @ ( power_power_real @ ( divide_divide_real @ one_one_real @ Y2 ) @ K2 ) @ X2 ) ) ) ) ).
% reals_power_lt_ex
thf(fact_1230_half__bounded__equal,axiom,
! [X2: real] :
( ( ord_less_eq_real @ one_one_real @ ( times_times_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real )
= ( X2
= ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% half_bounded_equal
thf(fact_1231_sum__le__prod1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% sum_le_prod1
thf(fact_1232_nat__add__1__add__1,axiom,
! [N2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ one_one_nat )
= ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% nat_add_1_add_1
thf(fact_1233_triangle__lemma,axiom,
! [X2: real,Y2: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y2 @ Z ) ) ) ) ) ) ).
% triangle_lemma
thf(fact_1234_ln__2__less__1,axiom,
ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% ln_2_less_1
thf(fact_1235_assms_I6_J,axiom,
! [I3: nat] :
( ( ord_less_nat @ I3 @ n )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ alpha )
@ ( sigma_measure_a2 @ m
@ ( collect_a
@ ^ [Omega: a] :
( ( member_a @ Omega @ ( sigma_space_a @ m ) )
& ( member_b @ ( x @ I3 @ Omega ) @ i ) ) ) ) ) ) ).
% assms(6)
thf(fact_1236_assms_I4_J,axiom,
( indepe3245197900929106295_nat_b @ m
@ ^ [Uu: nat] : borel_5459123734250506525orel_b
@ x
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ n ) ) ).
% assms(4)
thf(fact_1237_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X4: nat > real] :
( ( P @ X4 )
=> ( P @ ( F @ X4 ) ) )
=> ( ! [X4: nat > real] :
( ( P @ X4 )
=> ! [I2: nat] :
( ( Q @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I2 ) )
& ( ord_less_eq_real @ ( X4 @ I2 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I4: nat] : ( ord_less_eq_nat @ ( L2 @ X5 @ I4 ) @ one_one_nat )
& ! [X5: nat > real,I4: nat] :
( ( ( P @ X5 )
& ( Q @ I4 )
& ( ( X5 @ I4 )
= zero_zero_real ) )
=> ( ( L2 @ X5 @ I4 )
= zero_zero_nat ) )
& ! [X5: nat > real,I4: nat] :
( ( ( P @ X5 )
& ( Q @ I4 )
& ( ( X5 @ I4 )
= one_one_real ) )
=> ( ( L2 @ X5 @ I4 )
= one_one_nat ) )
& ! [X5: nat > real,I4: nat] :
( ( ( P @ X5 )
& ( Q @ I4 )
& ( ( L2 @ X5 @ I4 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I4 ) @ ( F @ X5 @ I4 ) ) )
& ! [X5: nat > real,I4: nat] :
( ( ( P @ X5 )
& ( Q @ I4 )
& ( ( L2 @ X5 @ I4 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X5 @ I4 ) @ ( X5 @ I4 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1238_set__bit__nonnegative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_1239_set__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% set_bit_negative_int_iff
thf(fact_1240_set__bit__greater__eq,axiom,
! [K: int,N2: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N2 @ K ) ) ).
% set_bit_greater_eq
thf(fact_1241_kuhn__lemma,axiom,
! [P2: nat,N2: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P2 )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( ( Label @ X4 @ I2 )
= zero_zero_nat )
| ( ( Label @ X4 @ I2 )
= one_one_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( ( X4 @ I2 )
= zero_zero_nat )
=> ( ( Label @ X4 @ I2 )
= zero_zero_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( ( X4 @ I2 )
= P2 )
=> ( ( Label @ X4 @ I2 )
= one_one_nat ) ) ) )
=> ~ ! [Q3: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_nat @ ( Q3 @ I4 ) @ P2 ) )
=> ~ ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ? [R2: nat > nat] :
( ! [J5: nat] :
( ( ord_less_nat @ J5 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q3 @ J5 ) @ ( R2 @ J5 ) )
& ( ord_less_eq_nat @ ( R2 @ J5 ) @ ( plus_plus_nat @ ( Q3 @ J5 ) @ one_one_nat ) ) ) )
& ? [S3: nat > nat] :
( ! [J5: nat] :
( ( ord_less_nat @ J5 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q3 @ J5 ) @ ( S3 @ J5 ) )
& ( ord_less_eq_nat @ ( S3 @ J5 ) @ ( plus_plus_nat @ ( Q3 @ J5 ) @ one_one_nat ) ) ) )
& ( ( Label @ R2 @ I4 )
!= ( Label @ S3 @ I4 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1242_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M6: nat] :
( ( P @ X2 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M6 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1243_one__less__numeral,axiom,
! [N2: num] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N2 ) )
= ( ord_less_num @ one @ N2 ) ) ).
% one_less_numeral
thf(fact_1244_verit__less__mono__div__int2,axiom,
! [A2: int,B3: int,N2: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N2 ) @ ( divide_divide_int @ A2 @ N2 ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1245_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1246_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or5832277885323065728an_int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_1247_numeral__eq__of__nat,axiom,
( numera4658534427948366547nnreal
= ( ^ [A4: num] : ( semiri6283507881447550617nnreal @ ( numeral_numeral_nat @ A4 ) ) ) ) ).
% numeral_eq_of_nat
thf(fact_1248_verit__la__generic,axiom,
! [A: int,X2: int] :
( ( ord_less_eq_int @ A @ X2 )
| ( A = X2 )
| ( ord_less_eq_int @ X2 @ A ) ) ).
% verit_la_generic
thf(fact_1249_nat__int__comparison_I1_J,axiom,
( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
= ( ^ [A4: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1250_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1251_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1252_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1253_int__ops_I3_J,axiom,
! [N2: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% int_ops(3)
thf(fact_1254_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1255_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1256_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1257_int__plus,axiom,
! [N2: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_1258_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1259_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1260_int__ops_I8_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_1261_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1262_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_1263_power__le__one__iff,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real )
= ( ( N2 = zero_zero_nat )
| ( ord_less_eq_real @ A @ one_one_real ) ) ) ) ).
% power_le_one_iff
thf(fact_1264_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1265_real__of__nat__ge__one__iff,axiom,
! [N2: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ).
% real_of_nat_ge_one_iff
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ ( ln_ln_real @ epsilon ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ alpha @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( semiri5074537144036343181t_real @ n ) ).
%------------------------------------------------------------------------------