TPTP Problem File: SLH0517^1.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_00281_010514__14772158_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1365 ( 547 unt; 88 typ; 0 def)
% Number of atoms : 3589 (1097 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9895 ( 328 ~; 100 |; 147 &;7787 @)
% ( 0 <=>;1533 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 406 ( 406 >; 0 *; 0 +; 0 <<)
% Number of symbols : 77 ( 74 usr; 17 con; 0-4 aty)
% Number of variables : 3316 ( 254 ^;2959 !; 103 ?;3316 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:45:01.626
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
set_real_real: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
formal3361831859752904756s_real: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
sigma_measure_real: $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_I_062_Itf__b_Mtf__b_J_J,type,
set_b_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
set_a_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (74)
thf(sy_c_Borel__Space_Ois__borel_001t__Real__Oreal_001t__Real__Oreal,type,
borel_236569967776329622l_real: ( real > real ) > sigma_measure_real > $o ).
thf(sy_c_Borel__Space_Ois__borel_001tf__a_001tf__b,type,
borel_is_borel_a_b: ( a > b ) > sigma_measure_a > $o ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Real__Oreal,type,
borel_5078946678739801102l_real: sigma_measure_real ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001tf__b,type,
borel_5459123734250506525orel_b: sigma_measure_b ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
invers68952373231134600s_real: formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
one_on8598947968683843321s_real: formal3361831859752904756s_real ).
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_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Harmonic__Numbers_Oharm_001t__Real__Oreal,type,
harmonic_harm_real: nat > real ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_If_001tf__b,type,
if_b: $o > b > b > b ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Median_Osort__primitive_001t__Nat__Onat_001tf__b,type,
sort_primitive_nat_b: nat > nat > ( nat > b ) > nat > b ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__b,type,
ord_less_b: b > b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__b,type,
ord_less_eq_b: b > b > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
ord_max_int: int > int > int ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
ord_max_real: real > real > real ).
thf(sy_c_Orderings_Oord__class_Omax_001tf__b,type,
ord_max_b: b > b > b ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
ord_min_int: int > int > int ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Real__Oreal,type,
ord_min_real: real > real > real ).
thf(sy_c_Orderings_Oord__class_Omin_001tf__b,type,
ord_min_b: b > b > b ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
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_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_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
collect_real_real: ( ( real > real ) > $o ) > set_real_real ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__b_J,type,
collect_a_b: ( ( a > b ) > $o ) > set_a_b ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Real__Oreal,type,
sigma_5267869275261027754l_real: sigma_measure_real > sigma_measure_real > set_real_real ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001tf__a,type,
sigma_measurable_a_a: sigma_measure_a > sigma_measure_a > set_a_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001tf__b,type,
sigma_measurable_a_b: sigma_measure_a > sigma_measure_b > set_a_b ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001tf__b,type,
sigma_measurable_b_b: sigma_measure_b > sigma_measure_b > set_b_b ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
member_real_real: ( real > real ) > set_real_real > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__b_J,type,
member_a_b: ( a > b ) > set_a_b > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__b_J,type,
member_b_b: ( b > b ) > set_b_b > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_M,type,
m: sigma_measure_a ).
thf(sy_v_X,type,
x: nat > a > b ).
thf(sy_v_g____,type,
g: nat > a > b ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_is__swap____,type,
is_swap: ( ( nat > b ) > nat > b ) > $o ).
thf(sy_v_j,type,
j: nat ).
thf(sy_v_ja____,type,
ja: nat ).
thf(sy_v_k____,type,
k: nat ).
thf(sy_v_meas__ptw____,type,
meas_ptw: ( nat > a > b ) > $o ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_x____,type,
x2: ( nat > b ) > nat > b ).
% Relevant facts (1264)
thf(fact_0__092_060open_062is__swap_Ax_092_060close_062,axiom,
is_swap @ x2 ).
% \<open>is_swap x\<close>
thf(fact_1_assms_I2_J,axiom,
ord_less_nat @ j @ n ).
% assms(2)
thf(fact_2__092_060open_062meas__ptw_Ag_092_060close_062,axiom,
meas_ptw @ g ).
% \<open>meas_ptw g\<close>
thf(fact_3_i__le,axiom,
ord_less_nat @ i @ n ).
% i_le
thf(fact_4_j__le,axiom,
ord_less_nat @ ja @ n ).
% j_le
thf(fact_5__092_060open_062k_A_060_An_092_060close_062,axiom,
ord_less_nat @ k @ n ).
% \<open>k < n\<close>
thf(fact_6_a,axiom,
! [K: nat] :
( ( ord_less_nat @ K @ n )
=> ( member_a_b @ ( g @ K ) @ ( sigma_measurable_a_b @ m @ borel_5459123734250506525orel_b ) ) ) ).
% a
thf(fact_7_assms_I3_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ n )
=> ( member_a_b @ ( x @ I ) @ ( sigma_measurable_a_b @ m @ borel_5459123734250506525orel_b ) ) ) ).
% assms(3)
thf(fact_8_x__def,axiom,
( x2
= ( sort_primitive_nat_b @ i @ ja ) ) ).
% x_def
thf(fact_9_max__min__same_I4_J,axiom,
! [Y: b,X: b] :
( ( ord_max_b @ Y @ ( ord_min_b @ X @ Y ) )
= Y ) ).
% max_min_same(4)
thf(fact_10_max__min__same_I4_J,axiom,
! [Y: nat,X: nat] :
( ( ord_max_nat @ Y @ ( ord_min_nat @ X @ Y ) )
= Y ) ).
% max_min_same(4)
thf(fact_11_max__min__same_I3_J,axiom,
! [X: b,Y: b] :
( ( ord_max_b @ ( ord_min_b @ X @ Y ) @ Y )
= Y ) ).
% max_min_same(3)
thf(fact_12_max__min__same_I3_J,axiom,
! [X: nat,Y: nat] :
( ( ord_max_nat @ ( ord_min_nat @ X @ Y ) @ Y )
= Y ) ).
% max_min_same(3)
thf(fact_13_max__min__same_I2_J,axiom,
! [X: b,Y: b] :
( ( ord_max_b @ ( ord_min_b @ X @ Y ) @ X )
= X ) ).
% max_min_same(2)
thf(fact_14_max__min__same_I2_J,axiom,
! [X: nat,Y: nat] :
( ( ord_max_nat @ ( ord_min_nat @ X @ Y ) @ X )
= X ) ).
% max_min_same(2)
thf(fact_15_max__min__same_I1_J,axiom,
! [X: b,Y: b] :
( ( ord_max_b @ X @ ( ord_min_b @ X @ Y ) )
= X ) ).
% max_min_same(1)
thf(fact_16_max__min__same_I1_J,axiom,
! [X: nat,Y: nat] :
( ( ord_max_nat @ X @ ( ord_min_nat @ X @ Y ) )
= X ) ).
% max_min_same(1)
thf(fact_17_min_Oabsorb3,axiom,
! [A: b,B: b] :
( ( ord_less_b @ A @ B )
=> ( ( ord_min_b @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_18_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_19_min_Oabsorb3,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_min_int @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_20_min_Oabsorb3,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_min_real @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_21_min_Oabsorb4,axiom,
! [B: b,A: b] :
( ( ord_less_b @ B @ A )
=> ( ( ord_min_b @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_22_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_23_min_Oabsorb4,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_min_int @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_24_min_Oabsorb4,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_min_real @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_25_min__less__iff__conj,axiom,
! [Z: b,X: b,Y: b] :
( ( ord_less_b @ Z @ ( ord_min_b @ X @ Y ) )
= ( ( ord_less_b @ Z @ X )
& ( ord_less_b @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_26_min__less__iff__conj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z @ X )
& ( ord_less_nat @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_27_min__less__iff__conj,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ Z @ ( ord_min_int @ X @ Y ) )
= ( ( ord_less_int @ Z @ X )
& ( ord_less_int @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_28_min__less__iff__conj,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ ( ord_min_real @ X @ Y ) )
= ( ( ord_less_real @ Z @ X )
& ( ord_less_real @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_29_max_Oabsorb3,axiom,
! [B: b,A: b] :
( ( ord_less_b @ B @ A )
=> ( ( ord_max_b @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_30_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_31_max_Oabsorb3,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_32_max_Oabsorb3,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_max_real @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_33_max_Oabsorb4,axiom,
! [A: b,B: b] :
( ( ord_less_b @ A @ B )
=> ( ( ord_max_b @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_34_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_35_max_Oabsorb4,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_36_max_Oabsorb4,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_max_real @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_37_max__less__iff__conj,axiom,
! [X: b,Y: b,Z: b] :
( ( ord_less_b @ ( ord_max_b @ X @ Y ) @ Z )
= ( ( ord_less_b @ X @ Z )
& ( ord_less_b @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_38_max__less__iff__conj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
= ( ( ord_less_nat @ X @ Z )
& ( ord_less_nat @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_39_max__less__iff__conj,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
= ( ( ord_less_int @ X @ Z )
& ( ord_less_int @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_40_max__less__iff__conj,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z )
= ( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Y @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_41_borel__measurable__min,axiom,
! [F: a > b,M: sigma_measure_a,G: a > b] :
( ( member_a_b @ F @ ( sigma_measurable_a_b @ M @ borel_5459123734250506525orel_b ) )
=> ( ( member_a_b @ G @ ( sigma_measurable_a_b @ M @ borel_5459123734250506525orel_b ) )
=> ( member_a_b
@ ^ [X2: a] : ( ord_min_b @ ( G @ X2 ) @ ( F @ X2 ) )
@ ( sigma_measurable_a_b @ M @ borel_5459123734250506525orel_b ) ) ) ) ).
% borel_measurable_min
thf(fact_42_borel__measurable__min,axiom,
! [F: real > real,M: sigma_measure_real,G: real > real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_real_real
@ ^ [X2: real] : ( ord_min_real @ ( G @ X2 ) @ ( F @ X2 ) )
@ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).
% borel_measurable_min
thf(fact_43_borel__measurable__max,axiom,
! [F: a > b,M: sigma_measure_a,G: a > b] :
( ( member_a_b @ F @ ( sigma_measurable_a_b @ M @ borel_5459123734250506525orel_b ) )
=> ( ( member_a_b @ G @ ( sigma_measurable_a_b @ M @ borel_5459123734250506525orel_b ) )
=> ( member_a_b
@ ^ [X2: a] : ( ord_max_b @ ( G @ X2 ) @ ( F @ X2 ) )
@ ( sigma_measurable_a_b @ M @ borel_5459123734250506525orel_b ) ) ) ) ).
% borel_measurable_max
thf(fact_44_borel__measurable__max,axiom,
! [F: real > real,M: sigma_measure_real,G: real > real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_real_real
@ ^ [X2: real] : ( ord_max_real @ ( G @ X2 ) @ ( F @ X2 ) )
@ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).
% borel_measurable_max
thf(fact_45_max_Oright__idem,axiom,
! [A: b,B: b] :
( ( ord_max_b @ ( ord_max_b @ A @ B ) @ B )
= ( ord_max_b @ A @ B ) ) ).
% max.right_idem
thf(fact_46_max_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
= ( ord_max_nat @ A @ B ) ) ).
% max.right_idem
thf(fact_47_max_Oleft__idem,axiom,
! [A: b,B: b] :
( ( ord_max_b @ A @ ( ord_max_b @ A @ B ) )
= ( ord_max_b @ A @ B ) ) ).
% max.left_idem
thf(fact_48_max_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
= ( ord_max_nat @ A @ B ) ) ).
% max.left_idem
thf(fact_49_max_Oidem,axiom,
! [A: b] :
( ( ord_max_b @ A @ A )
= A ) ).
% max.idem
thf(fact_50_max_Oidem,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ A )
= A ) ).
% max.idem
thf(fact_51_min_Oright__idem,axiom,
! [A: b,B: b] :
( ( ord_min_b @ ( ord_min_b @ A @ B ) @ B )
= ( ord_min_b @ A @ B ) ) ).
% min.right_idem
thf(fact_52_min_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ B )
= ( ord_min_nat @ A @ B ) ) ).
% min.right_idem
thf(fact_53_min_Oleft__idem,axiom,
! [A: b,B: b] :
( ( ord_min_b @ A @ ( ord_min_b @ A @ B ) )
= ( ord_min_b @ A @ B ) ) ).
% min.left_idem
thf(fact_54_min_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B ) )
= ( ord_min_nat @ A @ B ) ) ).
% min.left_idem
thf(fact_55_min_Oidem,axiom,
! [A: b] :
( ( ord_min_b @ A @ A )
= A ) ).
% min.idem
thf(fact_56_min_Oidem,axiom,
! [A: nat] :
( ( ord_min_nat @ A @ A )
= A ) ).
% min.idem
thf(fact_57__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062i_Aj_O_A_092_060lbrakk_062x_A_061_Asort__primitive_Ai_Aj_059_Ai_A_060_An_059_Aj_A_060_An_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [I2: nat,J: nat] :
( ( x2
= ( sort_primitive_nat_b @ I2 @ J ) )
=> ( ( ord_less_nat @ I2 @ n )
=> ~ ( ord_less_nat @ J @ n ) ) ) ).
% \<open>\<And>thesis. (\<And>i j. \<lbrakk>x = sort_primitive i j; i < n; j < n\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_58_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_59_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_60_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_61_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_62_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_63_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_64_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_65_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_66_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_67_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_68_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_69_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_70_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_71_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_72_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_73_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_74_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_75_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_76_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_77_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_78_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_79_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_80_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_81_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_82_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_83_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_84_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_85_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_86_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_87_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_88_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_89_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_90_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_91_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_92_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_93_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_94_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_95_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_96_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_97_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_98_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_99_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_100_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_101_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_102_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_103_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_104_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_105_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_106_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_107_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_108_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_109_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_110_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_111_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_112_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_113_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_114_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_115_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_116_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_117_order__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_118_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_119_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_120_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_121_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_122_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_123_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_124_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_125_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_126_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_127_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_128_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_129_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_130_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_131_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_132_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_133_mem__Collect__eq,axiom,
! [A: a > b,P: ( a > b ) > $o] :
( ( member_a_b @ A @ ( collect_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_134_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_135_mem__Collect__eq,axiom,
! [A: real > real,P: ( real > real ) > $o] :
( ( member_real_real @ A @ ( collect_real_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_136_Collect__mem__eq,axiom,
! [A2: set_a_b] :
( ( collect_a_b
@ ^ [X2: a > b] : ( member_a_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_137_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_138_Collect__mem__eq,axiom,
! [A2: set_real_real] :
( ( collect_real_real
@ ^ [X2: real > real] : ( member_real_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_139_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_140_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_141_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_142_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_143_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_144_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_145_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_146_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_147_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_148_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_149_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_150_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_151_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_152_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_153_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_154_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_155_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_156_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_157_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_158_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_159_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_160_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_161_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_162_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_163_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_164_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_165_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_166_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_167_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X3 )
=> ( P @ Y3 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_168_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_169_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_170_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_171_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_172_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_173_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_174_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_175_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_176_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_177_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_178_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_179_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_180_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z2: real] :
( ( ord_less_real @ X @ Z2 )
& ( ord_less_real @ Z2 @ Y ) ) ) ).
% dense
thf(fact_181_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_182_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_183_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_184_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_185_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_186_max_Oleft__commute,axiom,
! [B: b,A: b,C: b] :
( ( ord_max_b @ B @ ( ord_max_b @ A @ C ) )
= ( ord_max_b @ A @ ( ord_max_b @ B @ C ) ) ) ).
% max.left_commute
thf(fact_187_max_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% max.left_commute
thf(fact_188_max_Ocommute,axiom,
( ord_max_b
= ( ^ [A4: b,B3: b] : ( ord_max_b @ B3 @ A4 ) ) ) ).
% max.commute
thf(fact_189_max_Ocommute,axiom,
( ord_max_nat
= ( ^ [A4: nat,B3: nat] : ( ord_max_nat @ B3 @ A4 ) ) ) ).
% max.commute
thf(fact_190_max_Oassoc,axiom,
! [A: b,B: b,C: b] :
( ( ord_max_b @ ( ord_max_b @ A @ B ) @ C )
= ( ord_max_b @ A @ ( ord_max_b @ B @ C ) ) ) ).
% max.assoc
thf(fact_191_max_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% max.assoc
thf(fact_192_min_Oleft__commute,axiom,
! [B: b,A: b,C: b] :
( ( ord_min_b @ B @ ( ord_min_b @ A @ C ) )
= ( ord_min_b @ A @ ( ord_min_b @ B @ C ) ) ) ).
% min.left_commute
thf(fact_193_min_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_min_nat @ B @ ( ord_min_nat @ A @ C ) )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.left_commute
thf(fact_194_min_Ocommute,axiom,
( ord_min_b
= ( ^ [A4: b,B3: b] : ( ord_min_b @ B3 @ A4 ) ) ) ).
% min.commute
thf(fact_195_min_Ocommute,axiom,
( ord_min_nat
= ( ^ [A4: nat,B3: nat] : ( ord_min_nat @ B3 @ A4 ) ) ) ).
% min.commute
thf(fact_196_min_Oassoc,axiom,
! [A: b,B: b,C: b] :
( ( ord_min_b @ ( ord_min_b @ A @ B ) @ C )
= ( ord_min_b @ A @ ( ord_min_b @ B @ C ) ) ) ).
% min.assoc
thf(fact_197_min_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ C )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.assoc
thf(fact_198_sort__primitive_Osimps,axiom,
( sort_primitive_nat_b
= ( ^ [I3: nat,J2: nat,F2: nat > b,K2: nat] : ( if_b @ ( K2 = I3 ) @ ( ord_min_b @ ( F2 @ I3 ) @ ( F2 @ J2 ) ) @ ( if_b @ ( K2 = J2 ) @ ( ord_max_b @ ( F2 @ I3 ) @ ( F2 @ J2 ) ) @ ( F2 @ K2 ) ) ) ) ) ).
% sort_primitive.simps
thf(fact_199_sort__primitive_Oelims,axiom,
! [X: nat,Xa: nat,Xb: nat > b,Xc: nat,Y: b] :
( ( ( sort_primitive_nat_b @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( Xc = X )
=> ( Y
= ( ord_min_b @ ( Xb @ X ) @ ( Xb @ Xa ) ) ) )
& ( ( Xc != X )
=> ( ( ( Xc = Xa )
=> ( Y
= ( ord_max_b @ ( Xb @ X ) @ ( Xb @ Xa ) ) ) )
& ( ( Xc != Xa )
=> ( Y
= ( Xb @ Xc ) ) ) ) ) ) ) ).
% sort_primitive.elims
thf(fact_200_max_Ostrict__coboundedI2,axiom,
! [C: b,B: b,A: b] :
( ( ord_less_b @ C @ B )
=> ( ord_less_b @ C @ ( ord_max_b @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_201_max_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_202_max_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_203_max_Ostrict__coboundedI2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_204_max_Ostrict__coboundedI1,axiom,
! [C: b,A: b,B: b] :
( ( ord_less_b @ C @ A )
=> ( ord_less_b @ C @ ( ord_max_b @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_205_max_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_206_max_Ostrict__coboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ A )
=> ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_207_max_Ostrict__coboundedI1,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ A )
=> ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_208_max_Ostrict__order__iff,axiom,
( ord_less_b
= ( ^ [B3: b,A4: b] :
( ( A4
= ( ord_max_b @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_209_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( A4
= ( ord_max_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_210_max_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( A4
= ( ord_max_int @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_211_max_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( A4
= ( ord_max_real @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_212_max_Ostrict__boundedE,axiom,
! [B: b,C: b,A: b] :
( ( ord_less_b @ ( ord_max_b @ B @ C ) @ A )
=> ~ ( ( ord_less_b @ B @ A )
=> ~ ( ord_less_b @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_213_max_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_214_max_Ostrict__boundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
=> ~ ( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_215_max_Ostrict__boundedE,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
=> ~ ( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_216_less__max__iff__disj,axiom,
! [Z: b,X: b,Y: b] :
( ( ord_less_b @ Z @ ( ord_max_b @ X @ Y ) )
= ( ( ord_less_b @ Z @ X )
| ( ord_less_b @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_217_less__max__iff__disj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z @ X )
| ( ord_less_nat @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_218_less__max__iff__disj,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
= ( ( ord_less_int @ Z @ X )
| ( ord_less_int @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_219_less__max__iff__disj,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y ) )
= ( ( ord_less_real @ Z @ X )
| ( ord_less_real @ Z @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_220_min_Ostrict__coboundedI2,axiom,
! [B: b,C: b,A: b] :
( ( ord_less_b @ B @ C )
=> ( ord_less_b @ ( ord_min_b @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_221_min_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_222_min_Ostrict__coboundedI2,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ B @ C )
=> ( ord_less_int @ ( ord_min_int @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_223_min_Ostrict__coboundedI2,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ B @ C )
=> ( ord_less_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_224_min_Ostrict__coboundedI1,axiom,
! [A: b,C: b,B: b] :
( ( ord_less_b @ A @ C )
=> ( ord_less_b @ ( ord_min_b @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_225_min_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_226_min_Ostrict__coboundedI1,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ C )
=> ( ord_less_int @ ( ord_min_int @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_227_min_Ostrict__coboundedI1,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ C )
=> ( ord_less_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_228_min_Ostrict__order__iff,axiom,
( ord_less_b
= ( ^ [A4: b,B3: b] :
( ( A4
= ( ord_min_b @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% min.strict_order_iff
thf(fact_229_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( A4
= ( ord_min_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% min.strict_order_iff
thf(fact_230_min_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( A4
= ( ord_min_int @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% min.strict_order_iff
thf(fact_231_min_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( A4
= ( ord_min_real @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% min.strict_order_iff
thf(fact_232_min_Ostrict__boundedE,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_b @ A @ ( ord_min_b @ B @ C ) )
=> ~ ( ( ord_less_b @ A @ B )
=> ~ ( ord_less_b @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_233_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_234_min_Ostrict__boundedE,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ ( ord_min_int @ B @ C ) )
=> ~ ( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_235_min_Ostrict__boundedE,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( ord_min_real @ B @ C ) )
=> ~ ( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_236_min__less__iff__disj,axiom,
! [X: b,Y: b,Z: b] :
( ( ord_less_b @ ( ord_min_b @ X @ Y ) @ Z )
= ( ( ord_less_b @ X @ Z )
| ( ord_less_b @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_237_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ( ord_less_nat @ X @ Z )
| ( ord_less_nat @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_238_min__less__iff__disj,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ ( ord_min_int @ X @ Y ) @ Z )
= ( ( ord_less_int @ X @ Z )
| ( ord_less_int @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_239_min__less__iff__disj,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ ( ord_min_real @ X @ Y ) @ Z )
= ( ( ord_less_real @ X @ Z )
| ( ord_less_real @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_240_min__max__distrib2,axiom,
! [A: b,B: b,C: b] :
( ( ord_min_b @ A @ ( ord_max_b @ B @ C ) )
= ( ord_max_b @ ( ord_min_b @ A @ B ) @ ( ord_min_b @ A @ C ) ) ) ).
% min_max_distrib2
thf(fact_241_min__max__distrib2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_min_nat @ A @ ( ord_max_nat @ B @ C ) )
= ( ord_max_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ A @ C ) ) ) ).
% min_max_distrib2
thf(fact_242_min__max__distrib1,axiom,
! [B: b,C: b,A: b] :
( ( ord_min_b @ ( ord_max_b @ B @ C ) @ A )
= ( ord_max_b @ ( ord_min_b @ B @ A ) @ ( ord_min_b @ C @ A ) ) ) ).
% min_max_distrib1
thf(fact_243_min__max__distrib1,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_min_nat @ ( ord_max_nat @ B @ C ) @ A )
= ( ord_max_nat @ ( ord_min_nat @ B @ A ) @ ( ord_min_nat @ C @ A ) ) ) ).
% min_max_distrib1
thf(fact_244_max__min__distrib2,axiom,
! [A: b,B: b,C: b] :
( ( ord_max_b @ A @ ( ord_min_b @ B @ C ) )
= ( ord_min_b @ ( ord_max_b @ A @ B ) @ ( ord_max_b @ A @ C ) ) ) ).
% max_min_distrib2
thf(fact_245_max__min__distrib2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_max_nat @ A @ ( ord_min_nat @ B @ C ) )
= ( ord_min_nat @ ( ord_max_nat @ A @ B ) @ ( ord_max_nat @ A @ C ) ) ) ).
% max_min_distrib2
thf(fact_246_max__min__distrib1,axiom,
! [B: b,C: b,A: b] :
( ( ord_max_b @ ( ord_min_b @ B @ C ) @ A )
= ( ord_min_b @ ( ord_max_b @ B @ A ) @ ( ord_max_b @ C @ A ) ) ) ).
% max_min_distrib1
thf(fact_247_max__min__distrib1,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_max_nat @ ( ord_min_nat @ B @ C ) @ A )
= ( ord_min_nat @ ( ord_max_nat @ B @ A ) @ ( ord_max_nat @ C @ A ) ) ) ).
% max_min_distrib1
thf(fact_248_borel__measurable__const,axiom,
! [C: b,M: sigma_measure_a] :
( member_a_b
@ ^ [X2: a] : C
@ ( sigma_measurable_a_b @ M @ borel_5459123734250506525orel_b ) ) ).
% borel_measurable_const
thf(fact_249_borel__measurable__const,axiom,
! [C: real,M: sigma_measure_real] :
( member_real_real
@ ^ [X2: real] : C
@ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ).
% borel_measurable_const
thf(fact_250_meas__ptw__def,axiom,
( meas_ptw
= ( ^ [F2: nat > a > b] :
! [K2: nat] :
( ( ord_less_nat @ K2 @ n )
=> ( member_a_b @ ( F2 @ K2 ) @ ( sigma_measurable_a_b @ m @ borel_5459123734250506525orel_b ) ) ) ) ) ).
% meas_ptw_def
thf(fact_251_is__swap__def,axiom,
( is_swap
= ( ^ [Ts: ( nat > b ) > nat > b] :
? [I3: nat] :
( ( ord_less_nat @ I3 @ n )
& ? [J2: nat] :
( ( ord_less_nat @ J2 @ n )
& ( Ts
= ( sort_primitive_nat_b @ I3 @ J2 ) ) ) ) ) ) ).
% is_swap_def
thf(fact_252_is__borel__def,axiom,
( borel_is_borel_a_b
= ( ^ [F2: a > b,M3: sigma_measure_a] : ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M3 @ borel_5459123734250506525orel_b ) ) ) ) ).
% is_borel_def
thf(fact_253_is__borel__def,axiom,
( borel_236569967776329622l_real
= ( ^ [F2: real > real,M3: sigma_measure_real] : ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ M3 @ borel_5078946678739801102l_real ) ) ) ) ).
% is_borel_def
thf(fact_254_n__ge__0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% n_ge_0
thf(fact_255_measurable__compose__rev,axiom,
! [F: b > b,L: sigma_measure_b,N2: sigma_measure_b,G: a > b,M: sigma_measure_a] :
( ( member_b_b @ F @ ( sigma_measurable_b_b @ L @ N2 ) )
=> ( ( member_a_b @ G @ ( sigma_measurable_a_b @ M @ L ) )
=> ( member_a_b
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ ( sigma_measurable_a_b @ M @ N2 ) ) ) ) ).
% measurable_compose_rev
thf(fact_256_measurable__compose__rev,axiom,
! [F: a > b,L: sigma_measure_a,N2: sigma_measure_b,G: a > a,M: sigma_measure_a] :
( ( member_a_b @ F @ ( sigma_measurable_a_b @ L @ N2 ) )
=> ( ( member_a_a @ G @ ( sigma_measurable_a_a @ M @ L ) )
=> ( member_a_b
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ ( sigma_measurable_a_b @ M @ N2 ) ) ) ) ).
% measurable_compose_rev
thf(fact_257_measurable__compose__rev,axiom,
! [F: real > real,L: sigma_measure_real,N2: sigma_measure_real,G: real > real,M: sigma_measure_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ L @ N2 ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ L ) )
=> ( member_real_real
@ ^ [X2: real] : ( F @ ( G @ X2 ) )
@ ( sigma_5267869275261027754l_real @ M @ N2 ) ) ) ) ).
% measurable_compose_rev
thf(fact_258_measurable__compose,axiom,
! [F: a > a,M: sigma_measure_a,N2: sigma_measure_a,G: a > b,L: sigma_measure_b] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N2 ) )
=> ( ( member_a_b @ G @ ( sigma_measurable_a_b @ N2 @ L ) )
=> ( member_a_b
@ ^ [X2: a] : ( G @ ( F @ X2 ) )
@ ( sigma_measurable_a_b @ M @ L ) ) ) ) ).
% measurable_compose
thf(fact_259_measurable__compose,axiom,
! [F: a > b,M: sigma_measure_a,N2: sigma_measure_b,G: b > b,L: sigma_measure_b] :
( ( member_a_b @ F @ ( sigma_measurable_a_b @ M @ N2 ) )
=> ( ( member_b_b @ G @ ( sigma_measurable_b_b @ N2 @ L ) )
=> ( member_a_b
@ ^ [X2: a] : ( G @ ( F @ X2 ) )
@ ( sigma_measurable_a_b @ M @ L ) ) ) ) ).
% measurable_compose
thf(fact_260_measurable__compose,axiom,
! [F: real > real,M: sigma_measure_real,N2: sigma_measure_real,G: real > real,L: sigma_measure_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N2 ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N2 @ L ) )
=> ( member_real_real
@ ^ [X2: real] : ( G @ ( F @ X2 ) )
@ ( sigma_5267869275261027754l_real @ M @ L ) ) ) ) ).
% measurable_compose
thf(fact_261_measurable__id,axiom,
! [M: sigma_measure_real] :
( member_real_real
@ ^ [X2: real] : X2
@ ( sigma_5267869275261027754l_real @ M @ M ) ) ).
% measurable_id
thf(fact_262_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_263_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_264_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_265_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_266_max__0L,axiom,
! [N3: nat] :
( ( ord_max_nat @ zero_zero_nat @ N3 )
= N3 ) ).
% max_0L
thf(fact_267_max__0R,axiom,
! [N3: nat] :
( ( ord_max_nat @ N3 @ zero_zero_nat )
= N3 ) ).
% max_0R
thf(fact_268_min__0L,axiom,
! [N3: nat] :
( ( ord_min_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% min_0L
thf(fact_269_min__0R,axiom,
! [N3: nat] :
( ( ord_min_nat @ N3 @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_270_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_271_neq0__conv,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% neq0_conv
thf(fact_272_less__nat__zero__code,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_273_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_274_gr0I,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr0I
thf(fact_275_not__gr0,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_276_not__less0,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less0
thf(fact_277_less__zeroE,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_278_gr__implies__not0,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_279_infinite__descent0,axiom,
! [P: nat > $o,N3: 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 @ N3 ) ) ) ).
% infinite_descent0
thf(fact_280_nat__neq__iff,axiom,
! [M4: nat,N3: nat] :
( ( M4 != N3 )
= ( ( ord_less_nat @ M4 @ N3 )
| ( ord_less_nat @ N3 @ M4 ) ) ) ).
% nat_neq_iff
thf(fact_281_less__not__refl,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_not_refl
thf(fact_282_less__not__refl2,axiom,
! [N3: nat,M4: nat] :
( ( ord_less_nat @ N3 @ M4 )
=> ( M4 != N3 ) ) ).
% less_not_refl2
thf(fact_283_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_284_less__irrefl__nat,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_irrefl_nat
thf(fact_285_nat__less__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N3 ) ) ).
% nat_less_induct
thf(fact_286_infinite__descent,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) )
=> ( P @ N3 ) ) ).
% infinite_descent
thf(fact_287_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_288_not__gr__zero,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_289_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_290_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_291_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_292_gr__zeroI,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr_zeroI
thf(fact_293_not__less__zero,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_294_gr__implies__not__zero,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_295_zero__less__iff__neq__zero,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( N3 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_296_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_297_of__nat__0__less__iff,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N3 ) )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% of_nat_0_less_iff
thf(fact_298_of__nat__0__less__iff,axiom,
! [N3: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N3 ) )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% of_nat_0_less_iff
thf(fact_299_of__nat__0__less__iff,axiom,
! [N3: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N3 ) )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% of_nat_0_less_iff
thf(fact_300_of__nat__eq__iff,axiom,
! [M4: nat,N3: nat] :
( ( ( semiri1314217659103216013at_int @ M4 )
= ( semiri1314217659103216013at_int @ N3 ) )
= ( M4 = N3 ) ) ).
% of_nat_eq_iff
thf(fact_301_of__nat__eq__iff,axiom,
! [M4: nat,N3: nat] :
( ( ( semiri5074537144036343181t_real @ M4 )
= ( semiri5074537144036343181t_real @ N3 ) )
= ( M4 = N3 ) ) ).
% of_nat_eq_iff
thf(fact_302_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_303_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_304_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_305_of__nat__0__eq__iff,axiom,
! [N3: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N3 ) )
= ( zero_zero_nat = N3 ) ) ).
% of_nat_0_eq_iff
thf(fact_306_of__nat__0__eq__iff,axiom,
! [N3: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N3 ) )
= ( zero_zero_nat = N3 ) ) ).
% of_nat_0_eq_iff
thf(fact_307_of__nat__0__eq__iff,axiom,
! [N3: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N3 ) )
= ( zero_zero_nat = N3 ) ) ).
% of_nat_0_eq_iff
thf(fact_308_of__nat__eq__0__iff,axiom,
! [M4: nat] :
( ( ( semiri1316708129612266289at_nat @ M4 )
= zero_zero_nat )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_309_of__nat__eq__0__iff,axiom,
! [M4: nat] :
( ( ( semiri1314217659103216013at_int @ M4 )
= zero_zero_int )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_310_of__nat__eq__0__iff,axiom,
! [M4: nat] :
( ( ( semiri5074537144036343181t_real @ M4 )
= zero_zero_real )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_311_of__nat__less__iff,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N3 ) )
= ( ord_less_nat @ M4 @ N3 ) ) ).
% of_nat_less_iff
thf(fact_312_of__nat__less__iff,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) )
= ( ord_less_nat @ M4 @ N3 ) ) ).
% of_nat_less_iff
thf(fact_313_of__nat__less__iff,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N3 ) )
= ( ord_less_nat @ M4 @ N3 ) ) ).
% of_nat_less_iff
thf(fact_314_of__nat__max,axiom,
! [X: nat,Y: nat] :
( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
= ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% of_nat_max
thf(fact_315_of__nat__max,axiom,
! [X: nat,Y: nat] :
( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
= ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% of_nat_max
thf(fact_316_of__nat__max,axiom,
! [X: nat,Y: nat] :
( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
= ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).
% of_nat_max
thf(fact_317_of__nat__min,axiom,
! [X: nat,Y: nat] :
( ( semiri1316708129612266289at_nat @ ( ord_min_nat @ X @ Y ) )
= ( ord_min_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% of_nat_min
thf(fact_318_of__nat__min,axiom,
! [X: nat,Y: nat] :
( ( semiri1314217659103216013at_int @ ( ord_min_nat @ X @ Y ) )
= ( ord_min_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% of_nat_min
thf(fact_319_of__nat__min,axiom,
! [X: nat,Y: nat] :
( ( semiri5074537144036343181t_real @ ( ord_min_nat @ X @ Y ) )
= ( ord_min_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).
% of_nat_min
thf(fact_320_of__nat__less__0__iff,axiom,
! [M4: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_321_of__nat__less__0__iff,axiom,
! [M4: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M4 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_322_of__nat__less__0__iff,axiom,
! [M4: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M4 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_323_of__nat__less__imp__less,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N3 ) )
=> ( ord_less_nat @ M4 @ N3 ) ) ).
% of_nat_less_imp_less
thf(fact_324_of__nat__less__imp__less,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ord_less_nat @ M4 @ N3 ) ) ).
% of_nat_less_imp_less
thf(fact_325_of__nat__less__imp__less,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N3 ) )
=> ( ord_less_nat @ M4 @ N3 ) ) ).
% of_nat_less_imp_less
thf(fact_326_less__imp__of__nat__less,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ).
% less_imp_of_nat_less
thf(fact_327_less__imp__of__nat__less,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% less_imp_of_nat_less
thf(fact_328_less__imp__of__nat__less,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ).
% less_imp_of_nat_less
thf(fact_329_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_330_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_331_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_332_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_333_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_334_reals__Archimedean2,axiom,
! [X: real] :
? [N4: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% reals_Archimedean2
thf(fact_335_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_336_harm__pos__iff,axiom,
! [N3: nat] :
( ( ord_less_real @ zero_zero_real @ ( harmonic_harm_real @ N3 ) )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% harm_pos_iff
thf(fact_337_of__nat__zero__less__power__iff,axiom,
! [X: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N3 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_338_of__nat__zero__less__power__iff,axiom,
! [X: nat,N3: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N3 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_339_of__nat__zero__less__power__iff,axiom,
! [X: nat,N3: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N3 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_340_of__nat__le__0__iff,axiom,
! [M4: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ zero_zero_nat )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_341_of__nat__le__0__iff,axiom,
! [M4: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M4 ) @ zero_zero_int )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_342_of__nat__le__0__iff,axiom,
! [M4: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M4 ) @ zero_zero_real )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_343_harm__pos,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_real @ zero_zero_real @ ( harmonic_harm_real @ N3 ) ) ) ).
% harm_pos
thf(fact_344_ex__inverse__of__nat__less,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ X ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_345_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_346_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_347_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_348_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_349_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_350_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_351_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_352_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_353_assms_I1_J,axiom,
ord_less_eq_nat @ one_one_nat @ n ).
% assms(1)
thf(fact_354_le__zero__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_355_of__nat__le__iff,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N3 ) )
= ( ord_less_eq_nat @ M4 @ N3 ) ) ).
% of_nat_le_iff
thf(fact_356_of__nat__le__iff,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) )
= ( ord_less_eq_nat @ M4 @ N3 ) ) ).
% of_nat_le_iff
thf(fact_357_of__nat__le__iff,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N3 ) )
= ( ord_less_eq_nat @ M4 @ N3 ) ) ).
% of_nat_le_iff
thf(fact_358_of__nat__power,axiom,
! [M4: nat,N3: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M4 @ N3 ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ N3 ) ) ).
% of_nat_power
thf(fact_359_of__nat__power,axiom,
! [M4: nat,N3: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M4 @ N3 ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M4 ) @ N3 ) ) ).
% of_nat_power
thf(fact_360_of__nat__power,axiom,
! [M4: nat,N3: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M4 @ N3 ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M4 ) @ N3 ) ) ).
% of_nat_power
thf(fact_361_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_362_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_363_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_364_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_365_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_366_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_367_nat__zero__less__power__iff,axiom,
! [X: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N3 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_368_max_Oabsorb1,axiom,
! [B: b,A: b] :
( ( ord_less_eq_b @ B @ A )
=> ( ( ord_max_b @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_369_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_370_max_Oabsorb1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_371_max_Oabsorb1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_max_real @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_372_max_Oabsorb2,axiom,
! [A: b,B: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_max_b @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_373_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_374_max_Oabsorb2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_375_max_Oabsorb2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_max_real @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_376_max_Obounded__iff,axiom,
! [B: b,C: b,A: b] :
( ( ord_less_eq_b @ ( ord_max_b @ B @ C ) @ A )
= ( ( ord_less_eq_b @ B @ A )
& ( ord_less_eq_b @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_377_max_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_378_max_Obounded__iff,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_379_max_Obounded__iff,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
= ( ( ord_less_eq_real @ B @ A )
& ( ord_less_eq_real @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_380_min_Oabsorb1,axiom,
! [A: b,B: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_min_b @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_381_min_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_382_min_Oabsorb1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_min_int @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_383_min_Oabsorb1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_min_real @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_384_min_Oabsorb2,axiom,
! [B: b,A: b] :
( ( ord_less_eq_b @ B @ A )
=> ( ( ord_min_b @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_385_min_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_386_min_Oabsorb2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_min_int @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_387_min_Oabsorb2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_min_real @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_388_min_Obounded__iff,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_eq_b @ A @ ( ord_min_b @ B @ C ) )
= ( ( ord_less_eq_b @ A @ B )
& ( ord_less_eq_b @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_389_min_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_390_min_Obounded__iff,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_391_min_Obounded__iff,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( ord_min_real @ B @ C ) )
= ( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_392_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_393_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_394_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_395_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_396_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_397_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_398_power__eq__0__iff,axiom,
! [A: int,N3: nat] :
( ( ( power_power_int @ A @ N3 )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% power_eq_0_iff
thf(fact_399_power__eq__0__iff,axiom,
! [A: nat,N3: nat] :
( ( ( power_power_nat @ A @ N3 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% power_eq_0_iff
thf(fact_400_power__eq__0__iff,axiom,
! [A: real,N3: nat] :
( ( ( power_power_real @ A @ N3 )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% power_eq_0_iff
thf(fact_401_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_402_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_403_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_404_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_405_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_406_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_407_power__mono__iff,axiom,
! [A: nat,B: nat,N3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ B @ N3 ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_408_power__mono__iff,axiom,
! [A: int,B: int,N3: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ B @ N3 ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_409_power__mono__iff,axiom,
! [A: real,B: real,N3: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ B @ N3 ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_410_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_411_harm__nonneg,axiom,
! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( harmonic_harm_real @ N3 ) ) ).
% harm_nonneg
thf(fact_412_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_413_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_414_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_415_of__nat__mono,axiom,
! [I: nat,J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J3 ) ) ) ).
% of_nat_mono
thf(fact_416_of__nat__mono,axiom,
! [I: nat,J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J3 ) ) ) ).
% of_nat_mono
thf(fact_417_of__nat__mono,axiom,
! [I: nat,J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J3 ) ) ) ).
% of_nat_mono
thf(fact_418_zero__le__power,axiom,
! [A: nat,N3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N3 ) ) ) ).
% zero_le_power
thf(fact_419_zero__le__power,axiom,
! [A: int,N3: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N3 ) ) ) ).
% zero_le_power
thf(fact_420_zero__le__power,axiom,
! [A: real,N3: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N3 ) ) ) ).
% zero_le_power
thf(fact_421_power__mono,axiom,
! [A: nat,B: nat,N3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ B @ N3 ) ) ) ) ).
% power_mono
thf(fact_422_power__mono,axiom,
! [A: int,B: int,N3: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ B @ N3 ) ) ) ) ).
% power_mono
thf(fact_423_power__mono,axiom,
! [A: real,B: real,N3: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ B @ N3 ) ) ) ) ).
% power_mono
thf(fact_424_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_425_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_426_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_427_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_428_zle__int,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) )
= ( ord_less_eq_nat @ M4 @ N3 ) ) ).
% zle_int
thf(fact_429_int__int__eq,axiom,
! [M4: nat,N3: nat] :
( ( ( semiri1314217659103216013at_int @ M4 )
= ( semiri1314217659103216013at_int @ N3 ) )
= ( M4 = N3 ) ) ).
% int_int_eq
thf(fact_430_real__arch__simple,axiom,
! [X: real] :
? [N4: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% real_arch_simple
thf(fact_431_power__not__zero,axiom,
! [A: int,N3: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N3 )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_432_power__not__zero,axiom,
! [A: nat,N3: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N3 )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_433_power__not__zero,axiom,
! [A: real,N3: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N3 )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_434_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_435_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_436_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
= ( ord_less_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_437_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_438_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_439_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_440_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_441_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_442_le__cases3,axiom,
! [X: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_443_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_444_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_445_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_446_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_447_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_448_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_449_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_450_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_451_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_452_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_453_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_454_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_455_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_456_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_457_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_458_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_459_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_460_order__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_461_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_462_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_463_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_464_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_465_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_466_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_467_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_468_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_469_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_470_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_471_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_472_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_473_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_474_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_475_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_476_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_477_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_478_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_479_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_480_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_481_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_482_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_483_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_484_order__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_485_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_486_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_487_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_488_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_489_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_490_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_491_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_492_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_493_order__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_494_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_495_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_496_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_497_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_498_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_499_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_500_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_501_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_502_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_503_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_504_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_505_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_506_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_507_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_508_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_509_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_510_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_511_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_512_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_513_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_514_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_515_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_516_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_517_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_518_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_519_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_520_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_521_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_522_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_523_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_524_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_525_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_526_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_527_harm__mono,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_real @ ( harmonic_harm_real @ M4 ) @ ( harmonic_harm_real @ N3 ) ) ) ).
% harm_mono
thf(fact_528_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_529_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_530_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_531_power__inverse,axiom,
! [A: real,N3: nat] :
( ( power_power_real @ ( inverse_inverse_real @ A ) @ N3 )
= ( inverse_inverse_real @ ( power_power_real @ A @ N3 ) ) ) ).
% power_inverse
thf(fact_532_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_533_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_534_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_535_power__less__imp__less__base,axiom,
! [A: nat,N3: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ B @ N3 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_536_power__less__imp__less__base,axiom,
! [A: int,N3: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ B @ N3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_537_power__less__imp__less__base,axiom,
! [A: real,N3: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ B @ N3 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_538_max__def__raw,axiom,
( ord_max_b
= ( ^ [A4: b,B3: b] : ( if_b @ ( ord_less_eq_b @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_539_max__def__raw,axiom,
( ord_max_nat
= ( ^ [A4: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_540_max__def__raw,axiom,
( ord_max_int
= ( ^ [A4: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_541_max__def__raw,axiom,
( ord_max_real
= ( ^ [A4: real,B3: real] : ( if_real @ ( ord_less_eq_real @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_542_min__def__raw,axiom,
( ord_min_b
= ( ^ [A4: b,B3: b] : ( if_b @ ( ord_less_eq_b @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def_raw
thf(fact_543_min__def__raw,axiom,
( ord_min_nat
= ( ^ [A4: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def_raw
thf(fact_544_min__def__raw,axiom,
( ord_min_int
= ( ^ [A4: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def_raw
thf(fact_545_min__def__raw,axiom,
( ord_min_real
= ( ^ [A4: real,B3: real] : ( if_real @ ( ord_less_eq_real @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def_raw
thf(fact_546_power__eq__iff__eq__base,axiom,
! [N3: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N3 )
= ( power_power_nat @ B @ N3 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_547_power__eq__iff__eq__base,axiom,
! [N3: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N3 )
= ( power_power_int @ B @ N3 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_548_power__eq__iff__eq__base,axiom,
! [N3: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N3 )
= ( power_power_real @ B @ N3 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_549_power__eq__imp__eq__base,axiom,
! [A: nat,N3: nat,B: nat] :
( ( ( power_power_nat @ A @ N3 )
= ( power_power_nat @ B @ N3 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_550_power__eq__imp__eq__base,axiom,
! [A: int,N3: nat,B: int] :
( ( ( power_power_int @ A @ N3 )
= ( power_power_int @ B @ N3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_551_power__eq__imp__eq__base,axiom,
! [A: real,N3: nat,B: real] :
( ( ( power_power_real @ A @ N3 )
= ( power_power_real @ B @ N3 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_552_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
= ( ? [N: nat] :
( ( N != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_553_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D: real,E: real] :
( ( ord_less_real @ D @ E )
=> ( ( P @ D )
=> ( P @ E ) ) )
=> ( ! [N4: nat] :
( ( N4 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_554_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_555_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_556_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_557_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_558_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_559_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_560_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_561_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_562_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_563_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_564_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_565_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_566_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_567_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_568_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_569_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_570_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_571_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_572_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_573_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_574_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_575_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_576_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_577_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_578_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_579_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_580_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_581_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_582_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_583_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_584_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_585_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_586_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_587_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_588_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_589_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_590_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_591_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_592_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_593_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_594_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_595_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_596_dense__ge__bounded,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W2: real] :
( ( ord_less_real @ Z @ W2 )
=> ( ( ord_less_real @ W2 @ X )
=> ( ord_less_eq_real @ Y @ W2 ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_597_dense__le__bounded,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W2: real] :
( ( ord_less_real @ X @ W2 )
=> ( ( ord_less_real @ W2 @ Y )
=> ( ord_less_eq_real @ W2 @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_598_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_599_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_600_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_601_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_602_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_603_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_604_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_605_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_606_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_607_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_608_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_609_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_610_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_611_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_612_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_613_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_614_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_615_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_616_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_617_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_618_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_619_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_620_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_int @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_621_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_622_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_623_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_624_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_625_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_626_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_627_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_628_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_629_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_630_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_631_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_632_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_633_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_634_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_635_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_636_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_637_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_638_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_639_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_640_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_641_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_642_order__le__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_643_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_644_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_645_order__less__le__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_646_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_647_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_648_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_649_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_650_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_651_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_652_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_653_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_654_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_655_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_656_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_657_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_658_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_659_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_660_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_661_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_662_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_663_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_664_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_665_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_666_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_667_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_668_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_669_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_670_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_671_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_672_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_673_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_674_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_675_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_676_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_677_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_678_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_679_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_680_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_681_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_682_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_683_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_684_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_685_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_686_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_687_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_688_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_689_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_690_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_691_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_692_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( M2 != N ) ) ) ) ).
% nat_less_le
thf(fact_693_less__imp__le__nat,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_eq_nat @ M4 @ N3 ) ) ).
% less_imp_le_nat
thf(fact_694_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_695_less__or__eq__imp__le,axiom,
! [M4: nat,N3: nat] :
( ( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) )
=> ( ord_less_eq_nat @ M4 @ N3 ) ) ).
% less_or_eq_imp_le
thf(fact_696_le__neq__implies__less,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ( M4 != N3 )
=> ( ord_less_nat @ M4 @ N3 ) ) ) ).
% le_neq_implies_less
thf(fact_697_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J3: nat] :
( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_698_zero__less__power,axiom,
! [A: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N3 ) ) ) ).
% zero_less_power
thf(fact_699_zero__less__power,axiom,
! [A: int,N3: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N3 ) ) ) ).
% zero_less_power
thf(fact_700_zero__less__power,axiom,
! [A: real,N3: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N3 ) ) ) ).
% zero_less_power
thf(fact_701_max__def,axiom,
( ord_max_b
= ( ^ [A4: b,B3: b] : ( if_b @ ( ord_less_eq_b @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_702_max__def,axiom,
( ord_max_nat
= ( ^ [A4: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_703_max__def,axiom,
( ord_max_int
= ( ^ [A4: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_704_max__def,axiom,
( ord_max_real
= ( ^ [A4: real,B3: real] : ( if_real @ ( ord_less_eq_real @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_705_max_Omono,axiom,
! [C: b,A: b,D3: b,B: b] :
( ( ord_less_eq_b @ C @ A )
=> ( ( ord_less_eq_b @ D3 @ B )
=> ( ord_less_eq_b @ ( ord_max_b @ C @ D3 ) @ ( ord_max_b @ A @ B ) ) ) ) ).
% max.mono
thf(fact_706_max_Omono,axiom,
! [C: nat,A: nat,D3: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D3 @ B )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C @ D3 ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_707_max_Omono,axiom,
! [C: int,A: int,D3: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ( ord_less_eq_int @ D3 @ B )
=> ( ord_less_eq_int @ ( ord_max_int @ C @ D3 ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% max.mono
thf(fact_708_max_Omono,axiom,
! [C: real,A: real,D3: real,B: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ( ord_less_eq_real @ D3 @ B )
=> ( ord_less_eq_real @ ( ord_max_real @ C @ D3 ) @ ( ord_max_real @ A @ B ) ) ) ) ).
% max.mono
thf(fact_709_max_OorderE,axiom,
! [B: b,A: b] :
( ( ord_less_eq_b @ B @ A )
=> ( A
= ( ord_max_b @ A @ B ) ) ) ).
% max.orderE
thf(fact_710_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_711_max_OorderE,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( A
= ( ord_max_int @ A @ B ) ) ) ).
% max.orderE
thf(fact_712_max_OorderE,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( A
= ( ord_max_real @ A @ B ) ) ) ).
% max.orderE
thf(fact_713_max_OorderI,axiom,
! [A: b,B: b] :
( ( A
= ( ord_max_b @ A @ B ) )
=> ( ord_less_eq_b @ B @ A ) ) ).
% max.orderI
thf(fact_714_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_715_max_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( ord_max_int @ A @ B ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
% max.orderI
thf(fact_716_max_OorderI,axiom,
! [A: real,B: real] :
( ( A
= ( ord_max_real @ A @ B ) )
=> ( ord_less_eq_real @ B @ A ) ) ).
% max.orderI
thf(fact_717_max_OboundedE,axiom,
! [B: b,C: b,A: b] :
( ( ord_less_eq_b @ ( ord_max_b @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_b @ B @ A )
=> ~ ( ord_less_eq_b @ C @ A ) ) ) ).
% max.boundedE
thf(fact_718_max_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_719_max_OboundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_int @ B @ A )
=> ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% max.boundedE
thf(fact_720_max_OboundedE,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_real @ B @ A )
=> ~ ( ord_less_eq_real @ C @ A ) ) ) ).
% max.boundedE
thf(fact_721_max_OboundedI,axiom,
! [B: b,A: b,C: b] :
( ( ord_less_eq_b @ B @ A )
=> ( ( ord_less_eq_b @ C @ A )
=> ( ord_less_eq_b @ ( ord_max_b @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_722_max_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_723_max_OboundedI,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_724_max_OboundedI,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_725_max_Oorder__iff,axiom,
( ord_less_eq_b
= ( ^ [B3: b,A4: b] :
( A4
= ( ord_max_b @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_726_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( ord_max_nat @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_727_max_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( A4
= ( ord_max_int @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_728_max_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( A4
= ( ord_max_real @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_729_max_Ocobounded1,axiom,
! [A: b,B: b] : ( ord_less_eq_b @ A @ ( ord_max_b @ A @ B ) ) ).
% max.cobounded1
thf(fact_730_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_731_max_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded1
thf(fact_732_max_Ocobounded1,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ A @ ( ord_max_real @ A @ B ) ) ).
% max.cobounded1
thf(fact_733_max_Ocobounded2,axiom,
! [B: b,A: b] : ( ord_less_eq_b @ B @ ( ord_max_b @ A @ B ) ) ).
% max.cobounded2
thf(fact_734_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_735_max_Ocobounded2,axiom,
! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded2
thf(fact_736_max_Ocobounded2,axiom,
! [B: real,A: real] : ( ord_less_eq_real @ B @ ( ord_max_real @ A @ B ) ) ).
% max.cobounded2
thf(fact_737_le__max__iff__disj,axiom,
! [Z: b,X: b,Y: b] :
( ( ord_less_eq_b @ Z @ ( ord_max_b @ X @ Y ) )
= ( ( ord_less_eq_b @ Z @ X )
| ( ord_less_eq_b @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_738_le__max__iff__disj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_eq_nat @ Z @ X )
| ( ord_less_eq_nat @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_739_le__max__iff__disj,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
= ( ( ord_less_eq_int @ Z @ X )
| ( ord_less_eq_int @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_740_le__max__iff__disj,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_eq_real @ Z @ ( ord_max_real @ X @ Y ) )
= ( ( ord_less_eq_real @ Z @ X )
| ( ord_less_eq_real @ Z @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_741_max_Oabsorb__iff1,axiom,
( ord_less_eq_b
= ( ^ [B3: b,A4: b] :
( ( ord_max_b @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_742_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_max_nat @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_743_max_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_max_int @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_744_max_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( ord_max_real @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_745_max_Oabsorb__iff2,axiom,
( ord_less_eq_b
= ( ^ [A4: b,B3: b] :
( ( ord_max_b @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_746_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_max_nat @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_747_max_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_max_int @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_748_max_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( ord_max_real @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_749_max_OcoboundedI1,axiom,
! [C: b,A: b,B: b] :
( ( ord_less_eq_b @ C @ A )
=> ( ord_less_eq_b @ C @ ( ord_max_b @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_750_max_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_751_max_OcoboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_752_max_OcoboundedI1,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_753_max_OcoboundedI2,axiom,
! [C: b,B: b,A: b] :
( ( ord_less_eq_b @ C @ B )
=> ( ord_less_eq_b @ C @ ( ord_max_b @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_754_max_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_755_max_OcoboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_756_max_OcoboundedI2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_757_max__absorb1,axiom,
! [Y: b,X: b] :
( ( ord_less_eq_b @ Y @ X )
=> ( ( ord_max_b @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_758_max__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_max_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_759_max__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_max_int @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_760_max__absorb1,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_max_real @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_761_max__absorb2,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_max_b @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_762_max__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_max_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_763_max__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_max_int @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_764_max__absorb2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_max_real @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_765_min__def,axiom,
( ord_min_b
= ( ^ [A4: b,B3: b] : ( if_b @ ( ord_less_eq_b @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def
thf(fact_766_min__def,axiom,
( ord_min_nat
= ( ^ [A4: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def
thf(fact_767_min__def,axiom,
( ord_min_int
= ( ^ [A4: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def
thf(fact_768_min__def,axiom,
( ord_min_real
= ( ^ [A4: real,B3: real] : ( if_real @ ( ord_less_eq_real @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def
thf(fact_769_min_Omono,axiom,
! [A: b,C: b,B: b,D3: b] :
( ( ord_less_eq_b @ A @ C )
=> ( ( ord_less_eq_b @ B @ D3 )
=> ( ord_less_eq_b @ ( ord_min_b @ A @ B ) @ ( ord_min_b @ C @ D3 ) ) ) ) ).
% min.mono
thf(fact_770_min_Omono,axiom,
! [A: nat,C: nat,B: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D3 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C @ D3 ) ) ) ) ).
% min.mono
thf(fact_771_min_Omono,axiom,
! [A: int,C: int,B: int,D3: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ B @ D3 )
=> ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ ( ord_min_int @ C @ D3 ) ) ) ) ).
% min.mono
thf(fact_772_min_Omono,axiom,
! [A: real,C: real,B: real,D3: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ B @ D3 )
=> ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ ( ord_min_real @ C @ D3 ) ) ) ) ).
% min.mono
thf(fact_773_min_OorderE,axiom,
! [A: b,B: b] :
( ( ord_less_eq_b @ A @ B )
=> ( A
= ( ord_min_b @ A @ B ) ) ) ).
% min.orderE
thf(fact_774_min_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( ord_min_nat @ A @ B ) ) ) ).
% min.orderE
thf(fact_775_min_OorderE,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( A
= ( ord_min_int @ A @ B ) ) ) ).
% min.orderE
thf(fact_776_min_OorderE,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( A
= ( ord_min_real @ A @ B ) ) ) ).
% min.orderE
thf(fact_777_min_OorderI,axiom,
! [A: b,B: b] :
( ( A
= ( ord_min_b @ A @ B ) )
=> ( ord_less_eq_b @ A @ B ) ) ).
% min.orderI
thf(fact_778_min_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_min_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% min.orderI
thf(fact_779_min_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( ord_min_int @ A @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% min.orderI
thf(fact_780_min_OorderI,axiom,
! [A: real,B: real] :
( ( A
= ( ord_min_real @ A @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% min.orderI
thf(fact_781_min_OboundedE,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_eq_b @ A @ ( ord_min_b @ B @ C ) )
=> ~ ( ( ord_less_eq_b @ A @ B )
=> ~ ( ord_less_eq_b @ A @ C ) ) ) ).
% min.boundedE
thf(fact_782_min_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.boundedE
thf(fact_783_min_OboundedE,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) )
=> ~ ( ( ord_less_eq_int @ A @ B )
=> ~ ( ord_less_eq_int @ A @ C ) ) ) ).
% min.boundedE
thf(fact_784_min_OboundedE,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( ord_min_real @ B @ C ) )
=> ~ ( ( ord_less_eq_real @ A @ B )
=> ~ ( ord_less_eq_real @ A @ C ) ) ) ).
% min.boundedE
thf(fact_785_min_OboundedI,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_eq_b @ A @ C )
=> ( ord_less_eq_b @ A @ ( ord_min_b @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_786_min_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_787_min_OboundedI,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_788_min_OboundedI,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ A @ ( ord_min_real @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_789_min_Oorder__iff,axiom,
( ord_less_eq_b
= ( ^ [A4: b,B3: b] :
( A4
= ( ord_min_b @ A4 @ B3 ) ) ) ) ).
% min.order_iff
thf(fact_790_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( ord_min_nat @ A4 @ B3 ) ) ) ) ).
% min.order_iff
thf(fact_791_min_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( A4
= ( ord_min_int @ A4 @ B3 ) ) ) ) ).
% min.order_iff
thf(fact_792_min_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( A4
= ( ord_min_real @ A4 @ B3 ) ) ) ) ).
% min.order_iff
thf(fact_793_min_Ocobounded1,axiom,
! [A: b,B: b] : ( ord_less_eq_b @ ( ord_min_b @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_794_min_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_795_min_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_796_min_Ocobounded1,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_797_min_Ocobounded2,axiom,
! [A: b,B: b] : ( ord_less_eq_b @ ( ord_min_b @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_798_min_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_799_min_Ocobounded2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_800_min_Ocobounded2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_801_min_Oabsorb__iff1,axiom,
( ord_less_eq_b
= ( ^ [A4: b,B3: b] :
( ( ord_min_b @ A4 @ B3 )
= A4 ) ) ) ).
% min.absorb_iff1
thf(fact_802_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_min_nat @ A4 @ B3 )
= A4 ) ) ) ).
% min.absorb_iff1
thf(fact_803_min_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_min_int @ A4 @ B3 )
= A4 ) ) ) ).
% min.absorb_iff1
thf(fact_804_min_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( ord_min_real @ A4 @ B3 )
= A4 ) ) ) ).
% min.absorb_iff1
thf(fact_805_min_Oabsorb__iff2,axiom,
( ord_less_eq_b
= ( ^ [B3: b,A4: b] :
( ( ord_min_b @ A4 @ B3 )
= B3 ) ) ) ).
% min.absorb_iff2
thf(fact_806_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_min_nat @ A4 @ B3 )
= B3 ) ) ) ).
% min.absorb_iff2
thf(fact_807_min_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_min_int @ A4 @ B3 )
= B3 ) ) ) ).
% min.absorb_iff2
thf(fact_808_min_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( ord_min_real @ A4 @ B3 )
= B3 ) ) ) ).
% min.absorb_iff2
thf(fact_809_min_OcoboundedI1,axiom,
! [A: b,C: b,B: b] :
( ( ord_less_eq_b @ A @ C )
=> ( ord_less_eq_b @ ( ord_min_b @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_810_min_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_811_min_OcoboundedI1,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_812_min_OcoboundedI1,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_813_min_OcoboundedI2,axiom,
! [B: b,C: b,A: b] :
( ( ord_less_eq_b @ B @ C )
=> ( ord_less_eq_b @ ( ord_min_b @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_814_min_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_815_min_OcoboundedI2,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_816_min_OcoboundedI2,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ ( ord_min_real @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_817_min__le__iff__disj,axiom,
! [X: b,Y: b,Z: b] :
( ( ord_less_eq_b @ ( ord_min_b @ X @ Y ) @ Z )
= ( ( ord_less_eq_b @ X @ Z )
| ( ord_less_eq_b @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_818_min__le__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
| ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_819_min__le__iff__disj,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ ( ord_min_int @ X @ Y ) @ Z )
= ( ( ord_less_eq_int @ X @ Z )
| ( ord_less_eq_int @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_820_min__le__iff__disj,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ ( ord_min_real @ X @ Y ) @ Z )
= ( ( ord_less_eq_real @ X @ Z )
| ( ord_less_eq_real @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_821_min__absorb1,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_min_b @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_822_min__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_min_nat @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_823_min__absorb1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_min_int @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_824_min__absorb1,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_min_real @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_825_min__absorb2,axiom,
! [Y: b,X: b] :
( ( ord_less_eq_b @ Y @ X )
=> ( ( ord_min_b @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_826_min__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_min_nat @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_827_min__absorb2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_min_int @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_828_min__absorb2,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_min_real @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_829_nat__power__less__imp__less,axiom,
! [I: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M4 ) @ ( power_power_nat @ I @ N3 ) )
=> ( ord_less_nat @ M4 @ N3 ) ) ) ).
% nat_power_less_imp_less
thf(fact_830_power__strict__mono,axiom,
! [A: nat,B: nat,N3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ B @ N3 ) ) ) ) ) ).
% power_strict_mono
thf(fact_831_power__strict__mono,axiom,
! [A: int,B: int,N3: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ B @ N3 ) ) ) ) ) ).
% power_strict_mono
thf(fact_832_power__strict__mono,axiom,
! [A: real,B: real,N3: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ B @ N3 ) ) ) ) ) ).
% power_strict_mono
thf(fact_833_harm__expand_I1_J,axiom,
( ( harmonic_harm_real @ zero_zero_nat )
= zero_zero_real ) ).
% harm_expand(1)
thf(fact_834_zero__power,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( power_power_int @ zero_zero_int @ N3 )
= zero_zero_int ) ) ).
% zero_power
thf(fact_835_zero__power,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( power_power_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_836_zero__power,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( power_power_real @ zero_zero_real @ N3 )
= zero_zero_real ) ) ).
% zero_power
thf(fact_837_of__nat__0__le__iff,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N3 ) ) ).
% of_nat_0_le_iff
thf(fact_838_of__nat__0__le__iff,axiom,
! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N3 ) ) ).
% of_nat_0_le_iff
thf(fact_839_of__nat__0__le__iff,axiom,
! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% of_nat_0_le_iff
thf(fact_840_ex__least__nat__le,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_841_borel__measurable__inverse,axiom,
! [F: real > real,M: sigma_measure_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_real_real
@ ^ [X2: real] : ( inverse_inverse_real @ ( F @ X2 ) )
@ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurable_inverse
thf(fact_842_borel__measurable__power,axiom,
! [F: real > real,M: sigma_measure_real,N3: nat] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_real_real
@ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N3 )
@ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurable_power
thf(fact_843_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_844_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_845_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_846_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_847_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_848_inverse__le__iff__le,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le
thf(fact_849_inverse__le__iff__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_850_inverse__less__iff__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less
thf(fact_851_inverse__less__iff__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_852_inverse__negative__iff__negative,axiom,
! [A: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% inverse_negative_iff_negative
thf(fact_853_inverse__positive__iff__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% inverse_positive_iff_positive
thf(fact_854_inverse__nonpositive__iff__nonpositive,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_855_inverse__nonnegative__iff__nonnegative,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_856_inverse__eq__iff__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_857_inverse__inverse__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_858_power__one,axiom,
! [N3: nat] :
( ( power_power_int @ one_one_int @ N3 )
= one_one_int ) ).
% power_one
thf(fact_859_power__one,axiom,
! [N3: nat] :
( ( power_power_nat @ one_one_nat @ N3 )
= one_one_nat ) ).
% power_one
thf(fact_860_power__one,axiom,
! [N3: nat] :
( ( power_power_real @ one_one_real @ N3 )
= one_one_real ) ).
% power_one
thf(fact_861_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_862_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_863_inverse__eq__1__iff,axiom,
! [X: real] :
( ( ( inverse_inverse_real @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% inverse_eq_1_iff
thf(fact_864_inverse__1,axiom,
( ( inverse_inverse_real @ one_one_real )
= one_one_real ) ).
% inverse_1
thf(fact_865_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_866_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_867_power__inject__exp,axiom,
! [A: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M4 )
= ( power_power_nat @ A @ N3 ) )
= ( M4 = N3 ) ) ) ).
% power_inject_exp
thf(fact_868_power__inject__exp,axiom,
! [A: int,M4: nat,N3: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M4 )
= ( power_power_int @ A @ N3 ) )
= ( M4 = N3 ) ) ) ).
% power_inject_exp
thf(fact_869_power__inject__exp,axiom,
! [A: real,M4: nat,N3: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M4 )
= ( power_power_real @ A @ N3 ) )
= ( M4 = N3 ) ) ) ).
% power_inject_exp
thf(fact_870_max__0__1_I1_J,axiom,
( ( ord_max_int @ zero_zero_int @ one_one_int )
= one_one_int ) ).
% max_0_1(1)
thf(fact_871_max__0__1_I1_J,axiom,
( ( ord_max_real @ zero_zero_real @ one_one_real )
= one_one_real ) ).
% max_0_1(1)
thf(fact_872_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_873_max__0__1_I2_J,axiom,
( ( ord_max_int @ one_one_int @ zero_zero_int )
= one_one_int ) ).
% max_0_1(2)
thf(fact_874_max__0__1_I2_J,axiom,
( ( ord_max_real @ one_one_real @ zero_zero_real )
= one_one_real ) ).
% max_0_1(2)
thf(fact_875_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_876_min__0__1_I1_J,axiom,
( ( ord_min_int @ zero_zero_int @ one_one_int )
= zero_zero_int ) ).
% min_0_1(1)
thf(fact_877_min__0__1_I1_J,axiom,
( ( ord_min_real @ zero_zero_real @ one_one_real )
= zero_zero_real ) ).
% min_0_1(1)
thf(fact_878_min__0__1_I1_J,axiom,
( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
= zero_zero_nat ) ).
% min_0_1(1)
thf(fact_879_min__0__1_I2_J,axiom,
( ( ord_min_int @ one_one_int @ zero_zero_int )
= zero_zero_int ) ).
% min_0_1(2)
thf(fact_880_min__0__1_I2_J,axiom,
( ( ord_min_real @ one_one_real @ zero_zero_real )
= zero_zero_real ) ).
% min_0_1(2)
thf(fact_881_min__0__1_I2_J,axiom,
( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0_1(2)
thf(fact_882_of__nat__eq__1__iff,axiom,
! [N3: nat] :
( ( ( semiri1316708129612266289at_nat @ N3 )
= one_one_nat )
= ( N3 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_883_of__nat__eq__1__iff,axiom,
! [N3: nat] :
( ( ( semiri1314217659103216013at_int @ N3 )
= one_one_int )
= ( N3 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_884_of__nat__eq__1__iff,axiom,
! [N3: nat] :
( ( ( semiri5074537144036343181t_real @ N3 )
= one_one_real )
= ( N3 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_885_of__nat__1__eq__iff,axiom,
! [N3: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N3 ) )
= ( N3 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_886_of__nat__1__eq__iff,axiom,
! [N3: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N3 ) )
= ( N3 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_887_of__nat__1__eq__iff,axiom,
! [N3: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N3 ) )
= ( N3 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_888_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_889_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_890_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_891_less__one,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ one_one_nat )
= ( N3 = zero_zero_nat ) ) ).
% less_one
thf(fact_892_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_893_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_894_power__strict__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_895_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_896_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_897_power__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_898_power__strict__decreasing__iff,axiom,
! [B: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M4 ) @ ( power_power_nat @ B @ N3 ) )
= ( ord_less_nat @ N3 @ M4 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_899_power__strict__decreasing__iff,axiom,
! [B: int,M4: nat,N3: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M4 ) @ ( power_power_int @ B @ N3 ) )
= ( ord_less_nat @ N3 @ M4 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_900_power__strict__decreasing__iff,axiom,
! [B: real,M4: nat,N3: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M4 ) @ ( power_power_real @ B @ N3 ) )
= ( ord_less_nat @ N3 @ M4 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_901_power__decreasing__iff,axiom,
! [B: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M4 ) @ ( power_power_nat @ B @ N3 ) )
= ( ord_less_eq_nat @ N3 @ M4 ) ) ) ) ).
% power_decreasing_iff
thf(fact_902_power__decreasing__iff,axiom,
! [B: int,M4: nat,N3: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M4 ) @ ( power_power_int @ B @ N3 ) )
= ( ord_less_eq_nat @ N3 @ M4 ) ) ) ) ).
% power_decreasing_iff
thf(fact_903_power__decreasing__iff,axiom,
! [B: real,M4: nat,N3: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M4 ) @ ( power_power_real @ B @ N3 ) )
= ( ord_less_eq_nat @ N3 @ M4 ) ) ) ) ).
% power_decreasing_iff
thf(fact_904_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_905_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_906_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_907_le__trans,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_908_eq__imp__le,axiom,
! [M4: nat,N3: nat] :
( ( M4 = N3 )
=> ( ord_less_eq_nat @ M4 @ N3 ) ) ).
% eq_imp_le
thf(fact_909_le__antisym,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M4 )
=> ( M4 = N3 ) ) ) ).
% le_antisym
thf(fact_910_nat__le__linear,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
| ( ord_less_eq_nat @ N3 @ M4 ) ) ).
% nat_le_linear
thf(fact_911_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_912_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_913_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_914_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_915_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_916_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_917_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_918_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_919_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_920_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_921_inverse__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
= ( ( ord_less_eq_real @ X @ zero_zero_real )
| ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% inverse_le_1_iff
thf(fact_922_one__less__inverse__iff,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
= ( ( ord_less_real @ zero_zero_real @ X )
& ( ord_less_real @ X @ one_one_real ) ) ) ).
% one_less_inverse_iff
thf(fact_923_one__less__inverse,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% one_less_inverse
thf(fact_924_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_925_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_926_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_927_one__le__power,axiom,
! [A: nat,N3: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N3 ) ) ) ).
% one_le_power
thf(fact_928_one__le__power,axiom,
! [A: int,N3: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N3 ) ) ) ).
% one_le_power
thf(fact_929_one__le__power,axiom,
! [A: real,N3: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N3 ) ) ) ).
% one_le_power
thf(fact_930_power__increasing,axiom,
! [N3: nat,N2: nat,A: nat] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_increasing
thf(fact_931_power__increasing,axiom,
! [N3: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_increasing
thf(fact_932_power__increasing,axiom,
! [N3: nat,N2: nat,A: real] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_increasing
thf(fact_933_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_934_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_935_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_936_one__le__inverse__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
= ( ( ord_less_real @ zero_zero_real @ X )
& ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% one_le_inverse_iff
thf(fact_937_inverse__less__1__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
= ( ( ord_less_eq_real @ X @ zero_zero_real )
| ( ord_less_real @ one_one_real @ X ) ) ) ).
% inverse_less_1_iff
thf(fact_938_one__le__inverse,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% one_le_inverse
thf(fact_939_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_940_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X5 ) ).
% linordered_field_no_lb
thf(fact_941_power__le__one,axiom,
! [A: nat,N3: 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 @ N3 ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_942_power__le__one,axiom,
! [A: int,N3: 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 @ N3 ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_943_power__le__one,axiom,
! [A: real,N3: 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 @ N3 ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_944_power__decreasing,axiom,
! [N3: nat,N2: nat,A: nat] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ( 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 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ) ).
% power_decreasing
thf(fact_945_power__decreasing,axiom,
! [N3: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ( 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 ) @ ( power_power_int @ A @ N3 ) ) ) ) ) ).
% power_decreasing
thf(fact_946_power__decreasing,axiom,
! [N3: nat,N2: nat,A: real] :
( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ( 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 ) @ ( power_power_real @ A @ N3 ) ) ) ) ) ).
% power_decreasing
thf(fact_947_power__le__imp__le__exp,axiom,
! [A: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M4 ) @ ( power_power_nat @ A @ N3 ) )
=> ( ord_less_eq_nat @ M4 @ N3 ) ) ) ).
% power_le_imp_le_exp
thf(fact_948_power__le__imp__le__exp,axiom,
! [A: int,M4: nat,N3: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M4 ) @ ( power_power_int @ A @ N3 ) )
=> ( ord_less_eq_nat @ M4 @ N3 ) ) ) ).
% power_le_imp_le_exp
thf(fact_949_power__le__imp__le__exp,axiom,
! [A: real,M4: nat,N3: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M4 ) @ ( power_power_real @ A @ N3 ) )
=> ( ord_less_eq_nat @ M4 @ N3 ) ) ) ).
% power_le_imp_le_exp
thf(fact_950_power__0__left,axiom,
! [N3: nat] :
( ( ( N3 = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N3 )
= one_one_int ) )
& ( ( N3 != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N3 )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_951_power__0__left,axiom,
! [N3: nat] :
( ( ( N3 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N3 )
= one_one_nat ) )
& ( ( N3 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_952_power__0__left,axiom,
! [N3: nat] :
( ( ( N3 = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N3 )
= one_one_real ) )
& ( ( N3 != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N3 )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_953_power__strict__increasing,axiom,
! [N3: nat,N2: nat,A: nat] :
( ( ord_less_nat @ N3 @ N2 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_954_power__strict__increasing,axiom,
! [N3: nat,N2: nat,A: int] :
( ( ord_less_nat @ N3 @ N2 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_955_power__strict__increasing,axiom,
! [N3: nat,N2: nat,A: real] :
( ( ord_less_nat @ N3 @ N2 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_956_power__less__imp__less__exp,axiom,
! [A: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M4 ) @ ( power_power_nat @ A @ N3 ) )
=> ( ord_less_nat @ M4 @ N3 ) ) ) ).
% power_less_imp_less_exp
thf(fact_957_power__less__imp__less__exp,axiom,
! [A: int,M4: nat,N3: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M4 ) @ ( power_power_int @ A @ N3 ) )
=> ( ord_less_nat @ M4 @ N3 ) ) ) ).
% power_less_imp_less_exp
thf(fact_958_power__less__imp__less__exp,axiom,
! [A: real,M4: nat,N3: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M4 ) @ ( power_power_real @ A @ N3 ) )
=> ( ord_less_nat @ M4 @ N3 ) ) ) ).
% power_less_imp_less_exp
thf(fact_959_inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_960_power__strict__decreasing,axiom,
! [N3: nat,N2: nat,A: nat] :
( ( ord_less_nat @ N3 @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_961_power__strict__decreasing,axiom,
! [N3: nat,N2: nat,A: int] :
( ( ord_less_nat @ N3 @ N2 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_962_power__strict__decreasing,axiom,
! [N3: nat,N2: nat,A: real] :
( ( ord_less_nat @ N3 @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_963_self__le__power,axiom,
! [A: nat,N3: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% self_le_power
thf(fact_964_self__le__power,axiom,
! [A: int,N3: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N3 ) ) ) ) ).
% self_le_power
thf(fact_965_self__le__power,axiom,
! [A: real,N3: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N3 ) ) ) ) ).
% self_le_power
thf(fact_966_one__less__power,axiom,
! [A: nat,N3: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% one_less_power
thf(fact_967_one__less__power,axiom,
! [A: int,N3: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N3 ) ) ) ) ).
% one_less_power
thf(fact_968_one__less__power,axiom,
! [A: real,N3: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N3 ) ) ) ) ).
% one_less_power
thf(fact_969_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_970_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_971_nonzero__inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_972_nonzero__inverse__inverse__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_973_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_974_positive__imp__inverse__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% positive_imp_inverse_positive
thf(fact_975_negative__imp__inverse__negative,axiom,
! [A: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% negative_imp_inverse_negative
thf(fact_976_inverse__positive__imp__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
=> ( ( A != zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% inverse_positive_imp_positive
thf(fact_977_inverse__negative__imp__negative,axiom,
! [A: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
=> ( ( A != zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% inverse_negative_imp_negative
thf(fact_978_less__imp__inverse__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_979_inverse__less__imp__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_980_less__imp__inverse__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% less_imp_inverse_less
thf(fact_981_inverse__less__imp__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ B @ A ) ) ) ).
% inverse_less_imp_less
thf(fact_982_le__imp__inverse__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_983_inverse__le__imp__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_984_le__imp__inverse__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% le_imp_inverse_le
thf(fact_985_inverse__le__imp__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ B @ A ) ) ) ).
% inverse_le_imp_le
thf(fact_986_real__arch__invD,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [N4: nat] :
( ( N4 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ E2 ) ) ) ).
% real_arch_invD
thf(fact_987_one__less__of__natD,axiom,
! [N3: nat] :
( ( ord_less_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ N3 ) )
=> ( ord_less_nat @ one_one_nat @ N3 ) ) ).
% one_less_of_natD
thf(fact_988_one__less__of__natD,axiom,
! [N3: nat] :
( ( ord_less_int @ one_one_int @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ord_less_nat @ one_one_nat @ N3 ) ) ).
% one_less_of_natD
thf(fact_989_one__less__of__natD,axiom,
! [N3: nat] :
( ( ord_less_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
=> ( ord_less_nat @ one_one_nat @ N3 ) ) ).
% one_less_of_natD
thf(fact_990_realpow__pos__nth__unique,axiom,
! [N3: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N3 )
= A )
& ! [Y3: real] :
( ( ( ord_less_real @ zero_zero_real @ Y3 )
& ( ( power_power_real @ Y3 @ N3 )
= A ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_991_realpow__pos__nth,axiom,
! [N3: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N3 )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_992_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_993_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_994_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_995_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_996_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_997_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_998_complete__real,axiom,
! [S2: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S2 )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y2: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S2 )
=> ( ord_less_eq_real @ X5 @ Y2 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_999_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N4: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N4 ) ) ) ).
% real_arch_pow
thf(fact_1000_power__le__one__iff,axiom,
! [A: real,N3: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ one_one_real )
= ( ( N3 = zero_zero_nat )
| ( ord_less_eq_real @ A @ one_one_real ) ) ) ) ).
% power_le_one_iff
thf(fact_1001_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X @ N4 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1002_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1003_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1004_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1005_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_1006_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1007_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_1008_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_1009_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1010_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1011_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1012_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1013_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1014_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1015_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1016_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1017_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_1018_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I2: nat] :
( ( Q @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I2 ) )
& ( ord_less_eq_real @ ( X3 @ 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_1019_real__of__nat__ge__one__iff,axiom,
! [N3: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
= ( ord_less_eq_nat @ one_one_nat @ N3 ) ) ).
% real_of_nat_ge_one_iff
thf(fact_1020_seq__mono__lemma,axiom,
! [M4: nat,D3: nat > real,E2: nat > real] :
( ! [N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ord_less_real @ ( D3 @ N4 ) @ ( E2 @ N4 ) ) )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ord_less_eq_real @ ( E2 @ N4 ) @ ( E2 @ M4 ) ) )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ord_less_real @ ( D3 @ N5 ) @ ( E2 @ M4 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1021_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
@ ^ [X2: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X2 )
@ ^ [X2: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X2 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_1022_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [F: nat > nat > nat,Z: nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: nat,B: nat] :
( ( semila1623282765462674594er_nat @ F @ Z @ Less_eq @ Less )
=> ( ( Z
= ( F @ A @ B ) )
= ( ( A = Z )
& ( B = Z ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
thf(fact_1023_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [F: nat > nat > nat,Z: nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: nat,B: nat] :
( ( semila1623282765462674594er_nat @ F @ Z @ Less_eq @ Less )
=> ( ( ( F @ A @ B )
= Z )
= ( ( A = Z )
& ( B = Z ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
thf(fact_1024_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P4: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1025_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P4: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1026_nat__descend__induct,axiom,
! [N3: nat,P: nat > $o,M4: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N3 @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M4 ) ) ) ).
% nat_descend_induct
thf(fact_1027_fps__inverse__one_H,axiom,
( ( ( inverse_inverse_real @ one_one_real )
= one_one_real )
=> ( ( invers68952373231134600s_real @ one_on8598947968683843321s_real )
= one_on8598947968683843321s_real ) ) ).
% fps_inverse_one'
thf(fact_1028_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_1029_minf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_1030_minf_I7_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z2 )
=> ~ ( ord_less_real @ T @ X5 ) ) ).
% minf(7)
thf(fact_1031_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_1032_minf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_1033_minf_I5_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z2 )
=> ( ord_less_real @ X5 @ T ) ) ).
% minf(5)
thf(fact_1034_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1035_minf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1036_minf_I4_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1037_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1038_minf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1039_minf_I3_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1040_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1041_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1042_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1043_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1044_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1045_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1046_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1047_pinf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1048_pinf_I7_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ Z2 @ X5 )
=> ( ord_less_real @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1049_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1050_pinf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1051_pinf_I5_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ Z2 @ X5 )
=> ~ ( ord_less_real @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1052_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1053_pinf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1054_pinf_I4_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1055_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1056_pinf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1057_pinf_I3_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1058_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1059_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1060_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1061_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ Z2 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1062_reals__power__lt__ex,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ Y )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_real @ ( power_power_real @ ( divide_divide_real @ one_one_real @ Y ) @ K3 ) @ X ) ) ) ) ).
% reals_power_lt_ex
thf(fact_1063_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_1064_real__of__nat__div4,axiom,
! [N3: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1065_div__pos__pos__trivial,axiom,
! [K: int,L3: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L3 )
=> ( ( divide_divide_int @ K @ L3 )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1066_div__neg__neg__trivial,axiom,
! [K: int,L3: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L3 @ K )
=> ( ( divide_divide_int @ K @ L3 )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1067_div__less,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ( divide_divide_nat @ M4 @ N3 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1068_div__eq__dividend__iff,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ( ( divide_divide_nat @ M4 @ N3 )
= M4 )
= ( N3 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1069_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_1070_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_1071_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_1072_div__le__dividend,axiom,
! [M4: nat,N3: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M4 @ N3 ) @ M4 ) ).
% div_le_dividend
thf(fact_1073_div__le__mono,axiom,
! [M4: nat,N3: nat,K: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M4 @ K ) @ ( divide_divide_nat @ N3 @ K ) ) ) ).
% div_le_mono
thf(fact_1074_zdiv__int,axiom,
! [M4: nat,N3: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M4 @ N3 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zdiv_int
thf(fact_1075_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M4: nat,N3: nat] :
( ( ( divide_divide_nat @ M4 @ N3 )
= zero_zero_nat )
= ( ( ord_less_nat @ M4 @ N3 )
| ( N3 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1076_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1077_zdiv__mono1,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1078_zdiv__mono2,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1079_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1080_zdiv__mono1__neg,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1081_zdiv__mono2__neg,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1082_div__int__pos__iff,axiom,
! [K: int,L3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L3 ) )
= ( ( K = zero_zero_int )
| ( L3 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L3 ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L3 @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1083_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_1084_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_1085_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1086_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_1087_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_1088_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_1089_div__le__mono2,axiom,
! [M4: nat,N3: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N3 ) @ ( divide_divide_nat @ K @ M4 ) ) ) ) ).
% div_le_mono2
thf(fact_1090_div__greater__zero__iff,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M4 @ N3 ) )
= ( ( ord_less_eq_nat @ N3 @ M4 )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% div_greater_zero_iff
thf(fact_1091_div__less__dividend,axiom,
! [N3: nat,M4: nat] :
( ( ord_less_nat @ one_one_nat @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_nat @ ( divide_divide_nat @ M4 @ N3 ) @ M4 ) ) ) ).
% div_less_dividend
thf(fact_1092_real__root__increasing,axiom,
! [N3: nat,N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_1093_real__root__pow__pos2,axiom,
! [N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N3 @ X ) @ N3 )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_1094_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X2: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X2 )
@ ^ [X2: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X2 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1095_real__root__zero,axiom,
! [N3: nat] :
( ( root @ N3 @ zero_zero_real )
= zero_zero_real ) ).
% real_root_zero
thf(fact_1096_root__0,axiom,
! [X: real] :
( ( root @ zero_zero_nat @ X )
= zero_zero_real ) ).
% root_0
thf(fact_1097_real__root__eq__iff,axiom,
! [N3: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ( root @ N3 @ X )
= ( root @ N3 @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_1098_real__root__eq__0__iff,axiom,
! [N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ( root @ N3 @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_1099_real__root__le__iff,axiom,
! [N3: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_1100_real__root__less__iff,axiom,
! [N3: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_1101_real__root__one,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( root @ N3 @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_1102_real__root__eq__1__iff,axiom,
! [N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ( root @ N3 @ X )
= one_one_real )
= ( X = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_1103_real__root__ge__0__iff,axiom,
! [N3: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N3 @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_1104_real__root__le__0__iff,axiom,
! [N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_real @ ( root @ N3 @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_1105_real__root__lt__0__iff,axiom,
! [N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ ( root @ N3 @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_1106_real__root__gt__0__iff,axiom,
! [N3: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N3 @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_1107_real__root__ge__1__iff,axiom,
! [N3: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N3 @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_1108_real__root__le__1__iff,axiom,
! [N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_real @ ( root @ N3 @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_1109_real__root__lt__1__iff,axiom,
! [N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ ( root @ N3 @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_1110_real__root__gt__1__iff,axiom,
! [N3: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ one_one_real @ ( root @ N3 @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_1111_real__root__inverse,axiom,
! [N3: nat,X: real] :
( ( root @ N3 @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( root @ N3 @ X ) ) ) ).
% real_root_inverse
thf(fact_1112_borel__measurable__root,axiom,
! [N3: nat] : ( member_real_real @ ( root @ N3 ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ).
% borel_measurable_root
thf(fact_1113_real__root__pos__pos__le,axiom,
! [X: real,N3: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N3 @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_1114_real__root__le__mono,axiom,
! [N3: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_1115_real__root__less__mono,axiom,
! [N3: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_1116_real__root__power,axiom,
! [N3: nat,X: real,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( root @ N3 @ ( power_power_real @ X @ K ) )
= ( power_power_real @ ( root @ N3 @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_1117_real__root__gt__zero,axiom,
! [N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( root @ N3 @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_1118_real__root__strict__decreasing,axiom,
! [N3: nat,N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ N3 @ N2 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_1119_real__root__pos__pos,axiom,
! [N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N3 @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_1120_real__root__strict__increasing,axiom,
! [N3: nat,N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ N3 @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_1121_real__root__decreasing,axiom,
! [N3: nat,N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ N2 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_1122_real__root__power__cancel,axiom,
! [N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( root @ N3 @ ( power_power_real @ X @ N3 ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_1123_real__root__pos__unique,axiom,
! [N3: nat,Y: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ Y @ N3 )
= X )
=> ( ( root @ N3 @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_1124_real__root__pow__pos,axiom,
! [N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N3 @ X ) @ N3 )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_1125_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1126_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1127_Suc__le__mono,axiom,
! [N3: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( suc @ M4 ) )
= ( ord_less_eq_nat @ N3 @ M4 ) ) ).
% Suc_le_mono
thf(fact_1128_lessI,axiom,
! [N3: nat] : ( ord_less_nat @ N3 @ ( suc @ N3 ) ) ).
% lessI
thf(fact_1129_Suc__mono,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N3 ) ) ) ).
% Suc_mono
thf(fact_1130_Suc__less__eq,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N3 ) )
= ( ord_less_nat @ M4 @ N3 ) ) ).
% Suc_less_eq
thf(fact_1131_min__Suc__Suc,axiom,
! [M4: nat,N3: nat] :
( ( ord_min_nat @ ( suc @ M4 ) @ ( suc @ N3 ) )
= ( suc @ ( ord_min_nat @ M4 @ N3 ) ) ) ).
% min_Suc_Suc
thf(fact_1132_max__Suc__Suc,axiom,
! [M4: nat,N3: nat] :
( ( ord_max_nat @ ( suc @ M4 ) @ ( suc @ N3 ) )
= ( suc @ ( ord_max_nat @ M4 @ N3 ) ) ) ).
% max_Suc_Suc
thf(fact_1133_zero__less__Suc,axiom,
! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N3 ) ) ).
% zero_less_Suc
thf(fact_1134_less__Suc0,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( suc @ zero_zero_nat ) )
= ( N3 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1135_div__by__Suc__0,axiom,
! [M4: nat] :
( ( divide_divide_nat @ M4 @ ( suc @ zero_zero_nat ) )
= M4 ) ).
% div_by_Suc_0
thf(fact_1136_power__Suc__0,axiom,
! [N3: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N3 )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1137_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M4: nat] :
( ( ( power_power_nat @ X @ M4 )
= ( suc @ zero_zero_nat ) )
= ( ( M4 = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1138_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_1139_transitive__stepwise__le,axiom,
! [M4: nat,N3: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z2: nat] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z2 )
=> ( R2 @ X3 @ Z2 ) ) )
=> ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
=> ( R2 @ M4 @ N3 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1140_nat__induct__at__least,axiom,
! [M4: nat,N3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ( P @ M4 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1141_full__nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N3 ) ) ).
% full_nat_induct
thf(fact_1142_not__less__eq__eq,axiom,
! [M4: nat,N3: nat] :
( ( ~ ( ord_less_eq_nat @ M4 @ N3 ) )
= ( ord_less_eq_nat @ ( suc @ N3 ) @ M4 ) ) ).
% not_less_eq_eq
thf(fact_1143_Suc__n__not__le__n,axiom,
! [N3: nat] :
~ ( ord_less_eq_nat @ ( suc @ N3 ) @ N3 ) ).
% Suc_n_not_le_n
thf(fact_1144_le__Suc__eq,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N3 ) )
= ( ( ord_less_eq_nat @ M4 @ N3 )
| ( M4
= ( suc @ N3 ) ) ) ) ).
% le_Suc_eq
thf(fact_1145_Suc__le__D,axiom,
! [N3: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ M6 )
=> ? [M7: nat] :
( M6
= ( suc @ M7 ) ) ) ).
% Suc_le_D
thf(fact_1146_le__SucI,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_nat @ M4 @ ( suc @ N3 ) ) ) ).
% le_SucI
thf(fact_1147_le__SucE,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_eq_nat @ M4 @ N3 )
=> ( M4
= ( suc @ N3 ) ) ) ) ).
% le_SucE
thf(fact_1148_Suc__leD,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
=> ( ord_less_eq_nat @ M4 @ N3 ) ) ).
% Suc_leD
thf(fact_1149_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_1150_Suc__lessD,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M4 ) @ N3 )
=> ( ord_less_nat @ M4 @ N3 ) ) ).
% Suc_lessD
thf(fact_1151_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_1152_Suc__lessI,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ( ( suc @ M4 )
!= N3 )
=> ( ord_less_nat @ ( suc @ M4 ) @ N3 ) ) ) ).
% Suc_lessI
thf(fact_1153_less__SucE,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_nat @ M4 @ N3 )
=> ( M4 = N3 ) ) ) ).
% less_SucE
thf(fact_1154_less__SucI,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ M4 @ ( suc @ N3 ) ) ) ).
% less_SucI
thf(fact_1155_Ex__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
& ( P @ I3 ) ) )
= ( ( P @ N3 )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N3 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1156_less__Suc__eq,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N3 ) )
= ( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ).
% less_Suc_eq
thf(fact_1157_not__less__eq,axiom,
! [M4: nat,N3: nat] :
( ( ~ ( ord_less_nat @ M4 @ N3 ) )
= ( ord_less_nat @ N3 @ ( suc @ M4 ) ) ) ).
% not_less_eq
thf(fact_1158_All__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N3 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N3 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_1159_Suc__less__eq2,axiom,
! [N3: nat,M4: nat] :
( ( ord_less_nat @ ( suc @ N3 ) @ M4 )
= ( ? [M8: nat] :
( ( M4
= ( suc @ M8 ) )
& ( ord_less_nat @ N3 @ M8 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1160_less__antisym,axiom,
! [N3: nat,M4: nat] :
( ~ ( ord_less_nat @ N3 @ M4 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M4 ) )
=> ( M4 = N3 ) ) ) ).
% less_antisym
thf(fact_1161_Suc__less__SucD,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N3 ) )
=> ( ord_less_nat @ M4 @ N3 ) ) ).
% Suc_less_SucD
thf(fact_1162_less__trans__Suc,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1163_less__Suc__induct,axiom,
! [I: nat,J3: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J3 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K3 )
=> ( ( P @ I2 @ J )
=> ( ( P @ J @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J3 ) ) ) ) ).
% less_Suc_induct
thf(fact_1164_strict__inc__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J3 )
=> ( ! [I2: nat] :
( ( J3
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1165_not__less__less__Suc__eq,axiom,
! [N3: nat,M4: nat] :
( ~ ( ord_less_nat @ N3 @ M4 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M4 ) )
= ( N3 = M4 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1166_n__not__Suc__n,axiom,
! [N3: nat] :
( N3
!= ( suc @ N3 ) ) ).
% n_not_Suc_n
thf(fact_1167_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1168_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1169_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1170_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1171_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1172_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1173_nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N3 ) ) ) ).
% nat_induct
thf(fact_1174_diff__induct,axiom,
! [P: nat > nat > $o,M4: nat,N3: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M4 @ N3 ) ) ) ) ).
% diff_induct
thf(fact_1175_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1176_Suc__neq__Zero,axiom,
! [M4: nat] :
( ( suc @ M4 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1177_Zero__neq__Suc,axiom,
! [M4: nat] :
( zero_zero_nat
!= ( suc @ M4 ) ) ).
% Zero_neq_Suc
thf(fact_1178_Zero__not__Suc,axiom,
! [M4: nat] :
( zero_zero_nat
!= ( suc @ M4 ) ) ).
% Zero_not_Suc
thf(fact_1179_not0__implies__Suc,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ? [M7: nat] :
( N3
= ( suc @ M7 ) ) ) ).
% not0_implies_Suc
thf(fact_1180_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N4: nat] :
( ~ ( P @ N4 )
& ( P @ ( suc @ N4 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1181_less__Suc__eq__0__disj,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N3 ) )
= ( ( M4 = zero_zero_nat )
| ? [J2: nat] :
( ( M4
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N3 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1182_gr0__implies__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ? [M7: nat] :
( N3
= ( suc @ M7 ) ) ) ).
% gr0_implies_Suc
thf(fact_1183_All__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N3 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1184_gr0__conv__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( ? [M2: nat] :
( N3
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1185_Ex__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N3 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1186_Suc__leI,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 ) ) ).
% Suc_leI
thf(fact_1187_Suc__le__eq,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
= ( ord_less_nat @ M4 @ N3 ) ) ).
% Suc_le_eq
thf(fact_1188_dec__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( P @ I )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J3 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J3 ) ) ) ) ).
% dec_induct
thf(fact_1189_inc__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( P @ J3 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J3 )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1190_Suc__le__lessD,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
=> ( ord_less_nat @ M4 @ N3 ) ) ).
% Suc_le_lessD
thf(fact_1191_le__less__Suc__eq,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M4 ) )
= ( N3 = M4 ) ) ) ).
% le_less_Suc_eq
thf(fact_1192_less__Suc__eq__le,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N3 ) )
= ( ord_less_eq_nat @ M4 @ N3 ) ) ).
% less_Suc_eq_le
thf(fact_1193_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1194_le__imp__less__Suc,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_nat @ M4 @ ( suc @ N3 ) ) ) ).
% le_imp_less_Suc
thf(fact_1195_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1196_Suc__div__le__mono,axiom,
! [M4: nat,N3: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M4 @ N3 ) @ ( divide_divide_nat @ ( suc @ M4 ) @ N3 ) ) ).
% Suc_div_le_mono
thf(fact_1197_realpow__pos__nth2,axiom,
! [A: real,N3: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ ( suc @ N3 ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1198_ex__least__nat__less,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N3 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1199_nat__induct__non__zero,axiom,
! [N3: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1200_nat__one__le__power,axiom,
! [I: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N3 ) ) ) ).
% nat_one_le_power
thf(fact_1201_power__gt__expt,axiom,
! [N3: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
=> ( ord_less_nat @ K @ ( power_power_nat @ N3 @ K ) ) ) ).
% power_gt_expt
thf(fact_1202_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D: real,E: real] :
( ( ord_less_real @ D @ E )
=> ( ( P @ D )
=> ( P @ E ) ) )
=> ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_1203_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1204_mult__cancel2,axiom,
! [M4: nat,K: nat,N3: nat] :
( ( ( times_times_nat @ M4 @ K )
= ( times_times_nat @ N3 @ K ) )
= ( ( M4 = N3 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1205_mult__cancel1,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ( times_times_nat @ K @ M4 )
= ( times_times_nat @ K @ N3 ) )
= ( ( M4 = N3 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1206_mult__0__right,axiom,
! [M4: nat] :
( ( times_times_nat @ M4 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1207_mult__is__0,axiom,
! [M4: nat,N3: nat] :
( ( ( times_times_nat @ M4 @ N3 )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
| ( N3 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1208_nat__1__eq__mult__iff,axiom,
! [M4: nat,N3: nat] :
( ( one_one_nat
= ( times_times_nat @ M4 @ N3 ) )
= ( ( M4 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1209_nat__mult__eq__1__iff,axiom,
! [M4: nat,N3: nat] :
( ( ( times_times_nat @ M4 @ N3 )
= one_one_nat )
= ( ( M4 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1210_one__eq__mult__iff,axiom,
! [M4: nat,N3: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M4 @ N3 ) )
= ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1211_mult__eq__1__iff,axiom,
! [M4: nat,N3: nat] :
( ( ( times_times_nat @ M4 @ N3 )
= ( suc @ zero_zero_nat ) )
= ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1212_mult__less__cancel2,axiom,
! [M4: nat,K: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N3 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M4 @ N3 ) ) ) ).
% mult_less_cancel2
thf(fact_1213_nat__0__less__mult__iff,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M4 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M4 )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1214_one__le__mult__iff,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N3 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M4 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N3 ) ) ) ).
% one_le_mult_iff
thf(fact_1215_mult__le__cancel2,axiom,
! [M4: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N3 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M4 @ N3 ) ) ) ).
% mult_le_cancel2
thf(fact_1216_div__mult__self__is__m,axiom,
! [N3: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M4 @ N3 ) @ N3 )
= M4 ) ) ).
% div_mult_self_is_m
thf(fact_1217_div__mult__self1__is__m,axiom,
! [N3: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N3 @ M4 ) @ N3 )
= M4 ) ) ).
% div_mult_self1_is_m
thf(fact_1218_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_1219_nat__mult__1__right,axiom,
! [N3: nat] :
( ( times_times_nat @ N3 @ one_one_nat )
= N3 ) ).
% nat_mult_1_right
thf(fact_1220_nat__mult__1,axiom,
! [N3: nat] :
( ( times_times_nat @ one_one_nat @ N3 )
= N3 ) ).
% nat_mult_1
thf(fact_1221_mult__le__mono2,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J3 ) ) ) ).
% mult_le_mono2
thf(fact_1222_mult__le__mono1,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J3 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1223_mult__le__mono,axiom,
! [I: nat,J3: nat,K: nat,L3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_eq_nat @ K @ L3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J3 @ L3 ) ) ) ) ).
% mult_le_mono
thf(fact_1224_le__square,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% le_square
thf(fact_1225_le__cube,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% le_cube
thf(fact_1226_Suc__mult__cancel1,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M4 )
= ( times_times_nat @ ( suc @ K ) @ N3 ) )
= ( M4 = N3 ) ) ).
% Suc_mult_cancel1
thf(fact_1227_times__int__code_I2_J,axiom,
! [L3: int] :
( ( times_times_int @ zero_zero_int @ L3 )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1228_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1229_mult__0,axiom,
! [N3: nat] :
( ( times_times_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1230_nat__mult__max__right,axiom,
! [M4: nat,N3: nat,Q3: nat] :
( ( times_times_nat @ M4 @ ( ord_max_nat @ N3 @ Q3 ) )
= ( ord_max_nat @ ( times_times_nat @ M4 @ N3 ) @ ( times_times_nat @ M4 @ Q3 ) ) ) ).
% nat_mult_max_right
thf(fact_1231_nat__mult__max__left,axiom,
! [M4: nat,N3: nat,Q3: nat] :
( ( times_times_nat @ ( ord_max_nat @ M4 @ N3 ) @ Q3 )
= ( ord_max_nat @ ( times_times_nat @ M4 @ Q3 ) @ ( times_times_nat @ N3 @ Q3 ) ) ) ).
% nat_mult_max_left
thf(fact_1232_nat__mult__min__right,axiom,
! [M4: nat,N3: nat,Q3: nat] :
( ( times_times_nat @ M4 @ ( ord_min_nat @ N3 @ Q3 ) )
= ( ord_min_nat @ ( times_times_nat @ M4 @ N3 ) @ ( times_times_nat @ M4 @ Q3 ) ) ) ).
% nat_mult_min_right
thf(fact_1233_nat__mult__min__left,axiom,
! [M4: nat,N3: nat,Q3: nat] :
( ( times_times_nat @ ( ord_min_nat @ M4 @ N3 ) @ Q3 )
= ( ord_min_nat @ ( times_times_nat @ M4 @ Q3 ) @ ( times_times_nat @ N3 @ Q3 ) ) ) ).
% nat_mult_min_left
thf(fact_1234_mult__less__mono1,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J3 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1235_mult__less__mono2,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J3 ) ) ) ) ).
% mult_less_mono2
thf(fact_1236_Suc__mult__le__cancel1,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M4 ) @ ( times_times_nat @ ( suc @ K ) @ N3 ) )
= ( ord_less_eq_nat @ M4 @ N3 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1237_Suc__mult__less__cancel1,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M4 ) @ ( times_times_nat @ ( suc @ K ) @ N3 ) )
= ( ord_less_nat @ M4 @ N3 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1238_mult__eq__self__implies__10,axiom,
! [M4: nat,N3: nat] :
( ( M4
= ( times_times_nat @ M4 @ N3 ) )
=> ( ( N3 = one_one_nat )
| ( M4 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1239_zmult__zless__mono2,axiom,
! [I: int,J3: int,K: int] :
( ( ord_less_int @ I @ J3 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J3 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1240_div__times__less__eq__dividend,axiom,
! [M4: nat,N3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M4 @ N3 ) @ N3 ) @ M4 ) ).
% div_times_less_eq_dividend
thf(fact_1241_times__div__less__eq__dividend,axiom,
! [N3: nat,M4: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N3 @ ( divide_divide_nat @ M4 @ N3 ) ) @ M4 ) ).
% times_div_less_eq_dividend
thf(fact_1242_less__mult__imp__div__less,axiom,
! [M4: nat,I: nat,N3: nat] :
( ( ord_less_nat @ M4 @ ( times_times_nat @ I @ N3 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M4 @ N3 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1243_divide__real__def,axiom,
( divide_divide_real
= ( ^ [X2: real,Y5: real] : ( times_times_real @ X2 @ ( inverse_inverse_real @ Y5 ) ) ) ) ).
% divide_real_def
thf(fact_1244_n__less__n__mult__m,axiom,
! [N3: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
=> ( ord_less_nat @ N3 @ ( times_times_nat @ N3 @ M4 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1245_n__less__m__mult__n,axiom,
! [N3: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
=> ( ord_less_nat @ N3 @ ( times_times_nat @ M4 @ N3 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1246_one__less__mult,axiom,
! [N3: nat,M4: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N3 ) ) ) ) ).
% one_less_mult
thf(fact_1247_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y3: real] :
? [N4: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1248_div__less__iff__less__mult,axiom,
! [Q3: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M4 @ Q3 ) @ N3 )
= ( ord_less_nat @ M4 @ ( times_times_nat @ N3 @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1249_pos__zmult__eq__1__iff,axiom,
! [M4: int,N3: int] :
( ( ord_less_int @ zero_zero_int @ M4 )
=> ( ( ( times_times_int @ M4 @ N3 )
= one_one_int )
= ( ( M4 = one_one_int )
& ( N3 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1250_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_1251_zmult__zless__mono2__lemma,axiom,
! [I: int,J3: int,K: nat] :
( ( ord_less_int @ I @ J3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J3 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1252_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M4 @ ( divide_divide_nat @ N3 @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M4 @ Q3 ) @ N3 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1253_div__nat__eqI,axiom,
! [N3: nat,Q3: nat,M4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N3 @ Q3 ) @ M4 )
=> ( ( ord_less_nat @ M4 @ ( times_times_nat @ N3 @ ( suc @ Q3 ) ) )
=> ( ( divide_divide_nat @ M4 @ N3 )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_1254_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M7: nat] :
( ( ord_less_nat @ zero_zero_nat @ M7 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M7 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1255_split__div_H,axiom,
! [P: nat > $o,M4: nat,N3: nat] :
( ( P @ ( divide_divide_nat @ M4 @ N3 ) )
= ( ( ( N3 = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N3 @ Q4 ) @ M4 )
& ( ord_less_nat @ M4 @ ( times_times_nat @ N3 @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1256_nat__mult__le__cancel__disj,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M4 @ N3 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1257_nat__mult__less__cancel__disj,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M4 @ N3 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1258_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ( times_times_nat @ K @ M4 )
= ( times_times_nat @ K @ N3 ) )
= ( ( K = zero_zero_nat )
| ( M4 = N3 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1259_nat__mult__less__cancel1,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N3 ) )
= ( ord_less_nat @ M4 @ N3 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1260_nat__mult__eq__cancel1,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M4 )
= ( times_times_nat @ K @ N3 ) )
= ( M4 = N3 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1261_nat__mult__div__cancel__disj,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N3 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N3 ) )
= ( divide_divide_nat @ M4 @ N3 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1262_nat__mult__le__cancel1,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N3 ) )
= ( ord_less_eq_nat @ M4 @ N3 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1263_nat__mult__div__cancel1,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N3 ) )
= ( divide_divide_nat @ M4 @ N3 ) ) ) ).
% nat_mult_div_cancel1
% Helper facts (9)
thf(help_If_2_1_If_001tf__b_T,axiom,
! [X: b,Y: b] :
( ( if_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__b_T,axiom,
! [X: b,Y: b] :
( ( if_b @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (4)
thf(conj_0,hypothesis,
$true ).
thf(conj_1,hypothesis,
k != i ).
thf(conj_2,hypothesis,
k != ja ).
thf(conj_3,conjecture,
( member_a_b
@ ^ [Omega: a] : ( if_b @ ( k = i ) @ ( ord_min_b @ ( g @ i @ Omega ) @ ( g @ ja @ Omega ) ) @ ( if_b @ ( k = ja ) @ ( ord_max_b @ ( g @ i @ Omega ) @ ( g @ ja @ Omega ) ) @ ( g @ k @ Omega ) ) )
@ ( sigma_measurable_a_b @ m @ borel_5459123734250506525orel_b ) ) ).
%------------------------------------------------------------------------------