%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP097^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n010.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:31 AM UTC 2026
% Result : Theorem 2.70s 1.70s
% Output : Refutation 2.70s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 6
% Syntax : Number of formulae : 29 ( 23 unt; 0 typ; 0 def)
% Number of atoms : 69 ( 25 equ; 0 cnn)
% Maximal formula atoms : 2 ( 2 avg)
% Number of connectives : 389 ( 8 ~; 3 |; 0 &; 375 @)
% ( 0 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 42 ( 39 usr; 11 con; 0-3 aty)
% Number of variables : 13 ( 0 ^; 13 !; 0 ?; 13 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
real: $tType ).
thf(type_def_6,type,
int: $tType ).
thf(type_def_8,type,
nat: $tType ).
thf(type_def_9,type,
num: $tType ).
thf(type_def_10,type,
set_real: $tType ).
thf(func_def_0,type,
set_real: $tType ).
thf(func_def_1,type,
real: $tType ).
thf(func_def_2,type,
num: $tType ).
thf(func_def_3,type,
nat: $tType ).
thf(func_def_4,type,
int: $tType ).
thf(func_def_5,type,
archim1371465213g_real: real > int ).
thf(func_def_6,type,
one_one_int: int ).
thf(func_def_7,type,
one_one_nat: nat ).
thf(func_def_8,type,
one_one_real: real ).
thf(func_def_9,type,
zero_zero_int: int ).
thf(func_def_10,type,
zero_zero_nat: nat ).
thf(func_def_11,type,
zero_zero_real: real ).
thf(func_def_12,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_13,type,
nat2: int > nat ).
thf(func_def_14,type,
ring_1_of_int_int: int > int ).
thf(func_def_15,type,
ring_1_of_int_real: int > real ).
thf(func_def_16,type,
suc: nat > nat ).
thf(func_def_17,type,
semiri2019852685at_int: nat > int ).
thf(func_def_18,type,
semiri1382578993at_nat: nat > nat ).
thf(func_def_19,type,
semiri2110766477t_real: nat > real ).
thf(func_def_20,type,
neg_numeral_dbl_int: int > int ).
thf(func_def_21,type,
neg_numeral_dbl_real: real > real ).
thf(func_def_22,type,
bit0: num > num ).
thf(func_def_23,type,
one: num ).
thf(func_def_24,type,
numeral_numeral_int: num > int ).
thf(func_def_25,type,
numeral_numeral_nat: num > nat ).
thf(func_def_26,type,
numeral_numeral_real: num > real ).
thf(func_def_27,type,
pow: num > num > num ).
thf(func_def_28,type,
ord_less_int: int > int > $o ).
thf(func_def_29,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_30,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_31,type,
ord_less_eq_num: num > num > $o ).
thf(func_def_32,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_33,type,
power_power_int: int > nat > int ).
thf(func_def_34,type,
power_power_nat: nat > nat > nat ).
thf(func_def_35,type,
power_power_real: real > nat > real ).
thf(func_def_36,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_37,type,
log: real > real > real ).
thf(func_def_38,type,
powr_real: real > real > real ).
thf(func_def_39,type,
member_real: real > set_real > $o ).
thf(func_def_40,type,
d: int ).
thf(func_def_41,type,
n: nat ).
thf(func_def_53,type,
sK0: real > int ).
thf(func_def_54,type,
sK1: real > int ).
thf(f1411,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1410,f888]) ).
thf(f888,plain,
( ( ord_less_eq_int @ d @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f738]) ).
thf(f738,plain,
( ( ord_less_eq_int @ d @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) )
!= $true ),
inference(flattening,[status(thm)],[f609]) ).
thf(f609,plain,
( ( ord_less_eq_int @ d @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f608]) ).
thf(f608,plain,
~ ( ord_less_eq_int @ d @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ),
inference(rectify,[status(thm)],[f355]) ).
thf(f355,negated_conjecture,
~ ( ord_less_eq_int @ d @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f354]) ).
thf(f354,conjecture,
ord_less_eq_int @ d @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Z5U7N5EVnJ/DTF2THF_17924.p',conj_0) ).
thf(f1410,plain,
( ( ord_less_eq_int @ d @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) )
= $true ),
inference(backward_demodulation,[status(thm)],[f1376,f1399]) ).
thf(f1399,plain,
( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) )
= ( archim1371465213g_real @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) ) ),
inference(superposition,[status(thm)],[f864,f855]) ).
thf(f855,plain,
( ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) )
= ( ring_1_of_int_real @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f103]) ).
thf(f103,axiom,
( ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) )
= ( ring_1_of_int_real @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Z5U7N5EVnJ/DTF2THF_17924.p',fact_102__092_060open_0622_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_A_061_Areal__of__int_A_I2_A_094_Anat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062) ).
thf(f864,plain,
! [X0: int] :
( ( archim1371465213g_real @ ( ring_1_of_int_real @ X0 ) )
= X0 ),
inference(cnf_transformation,[status(thm)],[f742]) ).
thf(f742,plain,
! [X0: int] :
( ( archim1371465213g_real @ ( ring_1_of_int_real @ X0 ) )
= X0 ),
inference(rectify,[status(thm)],[f26]) ).
thf(f26,axiom,
! [X7: int] :
( ( archim1371465213g_real @ ( ring_1_of_int_real @ X7 ) )
= X7 ),
file('/export/starexec/sandbox/tmp/tmp.Z5U7N5EVnJ/DTF2THF_17924.p',fact_25_ceiling__of__int) ).
thf(f1376,plain,
( $true
= ( ord_less_eq_int @ d @ ( archim1371465213g_real @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f1375,f864]) ).
thf(f1375,plain,
( $true
= ( ord_less_eq_int @ ( archim1371465213g_real @ ( ring_1_of_int_real @ d ) ) @ ( archim1371465213g_real @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1374]) ).
thf(f1374,plain,
( ( $true != $true )
| ( $true
= ( ord_less_eq_int @ ( archim1371465213g_real @ ( ring_1_of_int_real @ d ) ) @ ( archim1371465213g_real @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) ) ) ) ),
inference(superposition,[status(thm)],[f875,f1019]) ).
thf(f1019,plain,
( $true
= ( ord_less_eq_real @ ( ring_1_of_int_real @ d ) @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) ) ),
inference(backward_demodulation,[status(thm)],[f869,f878]) ).
thf(f878,plain,
( ( ring_1_of_int_real @ d )
= ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ),
inference(cnf_transformation,[status(thm)],[f54]) ).
thf(f54,axiom,
( ( ring_1_of_int_real @ d )
= ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Z5U7N5EVnJ/DTF2THF_17924.p',fact_53__092_060open_062real__of__int_Ad_A_061_A2_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_092_060close_062) ).
thf(f869,plain,
( ( ord_less_eq_real @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f534]) ).
thf(f534,plain,
( ( ord_less_eq_real @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f533]) ).
thf(f533,plain,
ord_less_eq_real @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ),
inference(rectify,[status(thm)],[f104]) ).
thf(f104,axiom,
ord_less_eq_real @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Z5U7N5EVnJ/DTF2THF_17924.p',fact_103__092_060open_0622_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_A_092_060le_062_A2_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062) ).
thf(f875,plain,
! [X0: real,X1: real] :
( ( $true
!= ( ord_less_eq_real @ X0 @ X1 ) )
| ( $true
= ( ord_less_eq_int @ ( archim1371465213g_real @ X0 ) @ ( archim1371465213g_real @ X1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f783]) ).
thf(f783,plain,
! [X0: real,X1: real] :
( ( $true
= ( ord_less_eq_int @ ( archim1371465213g_real @ X0 ) @ ( archim1371465213g_real @ X1 ) ) )
| ( $true
!= ( ord_less_eq_real @ X0 @ X1 ) ) ),
inference(ennf_transformation,[status(thm)],[f392]) ).
thf(f392,plain,
! [X0: real,X1: real] :
( ( $true
= ( ord_less_eq_real @ X0 @ X1 ) )
=> ( $true
= ( ord_less_eq_int @ ( archim1371465213g_real @ X0 ) @ ( archim1371465213g_real @ X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f391]) ).
thf(f391,plain,
! [X0: real,X1: real] :
( ( ord_less_eq_real @ X0 @ X1 )
=> ( ord_less_eq_int @ ( archim1371465213g_real @ X0 ) @ ( archim1371465213g_real @ X1 ) ) ),
inference(rectify,[status(thm)],[f44]) ).
thf(f44,axiom,
! [X3: real,X0: real] :
( ( ord_less_eq_real @ X3 @ X0 )
=> ( ord_less_eq_int @ ( archim1371465213g_real @ X3 ) @ ( archim1371465213g_real @ X0 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Z5U7N5EVnJ/DTF2THF_17924.p',fact_43_ceiling__mono) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP097^1 : TPTP v9.2.1. Released v7.5.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n010.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:26:03 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : ITP097^1 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.46 % Domain : Interactive Theorem Proving
% 0.30/0.46 % Problem : Sledgehammer Liouville_Numbers problem prob_81__5865280_1
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Des21]
% 0.30/0.46 % Names : Liouville_Numbers/prob_81__5865280_1 [Des21]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.30/0.46 % Syntax : Number of formulae : 396 ( 279 unt; 42 typ; 0 def)
% 0.30/0.46 % Number of atoms : 778 ( 477 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 8 ( 2 avg)
% 0.30/0.46 % Number of connectives : 1870 ( 48 ~; 14 |; 13 &;1664 @)
% 0.30/0.46 % ( 0 <=>; 131 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 14 ( 4 avg)
% 0.30/0.46 % Number of types : 6 ( 5 usr)
% 0.30/0.46 % Number of type conns : 67 ( 67 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 40 ( 37 usr; 11 con; 0-3 aty)
% 0.30/0.46 % Number of variables : 590 ( 12 ^; 564 !; 14 ?; 590 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:39:12.418
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % Could-be-implicit typings (5)
% 0.30/0.46 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.30/0.46 set_real: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Real__Oreal,type,
% 0.30/0.46 real: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Num__Onum,type,
% 0.30/0.46 num: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Nat__Onat,type,
% 0.30/0.46 nat: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Int__Oint,type,
% 0.30/0.46 int: $tType ).
% 0.30/0.46
% 0.30/0.46 % Explicit typings (37)
% 0.30/0.46 thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
% 0.30/0.46 archim1371465213g_real: real > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
% 0.30/0.46 one_one_int: int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.30/0.46 one_one_nat: nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 0.30/0.46 one_one_real: real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 0.30/0.46 zero_zero_int: int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.30/0.46 zero_zero_nat: nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.30/0.46 zero_zero_real: real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_If_001t__Nat__Onat,type,
% 0.30/0.46 if_nat: $o > nat > nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Int_Onat,type,
% 0.30/0.46 nat2: int > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 0.30/0.46 ring_1_of_int_int: int > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 0.30/0.46 ring_1_of_int_real: int > real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Nat_OSuc,type,
% 0.30/0.46 suc: nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.30/0.46 semiri2019852685at_int: nat > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.30/0.46 semiri1382578993at_nat: nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 0.30/0.46 semiri2110766477t_real: nat > real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 0.30/0.46 neg_numeral_dbl_int: int > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 0.30/0.46 neg_numeral_dbl_real: real > real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Num_Onum_OBit0,type,
% 0.30/0.46 bit0: num > num ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Num_Onum_OOne,type,
% 0.30/0.46 one: num ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 0.30/0.46 numeral_numeral_int: num > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 0.30/0.46 numeral_numeral_nat: num > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 0.30/0.46 numeral_numeral_real: num > real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Num_Opow,type,
% 0.30/0.46 pow: num > num > num ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.30/0.46 ord_less_int: int > int > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.30/0.46 ord_less_eq_int: int > int > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.30/0.46 ord_less_eq_nat: nat > nat > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
% 0.30/0.46 ord_less_eq_num: num > num > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.30/0.46 ord_less_eq_real: real > real > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 0.30/0.46 power_power_int: int > nat > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 0.30/0.46 power_power_nat: nat > nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
% 0.30/0.46 power_power_real: real > nat > real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.30/0.46 collect_real: ( real > $o ) > set_real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Transcendental_Olog,type,
% 0.30/0.46 log: real > real > real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 0.30/0.46 powr_real: real > real > real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_member_001t__Real__Oreal,type,
% 0.30/0.46 member_real: real > set_real > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_d____,type,
% 0.30/0.46 d: int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_n____,type,
% 0.30/0.46 n: nat ).
% 0.30/0.46
% 0.30/0.46 % Relevant facts (350)
% 0.30/0.46 thf(fact_0_numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.46 ! [X: num,N: nat,A: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.30/0.46 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % numeral_power_le_of_int_cancel_iff
% 0.30/0.46 thf(fact_1_numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.46 ! [X: num,N: nat,A: int] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.30/0.46 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % numeral_power_le_of_int_cancel_iff
% 0.30/0.46 thf(fact_2_of__int__le__numeral__power__cancel__iff,axiom,
% 0.30/0.46 ! [A: int,X: num,N: nat] :
% 0.30/0.46 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.30/0.46 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_le_numeral_power_cancel_iff
% 0.30/0.46 thf(fact_3_of__int__le__numeral__power__cancel__iff,axiom,
% 0.30/0.46 ! [A: int,X: num,N: nat] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.30/0.46 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_le_numeral_power_cancel_iff
% 0.30/0.46 thf(fact_4_ceiling__numeral__power,axiom,
% 0.30/0.46 ! [X: num,N: nat] :
% 0.30/0.46 ( ( archim1371465213g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.30/0.46 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % ceiling_numeral_power
% 0.30/0.46 thf(fact_5_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.46 ! [X: num,N: nat,Y: int] :
% 0.30/0.46 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.30/0.46 = ( ring_1_of_int_int @ Y ) )
% 0.30/0.46 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.30/0.46 = Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % numeral_power_eq_of_int_cancel_iff
% 0.30/0.46 thf(fact_6_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.46 ! [X: num,N: nat,Y: int] :
% 0.30/0.46 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 0.30/0.46 = ( ring_1_of_int_real @ Y ) )
% 0.30/0.46 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.30/0.46 = Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % numeral_power_eq_of_int_cancel_iff
% 0.30/0.46 thf(fact_7_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.46 ! [Y: int,X: num,N: nat] :
% 0.30/0.46 ( ( ( ring_1_of_int_int @ Y )
% 0.30/0.46 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.30/0.46 = ( Y
% 0.30/0.46 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_eq_numeral_power_cancel_iff
% 0.30/0.46 thf(fact_8_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.46 ! [Y: int,X: num,N: nat] :
% 0.30/0.46 ( ( ( ring_1_of_int_real @ Y )
% 0.30/0.46 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.30/0.46 = ( Y
% 0.30/0.46 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_eq_numeral_power_cancel_iff
% 0.30/0.46 thf(fact_9_of__int__le__of__int__power__cancel__iff,axiom,
% 0.30/0.46 ! [B: int,W: nat,X: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.30/0.46 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_le_of_int_power_cancel_iff
% 0.30/0.46 thf(fact_10_of__int__le__of__int__power__cancel__iff,axiom,
% 0.30/0.46 ! [B: int,W: nat,X: int] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.30/0.46 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_le_of_int_power_cancel_iff
% 0.30/0.46 thf(fact_11_of__int__power__le__of__int__cancel__iff,axiom,
% 0.30/0.46 ! [X: int,B: int,W: nat] :
% 0.30/0.46 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.30/0.46 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_power_le_of_int_cancel_iff
% 0.30/0.46 thf(fact_12_of__int__power__le__of__int__cancel__iff,axiom,
% 0.30/0.46 ! [X: int,B: int,W: nat] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.30/0.46 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_power_le_of_int_cancel_iff
% 0.30/0.46 thf(fact_13_ceiling__le__numeral,axiom,
% 0.30/0.46 ! [X: real,V: num] :
% 0.30/0.46 ( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.46 = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ceiling_le_numeral
% 0.30/0.46 thf(fact_14_of__int__le__numeral__iff,axiom,
% 0.30/0.46 ! [Z: int,N: num] :
% 0.30/0.46 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.46 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_le_numeral_iff
% 0.30/0.46 thf(fact_15_of__int__le__numeral__iff,axiom,
% 0.30/0.46 ! [Z: int,N: num] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.46 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_le_numeral_iff
% 0.30/0.46 thf(fact_16_of__int__numeral__le__iff,axiom,
% 0.30/0.46 ! [N: num,Z: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.46 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_numeral_le_iff
% 0.30/0.46 thf(fact_17_of__int__numeral__le__iff,axiom,
% 0.30/0.46 ! [N: num,Z: int] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.46 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_numeral_le_iff
% 0.30/0.46 thf(fact_18_of__int__power,axiom,
% 0.30/0.46 ! [Z: int,N: nat] :
% 0.30/0.46 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 0.30/0.46 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_power
% 0.30/0.46 thf(fact_19_of__int__power,axiom,
% 0.30/0.46 ! [Z: int,N: nat] :
% 0.30/0.46 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 0.30/0.46 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_power
% 0.30/0.46 thf(fact_20_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.30/0.46 ! [B: int,W: nat,X: int] :
% 0.30/0.46 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 0.30/0.46 = ( ring_1_of_int_int @ X ) )
% 0.30/0.46 = ( ( power_power_int @ B @ W )
% 0.30/0.46 = X ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_eq_of_int_power_cancel_iff
% 0.30/0.46 thf(fact_21_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.30/0.46 ! [B: int,W: nat,X: int] :
% 0.30/0.46 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 0.30/0.46 = ( ring_1_of_int_real @ X ) )
% 0.30/0.46 = ( ( power_power_int @ B @ W )
% 0.30/0.46 = X ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_eq_of_int_power_cancel_iff
% 0.30/0.46 thf(fact_22_of__int__eq__iff,axiom,
% 0.30/0.46 ! [W: int,Z: int] :
% 0.30/0.46 ( ( ( ring_1_of_int_real @ W )
% 0.30/0.46 = ( ring_1_of_int_real @ Z ) )
% 0.30/0.46 = ( W = Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_eq_iff
% 0.30/0.46 thf(fact_23_of__int__eq__iff,axiom,
% 0.30/0.46 ! [W: int,Z: int] :
% 0.30/0.46 ( ( ( ring_1_of_int_int @ W )
% 0.30/0.46 = ( ring_1_of_int_int @ Z ) )
% 0.30/0.46 = ( W = Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_eq_iff
% 0.30/0.46 thf(fact_24_nat__numeral,axiom,
% 0.30/0.46 ! [K: num] :
% 0.30/0.46 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 0.30/0.46 = ( numeral_numeral_nat @ K ) ) ).
% 0.30/0.46
% 0.30/0.46 % nat_numeral
% 0.30/0.46 thf(fact_25_ceiling__of__int,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( archim1371465213g_real @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.46 = Z ) ).
% 0.30/0.46
% 0.30/0.46 % ceiling_of_int
% 0.30/0.46 thf(fact_26_of__int__ceiling__cancel,axiom,
% 0.30/0.46 ! [X: real] :
% 0.30/0.46 ( ( ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) )
% 0.30/0.46 = X )
% 0.30/0.46 = ( ? [N2: int] :
% 0.30/0.46 ( X
% 0.30/0.46 = ( ring_1_of_int_real @ N2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_ceiling_cancel
% 0.30/0.46 thf(fact_27_of__int__le__iff,axiom,
% 0.30/0.46 ! [W: int,Z: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.46 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_le_iff
% 0.30/0.46 thf(fact_28_of__int__le__iff,axiom,
% 0.30/0.46 ! [W: int,Z: int] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.46 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_le_iff
% 0.30/0.46 thf(fact_29_of__int__numeral,axiom,
% 0.30/0.46 ! [K: num] :
% 0.30/0.46 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 0.30/0.46 = ( numeral_numeral_int @ K ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_numeral
% 0.30/0.46 thf(fact_30_of__int__numeral,axiom,
% 0.30/0.46 ! [K: num] :
% 0.30/0.46 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 0.30/0.46 = ( numeral_numeral_real @ K ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_numeral
% 0.30/0.46 thf(fact_31_of__int__eq__numeral__iff,axiom,
% 0.30/0.46 ! [Z: int,N: num] :
% 0.30/0.46 ( ( ( ring_1_of_int_int @ Z )
% 0.30/0.46 = ( numeral_numeral_int @ N ) )
% 0.30/0.46 = ( Z
% 0.30/0.46 = ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_eq_numeral_iff
% 0.30/0.46 thf(fact_32_of__int__eq__numeral__iff,axiom,
% 0.30/0.46 ! [Z: int,N: num] :
% 0.30/0.46 ( ( ( ring_1_of_int_real @ Z )
% 0.30/0.46 = ( numeral_numeral_real @ N ) )
% 0.30/0.46 = ( Z
% 0.30/0.46 = ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_eq_numeral_iff
% 0.30/0.46 thf(fact_33_ceiling__numeral,axiom,
% 0.30/0.46 ! [V: num] :
% 0.30/0.46 ( ( archim1371465213g_real @ ( numeral_numeral_real @ V ) )
% 0.30/0.46 = ( numeral_numeral_int @ V ) ) ).
% 0.30/0.46
% 0.30/0.46 % ceiling_numeral
% 0.30/0.46 thf(fact_34_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.46 ! [X: int,B: int,W: nat] :
% 0.30/0.46 ( ( ( ring_1_of_int_int @ X )
% 0.30/0.46 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.30/0.46 = ( X
% 0.30/0.46 = ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_power_eq_of_int_cancel_iff
% 0.30/0.46 thf(fact_35_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.46 ! [X: int,B: int,W: nat] :
% 0.30/0.46 ( ( ( ring_1_of_int_real @ X )
% 0.30/0.46 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.30/0.46 = ( X
% 0.30/0.46 = ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_power_eq_of_int_cancel_iff
% 0.30/0.46 thf(fact_36_numeral__power__eq__nat__cancel__iff,axiom,
% 0.30/0.46 ! [X: num,N: nat,Y: int] :
% 0.30/0.46 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.30/0.46 = ( nat2 @ Y ) )
% 0.30/0.46 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.30/0.46 = Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % numeral_power_eq_nat_cancel_iff
% 0.30/0.46 thf(fact_37_nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.46 ! [Y: int,X: num,N: nat] :
% 0.30/0.46 ( ( ( nat2 @ Y )
% 0.30/0.46 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.30/0.46 = ( Y
% 0.30/0.46 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nat_eq_numeral_power_cancel_iff
% 0.30/0.46 thf(fact_38_numeral__power__le__nat__cancel__iff,axiom,
% 0.30/0.46 ! [X: num,N: nat,A: int] :
% 0.30/0.46 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 0.30/0.46 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % numeral_power_le_nat_cancel_iff
% 0.30/0.46 thf(fact_39_nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.46 ! [A: int,X: num,N: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.30/0.46 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nat_le_numeral_power_cancel_iff
% 0.30/0.46 thf(fact_40_nat__mono,axiom,
% 0.30/0.46 ! [X: int,Y: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ X @ Y )
% 0.30/0.46 => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nat_mono
% 0.30/0.46 thf(fact_41_ex__le__of__int,axiom,
% 0.30/0.46 ! [X: real] :
% 0.30/0.46 ? [Z2: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % ex_le_of_int
% 0.30/0.46 thf(fact_42_le__of__int__ceiling,axiom,
% 0.30/0.46 ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_of_int_ceiling
% 0.30/0.46 thf(fact_43_ceiling__mono,axiom,
% 0.30/0.46 ! [Y: real,X: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ Y @ X )
% 0.30/0.46 => ( ord_less_eq_int @ ( archim1371465213g_real @ Y ) @ ( archim1371465213g_real @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ceiling_mono
% 0.30/0.46 thf(fact_44_ceiling__le__iff,axiom,
% 0.30/0.46 ! [X: real,Z: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ Z )
% 0.30/0.46 = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ceiling_le_iff
% 0.30/0.46 thf(fact_45_ceiling__le,axiom,
% 0.30/0.46 ! [X: real,A: int] :
% 0.30/0.46 ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
% 0.30/0.46 => ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % ceiling_le
% 0.30/0.46 thf(fact_46__092_060open_062log_A2_A1_A_092_060le_062_Alog_A2_A_Ireal__of__int_Ad_J_092_060close_062,axiom,
% 0.30/0.46 ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ one_one_real ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ).
% 0.30/0.46
% 0.30/0.46 % \<open>log 2 1 \<le> log 2 (real_of_int d)\<close>
% 0.30/0.46 thf(fact_47_semiring__norm_I69_J,axiom,
% 0.30/0.46 ! [M: num] :
% 0.30/0.46 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 0.30/0.46
% 0.30/0.46 % semiring_norm(69)
% 0.30/0.46 thf(fact_48_semiring__norm_I83_J,axiom,
% 0.30/0.46 ! [N: num] :
% 0.30/0.46 ( one
% 0.30/0.46 != ( bit0 @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring_norm(83)
% 0.30/0.46 thf(fact_49_semiring__norm_I85_J,axiom,
% 0.30/0.46 ! [M: num] :
% 0.30/0.46 ( ( bit0 @ M )
% 0.30/0.46 != one ) ).
% 0.30/0.46
% 0.30/0.46 % semiring_norm(85)
% 0.30/0.46 thf(fact_50_numeral__le__iff,axiom,
% 0.30/0.46 ! [M: num,N: num] :
% 0.30/0.46 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.46 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % numeral_le_iff
% 0.30/0.46 thf(fact_51_numeral__le__iff,axiom,
% 0.30/0.46 ! [M: num,N: num] :
% 0.30/0.46 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.46 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % numeral_le_iff
% 0.30/0.46 thf(fact_52_numeral__le__iff,axiom,
% 0.30/0.46 ! [M: num,N: num] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.46 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % numeral_le_iff
% 0.30/0.46 thf(fact_53__092_060open_062real__of__int_Ad_A_061_A2_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_092_060close_062,axiom,
% 0.30/0.46 ( ( ring_1_of_int_real @ d )
% 0.30/0.46 = ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % \<open>real_of_int d = 2 powr log 2 (real_of_int d)\<close>
% 0.30/0.46 thf(fact_54_n__def,axiom,
% 0.30/0.46 ( n
% 0.30/0.46 = ( suc @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % n_def
% 0.30/0.46 thf(fact_55_semiring__norm_I68_J,axiom,
% 0.30/0.46 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 0.30/0.46
% 0.30/0.46 % semiring_norm(68)
% 0.30/0.46 thf(fact_56_semiring__norm_I71_J,axiom,
% 0.30/0.46 ! [M: num,N: num] :
% 0.30/0.46 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.46 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring_norm(71)
% 0.30/0.46 thf(fact_57_semiring__norm_I87_J,axiom,
% 0.30/0.46 ! [M: num,N: num] :
% 0.30/0.46 ( ( ( bit0 @ M )
% 0.30/0.46 = ( bit0 @ N ) )
% 0.30/0.46 = ( M = N ) ) ).
% 0.30/0.46
% 0.30/0.46 % semiring_norm(87)
% 0.30/0.46 thf(fact_58_verit__eq__simplify_I8_J,axiom,
% 0.30/0.46 ! [X2: num,Y2: num] :
% 0.30/0.46 ( ( ( bit0 @ X2 )
% 0.30/0.46 = ( bit0 @ Y2 ) )
% 0.30/0.46 = ( X2 = Y2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % verit_eq_simplify(8)
% 0.30/0.46 thf(fact_59_numeral__eq__iff,axiom,
% 0.30/0.46 ! [M: num,N: num] :
% 0.30/0.46 ( ( ( numeral_numeral_int @ M )
% 0.30/0.46 = ( numeral_numeral_int @ N ) )
% 0.30/0.46 = ( M = N ) ) ).
% 0.30/0.46
% 0.30/0.46 % numeral_eq_iff
% 0.30/0.46 thf(fact_60_numeral__eq__iff,axiom,
% 0.30/0.46 ! [M: num,N: num] :
% 0.30/0.46 ( ( ( numeral_numeral_real @ M )
% 0.30/0.46 = ( numeral_numeral_real @ N ) )
% 0.30/0.46 = ( M = N ) ) ).
% 0.30/0.46
% 0.30/0.46 % numeral_eq_iff
% 0.30/0.46 thf(fact_61_numeral__eq__iff,axiom,
% 0.30/0.46 ! [M: num,N: num] :
% 0.30/0.46 ( ( ( numeral_numeral_nat @ M )
% 0.30/0.46 = ( numeral_numeral_nat @ N ) )
% 0.30/0.46 = ( M = N ) ) ).
% 0.30/0.46
% 0.30/0.46 % numeral_eq_iff
% 0.30/0.46 thf(fact_62_of__int__eq__1__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ( ring_1_of_int_real @ Z )
% 0.30/0.46 = one_one_real )
% 0.30/0.46 = ( Z = one_one_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_eq_1_iff
% 0.30/0.46 thf(fact_63_of__int__eq__1__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ( ring_1_of_int_int @ Z )
% 0.30/0.46 = one_one_int )
% 0.30/0.46 = ( Z = one_one_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_eq_1_iff
% 0.30/0.46 thf(fact_64_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: real,P: real > $o] :
% 0.30/0.46 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.30/0.46 = ( P @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_65_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A2: set_real] :
% 0.30/0.46 ( ( collect_real
% 0.30/0.46 @ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
% 0.30/0.46 = A2 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_66_of__int__1,axiom,
% 0.30/0.46 ( ( ring_1_of_int_real @ one_one_int )
% 0.30/0.46 = one_one_real ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_1
% 0.30/0.46 thf(fact_67_of__int__1,axiom,
% 0.30/0.46 ( ( ring_1_of_int_int @ one_one_int )
% 0.30/0.46 = one_one_int ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_1
% 0.30/0.46 thf(fact_68_ceiling__one,axiom,
% 0.30/0.46 ( ( archim1371465213g_real @ one_one_real )
% 0.30/0.46 = one_one_int ) ).
% 0.30/0.47
% 0.30/0.47 % ceiling_one
% 0.30/0.47 thf(fact_69_numeral__eq__one__iff,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( ( ( numeral_numeral_int @ N )
% 0.30/0.47 = one_one_int )
% 0.30/0.47 = ( N = one ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_eq_one_iff
% 0.30/0.47 thf(fact_70_numeral__eq__one__iff,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( ( ( numeral_numeral_real @ N )
% 0.30/0.47 = one_one_real )
% 0.30/0.47 = ( N = one ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_eq_one_iff
% 0.30/0.47 thf(fact_71_numeral__eq__one__iff,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( ( ( numeral_numeral_nat @ N )
% 0.30/0.47 = one_one_nat )
% 0.30/0.47 = ( N = one ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_eq_one_iff
% 0.30/0.47 thf(fact_72_one__eq__numeral__iff,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( ( one_one_int
% 0.30/0.47 = ( numeral_numeral_int @ N ) )
% 0.30/0.47 = ( one = N ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_eq_numeral_iff
% 0.30/0.47 thf(fact_73_one__eq__numeral__iff,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( ( one_one_real
% 0.30/0.47 = ( numeral_numeral_real @ N ) )
% 0.30/0.47 = ( one = N ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_eq_numeral_iff
% 0.30/0.47 thf(fact_74_one__eq__numeral__iff,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( ( one_one_nat
% 0.30/0.47 = ( numeral_numeral_nat @ N ) )
% 0.30/0.47 = ( one = N ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_eq_numeral_iff
% 0.30/0.47 thf(fact_75_of__int__1__le__iff,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.47 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_1_le_iff
% 0.30/0.47 thf(fact_76_of__int__1__le__iff,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.47 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_1_le_iff
% 0.30/0.47 thf(fact_77_of__int__le__1__iff,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.30/0.47 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_le_1_iff
% 0.30/0.47 thf(fact_78_of__int__le__1__iff,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.30/0.47 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_le_1_iff
% 0.30/0.47 thf(fact_79_ceiling__le__one,axiom,
% 0.30/0.47 ! [X: real] :
% 0.30/0.47 ( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ one_one_int )
% 0.30/0.47 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % ceiling_le_one
% 0.30/0.47 thf(fact_80_numeral__le__one__iff,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.30/0.47 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_le_one_iff
% 0.30/0.47 thf(fact_81_numeral__le__one__iff,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.30/0.47 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_le_one_iff
% 0.30/0.47 thf(fact_82_numeral__le__one__iff,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.30/0.47 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_le_one_iff
% 0.30/0.47 thf(fact_83_complete__real,axiom,
% 0.30/0.47 ! [S: set_real] :
% 0.30/0.47 ( ? [X4: real] : ( member_real @ X4 @ S )
% 0.30/0.47 => ( ? [Z3: real] :
% 0.30/0.47 ! [X5: real] :
% 0.30/0.47 ( ( member_real @ X5 @ S )
% 0.30/0.47 => ( ord_less_eq_real @ X5 @ Z3 ) )
% 0.30/0.47 => ? [Y3: real] :
% 0.30/0.47 ( ! [X4: real] :
% 0.30/0.47 ( ( member_real @ X4 @ S )
% 0.30/0.47 => ( ord_less_eq_real @ X4 @ Y3 ) )
% 0.30/0.47 & ! [Z3: real] :
% 0.30/0.47 ( ! [X5: real] :
% 0.30/0.47 ( ( member_real @ X5 @ S )
% 0.30/0.47 => ( ord_less_eq_real @ X5 @ Z3 ) )
% 0.30/0.47 => ( ord_less_eq_real @ Y3 @ Z3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % complete_real
% 0.30/0.47 thf(fact_84_le__numeral__extra_I4_J,axiom,
% 0.30/0.47 ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.30/0.47
% 0.30/0.47 % le_numeral_extra(4)
% 0.30/0.47 thf(fact_85_le__numeral__extra_I4_J,axiom,
% 0.30/0.47 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.30/0.47
% 0.30/0.47 % le_numeral_extra(4)
% 0.30/0.47 thf(fact_86_le__numeral__extra_I4_J,axiom,
% 0.30/0.47 ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.30/0.47
% 0.30/0.47 % le_numeral_extra(4)
% 0.30/0.47 thf(fact_87_one__le__numeral,axiom,
% 0.30/0.47 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_le_numeral
% 0.30/0.47 thf(fact_88_one__le__numeral,axiom,
% 0.30/0.47 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_le_numeral
% 0.30/0.47 thf(fact_89_one__le__numeral,axiom,
% 0.30/0.47 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_le_numeral
% 0.30/0.47 thf(fact_90_numeral__One,axiom,
% 0.30/0.47 ( ( numeral_numeral_int @ one )
% 0.30/0.47 = one_one_int ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_One
% 0.30/0.47 thf(fact_91_numeral__One,axiom,
% 0.30/0.47 ( ( numeral_numeral_real @ one )
% 0.30/0.47 = one_one_real ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_One
% 0.30/0.47 thf(fact_92_numeral__One,axiom,
% 0.30/0.47 ( ( numeral_numeral_nat @ one )
% 0.30/0.47 = one_one_nat ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_One
% 0.30/0.47 thf(fact_93_verit__la__disequality,axiom,
% 0.30/0.47 ! [A: int,B: int] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 | ~ ( ord_less_eq_int @ A @ B )
% 0.30/0.47 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % verit_la_disequality
% 0.30/0.47 thf(fact_94_verit__la__disequality,axiom,
% 0.30/0.47 ! [A: nat,B: nat] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 | ~ ( ord_less_eq_nat @ A @ B )
% 0.30/0.47 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % verit_la_disequality
% 0.30/0.47 thf(fact_95_verit__la__disequality,axiom,
% 0.30/0.47 ! [A: real,B: real] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 | ~ ( ord_less_eq_real @ A @ B )
% 0.30/0.47 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % verit_la_disequality
% 0.30/0.47 thf(fact_96_verit__la__disequality,axiom,
% 0.30/0.47 ! [A: num,B: num] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 | ~ ( ord_less_eq_num @ A @ B )
% 0.30/0.47 | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % verit_la_disequality
% 0.30/0.47 thf(fact_97_le__num__One__iff,axiom,
% 0.30/0.47 ! [X: num] :
% 0.30/0.47 ( ( ord_less_eq_num @ X @ one )
% 0.30/0.47 = ( X = one ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_num_One_iff
% 0.30/0.47 thf(fact_98_two__realpow__ge__one,axiom,
% 0.30/0.47 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % two_realpow_ge_one
% 0.30/0.47 thf(fact_99_verit__la__generic,axiom,
% 0.30/0.47 ! [A: int,X: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ A @ X )
% 0.30/0.47 | ( A = X )
% 0.30/0.47 | ( ord_less_eq_int @ X @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % verit_la_generic
% 0.30/0.47 thf(fact_100_verit__eq__simplify_I10_J,axiom,
% 0.30/0.47 ! [X2: num] :
% 0.30/0.47 ( one
% 0.30/0.47 != ( bit0 @ X2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % verit_eq_simplify(10)
% 0.30/0.47 thf(fact_101_nat__numeral__as__int,axiom,
% 0.30/0.47 ( numeral_numeral_nat
% 0.30/0.47 = ( ^ [I: num] : ( nat2 @ ( numeral_numeral_int @ I ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_numeral_as_int
% 0.30/0.47 thf(fact_102__092_060open_0622_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_A_061_Areal__of__int_A_I2_A_094_Anat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062,axiom,
% 0.30/0.47 ( ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) )
% 0.30/0.47 = ( ring_1_of_int_real @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>2 powr real (nat \<lceil>log 2 (real_of_int d)\<rceil>) = real_of_int (2 ^ nat \<lceil>log 2 (real_of_int d)\<rceil>)\<close>
% 0.30/0.47 thf(fact_103__092_060open_0622_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_A_092_060le_062_A2_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062,axiom,
% 0.30/0.47 ord_less_eq_real @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>2 powr log 2 (real_of_int d) \<le> 2 powr real (nat \<lceil>log 2 (real_of_int d)\<rceil>)\<close>
% 0.30/0.47 thf(fact_104_numeral__powr__numeral__real,axiom,
% 0.30/0.47 ! [M: num,N: num] :
% 0.30/0.47 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.47 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_powr_numeral_real
% 0.30/0.47 thf(fact_105_powr__one__eq__one,axiom,
% 0.30/0.47 ! [A: real] :
% 0.30/0.47 ( ( powr_real @ one_one_real @ A )
% 0.30/0.47 = one_one_real ) ).
% 0.30/0.47
% 0.30/0.47 % powr_one_eq_one
% 0.30/0.47 thf(fact_106_self__le__ge2__pow,axiom,
% 0.30/0.47 ! [K: nat,M: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.47 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % self_le_ge2_pow
% 0.30/0.47 thf(fact_107_power2__nat__le__eq__le,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.47 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % power2_nat_le_eq_le
% 0.30/0.47 thf(fact_108_power2__nat__le__imp__le,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.30/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % power2_nat_le_imp_le
% 0.30/0.47 thf(fact_109_Suc__le__mono,axiom,
% 0.30/0.47 ! [N: nat,M: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.30/0.47 = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.30/0.47
% 0.30/0.47 % Suc_le_mono
% 0.30/0.47 thf(fact_110_power__one__right,axiom,
% 0.30/0.47 ! [A: int] :
% 0.30/0.47 ( ( power_power_int @ A @ one_one_nat )
% 0.30/0.47 = A ) ).
% 0.30/0.47
% 0.30/0.47 % power_one_right
% 0.30/0.47 thf(fact_111_power__one__right,axiom,
% 0.30/0.47 ! [A: nat] :
% 0.30/0.47 ( ( power_power_nat @ A @ one_one_nat )
% 0.30/0.47 = A ) ).
% 0.30/0.47
% 0.30/0.47 % power_one_right
% 0.30/0.47 thf(fact_112_power__one__right,axiom,
% 0.30/0.47 ! [A: real] :
% 0.30/0.47 ( ( power_power_real @ A @ one_one_nat )
% 0.30/0.47 = A ) ).
% 0.30/0.47
% 0.30/0.47 % power_one_right
% 0.30/0.47 thf(fact_113_nat_Oinject,axiom,
% 0.30/0.47 ! [X2: nat,Y2: nat] :
% 0.30/0.47 ( ( ( suc @ X2 )
% 0.30/0.47 = ( suc @ Y2 ) )
% 0.30/0.47 = ( X2 = Y2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat.inject
% 0.30/0.47 thf(fact_114_old_Onat_Oinject,axiom,
% 0.30/0.47 ! [Nat: nat,Nat2: nat] :
% 0.30/0.47 ( ( ( suc @ Nat )
% 0.30/0.47 = ( suc @ Nat2 ) )
% 0.30/0.47 = ( Nat = Nat2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.nat.inject
% 0.30/0.47 thf(fact_115_of__nat__eq__iff,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ( semiri2110766477t_real @ M )
% 0.30/0.47 = ( semiri2110766477t_real @ N ) )
% 0.30/0.47 = ( M = N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_eq_iff
% 0.30/0.47 thf(fact_116_of__nat__eq__iff,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ( semiri2019852685at_int @ M )
% 0.30/0.47 = ( semiri2019852685at_int @ N ) )
% 0.30/0.47 = ( M = N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_eq_iff
% 0.30/0.47 thf(fact_117_power__one,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( power_power_int @ one_one_int @ N )
% 0.30/0.47 = one_one_int ) ).
% 0.30/0.47
% 0.30/0.47 % power_one
% 0.30/0.47 thf(fact_118_power__one,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( power_power_nat @ one_one_nat @ N )
% 0.30/0.47 = one_one_nat ) ).
% 0.30/0.47
% 0.30/0.47 % power_one
% 0.30/0.47 thf(fact_119_power__one,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( power_power_real @ one_one_real @ N )
% 0.30/0.47 = one_one_real ) ).
% 0.30/0.47
% 0.30/0.47 % power_one
% 0.30/0.47 thf(fact_120_of__nat__eq__1__iff,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ( semiri1382578993at_nat @ N )
% 0.30/0.47 = one_one_nat )
% 0.30/0.47 = ( N = one_one_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_eq_1_iff
% 0.30/0.47 thf(fact_121_of__nat__eq__1__iff,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ( semiri2110766477t_real @ N )
% 0.30/0.47 = one_one_real )
% 0.30/0.47 = ( N = one_one_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_eq_1_iff
% 0.30/0.47 thf(fact_122_of__nat__eq__1__iff,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ( semiri2019852685at_int @ N )
% 0.30/0.47 = one_one_int )
% 0.30/0.47 = ( N = one_one_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_eq_1_iff
% 0.30/0.47 thf(fact_123_of__nat__1__eq__iff,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( one_one_nat
% 0.30/0.47 = ( semiri1382578993at_nat @ N ) )
% 0.30/0.47 = ( N = one_one_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_1_eq_iff
% 0.30/0.47 thf(fact_124_of__nat__1__eq__iff,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( one_one_real
% 0.30/0.47 = ( semiri2110766477t_real @ N ) )
% 0.30/0.47 = ( N = one_one_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_1_eq_iff
% 0.30/0.47 thf(fact_125_of__nat__1__eq__iff,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( one_one_int
% 0.30/0.47 = ( semiri2019852685at_int @ N ) )
% 0.30/0.47 = ( N = one_one_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_1_eq_iff
% 0.30/0.47 thf(fact_126_of__nat__1,axiom,
% 0.30/0.47 ( ( semiri1382578993at_nat @ one_one_nat )
% 0.30/0.47 = one_one_nat ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_1
% 0.30/0.47 thf(fact_127_of__nat__1,axiom,
% 0.30/0.47 ( ( semiri2110766477t_real @ one_one_nat )
% 0.30/0.47 = one_one_real ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_1
% 0.30/0.47 thf(fact_128_of__nat__1,axiom,
% 0.30/0.47 ( ( semiri2019852685at_int @ one_one_nat )
% 0.30/0.47 = one_one_int ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_1
% 0.30/0.47 thf(fact_129_of__int__of__nat__eq,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ring_1_of_int_real @ ( semiri2019852685at_int @ N ) )
% 0.30/0.47 = ( semiri2110766477t_real @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_of_nat_eq
% 0.30/0.47 thf(fact_130_of__int__of__nat__eq,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ring_1_of_int_int @ ( semiri2019852685at_int @ N ) )
% 0.30/0.47 = ( semiri2019852685at_int @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_of_nat_eq
% 0.30/0.47 thf(fact_131_ceiling__of__nat,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( archim1371465213g_real @ ( semiri2110766477t_real @ N ) )
% 0.30/0.47 = ( semiri2019852685at_int @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % ceiling_of_nat
% 0.30/0.47 thf(fact_132_of__nat__le__iff,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 0.30/0.47 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_le_iff
% 0.30/0.47 thf(fact_133_of__nat__le__iff,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 0.30/0.47 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_le_iff
% 0.30/0.47 thf(fact_134_of__nat__le__iff,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) )
% 0.30/0.47 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_le_iff
% 0.30/0.47 thf(fact_135_of__nat__numeral,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( ( semiri1382578993at_nat @ ( numeral_numeral_nat @ N ) )
% 0.30/0.47 = ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_numeral
% 0.30/0.47 thf(fact_136_of__nat__numeral,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( ( semiri2110766477t_real @ ( numeral_numeral_nat @ N ) )
% 0.30/0.47 = ( numeral_numeral_real @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_numeral
% 0.30/0.47 thf(fact_137_of__nat__numeral,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( ( semiri2019852685at_int @ ( numeral_numeral_nat @ N ) )
% 0.30/0.47 = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_numeral
% 0.30/0.47 thf(fact_138_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.47 ! [X: nat,B: nat,W: nat] :
% 0.30/0.47 ( ( ( semiri1382578993at_nat @ X )
% 0.30/0.47 = ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) )
% 0.30/0.47 = ( X
% 0.30/0.47 = ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_power_eq_of_nat_cancel_iff
% 0.30/0.47 thf(fact_139_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.47 ! [X: nat,B: nat,W: nat] :
% 0.30/0.47 ( ( ( semiri2110766477t_real @ X )
% 0.30/0.47 = ( power_power_real @ ( semiri2110766477t_real @ B ) @ W ) )
% 0.30/0.47 = ( X
% 0.30/0.47 = ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_power_eq_of_nat_cancel_iff
% 0.30/0.47 thf(fact_140_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.47 ! [X: nat,B: nat,W: nat] :
% 0.30/0.47 ( ( ( semiri2019852685at_int @ X )
% 0.30/0.47 = ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) )
% 0.30/0.47 = ( X
% 0.30/0.47 = ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_power_eq_of_nat_cancel_iff
% 0.30/0.47 thf(fact_141_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.30/0.47 ! [B: nat,W: nat,X: nat] :
% 0.30/0.47 ( ( ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W )
% 0.30/0.47 = ( semiri1382578993at_nat @ X ) )
% 0.30/0.47 = ( ( power_power_nat @ B @ W )
% 0.30/0.47 = X ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_eq_of_nat_power_cancel_iff
% 0.30/0.47 thf(fact_142_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.30/0.47 ! [B: nat,W: nat,X: nat] :
% 0.30/0.47 ( ( ( power_power_real @ ( semiri2110766477t_real @ B ) @ W )
% 0.30/0.47 = ( semiri2110766477t_real @ X ) )
% 0.30/0.47 = ( ( power_power_nat @ B @ W )
% 0.30/0.47 = X ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_eq_of_nat_power_cancel_iff
% 0.30/0.47 thf(fact_143_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.30/0.47 ! [B: nat,W: nat,X: nat] :
% 0.30/0.47 ( ( ( power_power_int @ ( semiri2019852685at_int @ B ) @ W )
% 0.30/0.47 = ( semiri2019852685at_int @ X ) )
% 0.30/0.47 = ( ( power_power_nat @ B @ W )
% 0.30/0.47 = X ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_eq_of_nat_power_cancel_iff
% 0.30/0.47 thf(fact_144_of__nat__power,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( semiri1382578993at_nat @ ( power_power_nat @ M @ N ) )
% 0.30/0.47 = ( power_power_nat @ ( semiri1382578993at_nat @ M ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_power
% 0.30/0.47 thf(fact_145_of__nat__power,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( semiri2110766477t_real @ ( power_power_nat @ M @ N ) )
% 0.30/0.47 = ( power_power_real @ ( semiri2110766477t_real @ M ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_power
% 0.30/0.47 thf(fact_146_of__nat__power,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( semiri2019852685at_int @ ( power_power_nat @ M @ N ) )
% 0.30/0.47 = ( power_power_int @ ( semiri2019852685at_int @ M ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_power
% 0.30/0.47 thf(fact_147_Suc__1,axiom,
% 0.30/0.47 ( ( suc @ one_one_nat )
% 0.30/0.47 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Suc_1
% 0.30/0.47 thf(fact_148_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.47 ! [X: nat,B: nat,W: nat] :
% 0.30/0.47 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ X ) @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) )
% 0.30/0.47 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.47 thf(fact_149_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.47 ! [X: nat,B: nat,W: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ X ) @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) )
% 0.30/0.47 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.47 thf(fact_150_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.47 ! [X: nat,B: nat,W: nat] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( semiri2110766477t_real @ X ) @ ( power_power_real @ ( semiri2110766477t_real @ B ) @ W ) )
% 0.30/0.47 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.47 thf(fact_151_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.47 ! [B: nat,W: nat,X: nat] :
% 0.30/0.47 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) @ ( semiri2019852685at_int @ X ) )
% 0.30/0.47 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.47 thf(fact_152_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.47 ! [B: nat,W: nat,X: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) @ ( semiri1382578993at_nat @ X ) )
% 0.30/0.47 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.47 thf(fact_153_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.47 ! [B: nat,W: nat,X: nat] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri2110766477t_real @ B ) @ W ) @ ( semiri2110766477t_real @ X ) )
% 0.30/0.47 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.47 thf(fact_154_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.47 ! [Y: nat,X: num,N: nat] :
% 0.30/0.47 ( ( ( semiri1382578993at_nat @ Y )
% 0.30/0.47 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.30/0.47 = ( Y
% 0.30/0.47 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.47 thf(fact_155_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.47 ! [Y: nat,X: num,N: nat] :
% 0.30/0.47 ( ( ( semiri2110766477t_real @ Y )
% 0.30/0.47 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.30/0.47 = ( Y
% 0.30/0.47 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.47 thf(fact_156_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.47 ! [Y: nat,X: num,N: nat] :
% 0.30/0.47 ( ( ( semiri2019852685at_int @ Y )
% 0.30/0.47 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.30/0.47 = ( Y
% 0.30/0.47 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.47 thf(fact_157_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.47 ! [X: num,N: nat,Y: nat] :
% 0.30/0.47 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.30/0.47 = ( semiri1382578993at_nat @ Y ) )
% 0.30/0.47 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.30/0.47 = Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.47 thf(fact_158_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.47 ! [X: num,N: nat,Y: nat] :
% 0.30/0.47 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 0.30/0.47 = ( semiri2110766477t_real @ Y ) )
% 0.30/0.47 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.30/0.47 = Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.47 thf(fact_159_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.47 ! [X: num,N: nat,Y: nat] :
% 0.30/0.47 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.30/0.47 = ( semiri2019852685at_int @ Y ) )
% 0.30/0.47 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.30/0.47 = Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.47 thf(fact_160_numeral__le__real__of__nat__iff,axiom,
% 0.30/0.47 ! [N: num,M: nat] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri2110766477t_real @ M ) )
% 0.30/0.47 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_le_real_of_nat_iff
% 0.30/0.47 thf(fact_161_nat__ceiling__le__eq,axiom,
% 0.30/0.47 ! [X: real,A: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( nat2 @ ( archim1371465213g_real @ X ) ) @ A )
% 0.30/0.47 = ( ord_less_eq_real @ X @ ( semiri2110766477t_real @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_ceiling_le_eq
% 0.30/0.47 thf(fact_162_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.47 ! [X: nat,I2: num,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
% 0.30/0.47 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_le_numeral_power_cancel_iff
% 0.30/0.47 thf(fact_163_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.47 ! [X: nat,I2: num,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
% 0.30/0.47 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_le_numeral_power_cancel_iff
% 0.30/0.47 thf(fact_164_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.47 ! [X: nat,I2: num,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( semiri2110766477t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) )
% 0.30/0.47 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_le_numeral_power_cancel_iff
% 0.30/0.47 thf(fact_165_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.47 ! [I2: num,N: nat,X: nat] :
% 0.30/0.47 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri2019852685at_int @ X ) )
% 0.30/0.47 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_power_le_of_nat_cancel_iff
% 0.30/0.47 thf(fact_166_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.47 ! [I2: num,N: nat,X: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1382578993at_nat @ X ) )
% 0.30/0.47 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_power_le_of_nat_cancel_iff
% 0.30/0.47 thf(fact_167_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.47 ! [I2: num,N: nat,X: nat] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) @ ( semiri2110766477t_real @ X ) )
% 0.30/0.47 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % numeral_power_le_of_nat_cancel_iff
% 0.30/0.47 thf(fact_168_nat__one__as__int,axiom,
% 0.30/0.47 ( one_one_nat
% 0.30/0.47 = ( nat2 @ one_one_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_one_as_int
% 0.30/0.47 thf(fact_169_real__arch__simple,axiom,
% 0.30/0.47 ! [X: real] :
% 0.30/0.47 ? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri2110766477t_real @ N3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % real_arch_simple
% 0.30/0.47 thf(fact_170_of__nat__mono,axiom,
% 0.30/0.47 ! [I2: nat,J: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.47 => ( ord_less_eq_int @ ( semiri2019852685at_int @ I2 ) @ ( semiri2019852685at_int @ J ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_mono
% 0.30/0.47 thf(fact_171_of__nat__mono,axiom,
% 0.30/0.47 ! [I2: nat,J: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.47 => ( ord_less_eq_nat @ ( semiri1382578993at_nat @ I2 ) @ ( semiri1382578993at_nat @ J ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_mono
% 0.30/0.47 thf(fact_172_of__nat__mono,axiom,
% 0.30/0.47 ! [I2: nat,J: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.47 => ( ord_less_eq_real @ ( semiri2110766477t_real @ I2 ) @ ( semiri2110766477t_real @ J ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_mono
% 0.30/0.47 thf(fact_173_numerals_I1_J,axiom,
% 0.30/0.47 ( ( numeral_numeral_nat @ one )
% 0.30/0.47 = one_one_nat ) ).
% 0.30/0.47
% 0.30/0.47 % numerals(1)
% 0.30/0.47 thf(fact_174_of__nat__ceiling,axiom,
% 0.30/0.47 ! [R: real] : ( ord_less_eq_real @ R @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ R ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_ceiling
% 0.30/0.47 thf(fact_175_Suc__inject,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( ( suc @ X )
% 0.30/0.47 = ( suc @ Y ) )
% 0.30/0.47 => ( X = Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % Suc_inject
% 0.30/0.47 thf(fact_176_n__not__Suc__n,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( N
% 0.30/0.47 != ( suc @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % n_not_Suc_n
% 0.30/0.47 thf(fact_177_Nat_Oex__has__greatest__nat,axiom,
% 0.30/0.47 ! [P: nat > $o,K: nat,B: nat] :
% 0.30/0.47 ( ( P @ K )
% 0.30/0.47 => ( ! [Y3: nat] :
% 0.30/0.47 ( ( P @ Y3 )
% 0.30/0.47 => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.30/0.47 => ? [X5: nat] :
% 0.30/0.47 ( ( P @ X5 )
% 0.30/0.47 & ! [Y4: nat] :
% 0.30/0.47 ( ( P @ Y4 )
% 0.30/0.47 => ( ord_less_eq_nat @ Y4 @ X5 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Nat.ex_has_greatest_nat
% 0.30/0.47 thf(fact_178_nat__le__linear,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.47 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_le_linear
% 0.30/0.47 thf(fact_179_le__antisym,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.47 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.47 => ( M = N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_antisym
% 0.30/0.47 thf(fact_180_eq__imp__le,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( M = N )
% 0.30/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_imp_le
% 0.30/0.47 thf(fact_181_le__trans,axiom,
% 0.30/0.47 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.47 => ( ( ord_less_eq_nat @ J @ K )
% 0.30/0.47 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_trans
% 0.30/0.47 thf(fact_182_le__refl,axiom,
% 0.30/0.47 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.30/0.47
% 0.30/0.47 % le_refl
% 0.30/0.47 thf(fact_183_real__nat__ceiling__ge,axiom,
% 0.30/0.47 ! [X: real] : ( ord_less_eq_real @ X @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % real_nat_ceiling_ge
% 0.30/0.47 thf(fact_184_powr__powr__swap,axiom,
% 0.30/0.47 ! [X: real,A: real,B: real] :
% 0.30/0.47 ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 0.30/0.47 = ( powr_real @ ( powr_real @ X @ B ) @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % powr_powr_swap
% 0.30/0.47 thf(fact_185_log2__of__power__eq,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( M
% 0.30/0.47 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.47 => ( ( semiri2110766477t_real @ N )
% 0.30/0.47 = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % log2_of_power_eq
% 0.30/0.47 thf(fact_186_le__log2__of__power,axiom,
% 0.30/0.47 ! [N: nat,M: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.30/0.47 => ( ord_less_eq_real @ ( semiri2110766477t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_log2_of_power
% 0.30/0.47 thf(fact_187_transitive__stepwise__le,axiom,
% 0.30/0.47 ! [M: nat,N: nat,R2: nat > nat > $o] :
% 0.30/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.47 => ( ! [X5: nat] : ( R2 @ X5 @ X5 )
% 0.30/0.47 => ( ! [X5: nat,Y3: nat,Z2: nat] :
% 0.30/0.47 ( ( R2 @ X5 @ Y3 )
% 0.30/0.47 => ( ( R2 @ Y3 @ Z2 )
% 0.30/0.47 => ( R2 @ X5 @ Z2 ) ) )
% 0.30/0.47 => ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
% 0.30/0.47 => ( R2 @ M @ N ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % transitive_stepwise_le
% 0.30/0.47 thf(fact_188_nat__induct__at__least,axiom,
% 0.30/0.47 ! [M: nat,N: nat,P: nat > $o] :
% 0.30/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.47 => ( ( P @ M )
% 0.30/0.47 => ( ! [N3: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ M @ N3 )
% 0.30/0.47 => ( ( P @ N3 )
% 0.30/0.47 => ( P @ ( suc @ N3 ) ) ) )
% 0.30/0.47 => ( P @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_induct_at_least
% 0.30/0.47 thf(fact_189_full__nat__induct,axiom,
% 0.30/0.47 ! [P: nat > $o,N: nat] :
% 0.30/0.47 ( ! [N3: nat] :
% 0.30/0.47 ( ! [M2: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
% 0.30/0.47 => ( P @ M2 ) )
% 0.30/0.47 => ( P @ N3 ) )
% 0.30/0.47 => ( P @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % full_nat_induct
% 0.30/0.47 thf(fact_190_not__less__eq__eq,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.30/0.47 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_less_eq_eq
% 0.30/0.47 thf(fact_191_Suc__n__not__le__n,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.30/0.47
% 0.30/0.47 % Suc_n_not_le_n
% 0.30/0.47 thf(fact_192_le__Suc__eq,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.47 = ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.47 | ( M
% 0.30/0.47 = ( suc @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_Suc_eq
% 0.30/0.47 thf(fact_193_Suc__le__D,axiom,
% 0.30/0.47 ! [N: nat,M3: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
% 0.30/0.47 => ? [M4: nat] :
% 0.30/0.47 ( M3
% 0.30/0.47 = ( suc @ M4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Suc_le_D
% 0.30/0.47 thf(fact_194_le__SucI,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.47 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_SucI
% 0.30/0.47 thf(fact_195_le__SucE,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.47 => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.47 => ( M
% 0.30/0.47 = ( suc @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_SucE
% 0.30/0.47 thf(fact_196_Suc__leD,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.30/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % Suc_leD
% 0.30/0.47 thf(fact_197_one__le__power,axiom,
% 0.30/0.47 ! [A: int,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.30/0.47 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_le_power
% 0.30/0.47 thf(fact_198_one__le__power,axiom,
% 0.30/0.47 ! [A: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.30/0.47 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_le_power
% 0.30/0.47 thf(fact_199_one__le__power,axiom,
% 0.30/0.47 ! [A: real,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.30/0.47 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_le_power
% 0.30/0.47 thf(fact_200_lift__Suc__antimono__le,axiom,
% 0.30/0.47 ! [F: nat > int,N: nat,N4: nat] :
% 0.30/0.47 ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.30/0.47 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.47 => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lift_Suc_antimono_le
% 0.30/0.47 thf(fact_201_lift__Suc__antimono__le,axiom,
% 0.30/0.47 ! [F: nat > nat,N: nat,N4: nat] :
% 0.30/0.47 ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.30/0.47 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.47 => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lift_Suc_antimono_le
% 0.30/0.47 thf(fact_202_lift__Suc__antimono__le,axiom,
% 0.30/0.47 ! [F: nat > real,N: nat,N4: nat] :
% 0.30/0.47 ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.30/0.47 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.47 => ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lift_Suc_antimono_le
% 0.30/0.47 thf(fact_203_lift__Suc__antimono__le,axiom,
% 0.30/0.47 ! [F: nat > num,N: nat,N4: nat] :
% 0.30/0.47 ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.30/0.47 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.47 => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lift_Suc_antimono_le
% 0.30/0.47 thf(fact_204_lift__Suc__mono__le,axiom,
% 0.30/0.47 ! [F: nat > int,N: nat,N4: nat] :
% 0.30/0.47 ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.47 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.47 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lift_Suc_mono_le
% 0.30/0.47 thf(fact_205_lift__Suc__mono__le,axiom,
% 0.30/0.47 ! [F: nat > nat,N: nat,N4: nat] :
% 0.30/0.47 ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.47 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.47 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lift_Suc_mono_le
% 0.30/0.47 thf(fact_206_lift__Suc__mono__le,axiom,
% 0.30/0.47 ! [F: nat > real,N: nat,N4: nat] :
% 0.30/0.47 ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.47 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.47 => ( ord_less_eq_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lift_Suc_mono_le
% 0.30/0.47 thf(fact_207_lift__Suc__mono__le,axiom,
% 0.30/0.47 ! [F: nat > num,N: nat,N4: nat] :
% 0.30/0.47 ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.47 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.47 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lift_Suc_mono_le
% 0.30/0.47 thf(fact_208_powr__mono,axiom,
% 0.30/0.47 ! [A: real,B: real,X: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.47 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.30/0.47 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % powr_mono
% 0.30/0.47 thf(fact_209_power__increasing,axiom,
% 0.30/0.47 ! [N: nat,N5: nat,A: int] :
% 0.30/0.47 ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.47 => ( ( ord_less_eq_int @ one_one_int @ A )
% 0.30/0.47 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_increasing
% 0.30/0.47 thf(fact_210_power__increasing,axiom,
% 0.30/0.47 ! [N: nat,N5: nat,A: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.47 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.30/0.47 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_increasing
% 0.30/0.47 thf(fact_211_power__increasing,axiom,
% 0.30/0.47 ! [N: nat,N5: nat,A: real] :
% 0.30/0.47 ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.47 => ( ( ord_less_eq_real @ one_one_real @ A )
% 0.30/0.47 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_increasing
% 0.30/0.47 thf(fact_212_one__power2,axiom,
% 0.30/0.47 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.47 = one_one_int ) ).
% 0.30/0.47
% 0.30/0.47 % one_power2
% 0.30/0.47 thf(fact_213_one__power2,axiom,
% 0.30/0.47 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.47 = one_one_nat ) ).
% 0.30/0.47
% 0.30/0.47 % one_power2
% 0.30/0.47 thf(fact_214_one__power2,axiom,
% 0.30/0.47 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.47 = one_one_real ) ).
% 0.30/0.47
% 0.30/0.47 % one_power2
% 0.30/0.47 thf(fact_215_dbl__simps_I3_J,axiom,
% 0.30/0.47 ( ( neg_numeral_dbl_int @ one_one_int )
% 0.30/0.47 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dbl_simps(3)
% 0.30/0.47 thf(fact_216_dbl__simps_I3_J,axiom,
% 0.30/0.47 ( ( neg_numeral_dbl_real @ one_one_real )
% 0.30/0.47 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dbl_simps(3)
% 0.30/0.47 thf(fact_217_power__numeral,axiom,
% 0.30/0.47 ! [K: num,L: num] :
% 0.30/0.47 ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.30/0.47 = ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_numeral
% 0.30/0.47 thf(fact_218_power__numeral,axiom,
% 0.30/0.47 ! [K: num,L: num] :
% 0.30/0.47 ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.30/0.47 = ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_numeral
% 0.30/0.47 thf(fact_219_power__numeral,axiom,
% 0.30/0.47 ! [K: num,L: num] :
% 0.30/0.47 ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.30/0.47 = ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_numeral
% 0.30/0.47 thf(fact_220_order__refl,axiom,
% 0.30/0.47 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 0.30/0.47
% 0.30/0.47 % order_refl
% 0.30/0.47 thf(fact_221_order__refl,axiom,
% 0.30/0.47 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.30/0.47
% 0.30/0.47 % order_refl
% 0.30/0.47 thf(fact_222_order__refl,axiom,
% 0.30/0.47 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 0.30/0.47
% 0.30/0.47 % order_refl
% 0.30/0.47 thf(fact_223_order__refl,axiom,
% 0.30/0.47 ! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% 0.30/0.47
% 0.30/0.47 % order_refl
% 0.30/0.47 thf(fact_224_powr__numeral,axiom,
% 0.30/0.47 ! [X: real,N: num] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.47 => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
% 0.30/0.47 = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % powr_numeral
% 0.30/0.47 thf(fact_225_power2__eq__iff__nonneg,axiom,
% 0.30/0.47 ! [X: int,Y: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.30/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.30/0.47 => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.47 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.47 = ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power2_eq_iff_nonneg
% 0.30/0.47 thf(fact_226_power2__eq__iff__nonneg,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.30/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.30/0.47 => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.47 = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.47 = ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power2_eq_iff_nonneg
% 0.30/0.47 thf(fact_227_power2__eq__iff__nonneg,axiom,
% 0.30/0.47 ! [X: real,Y: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.30/0.47 => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.47 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.47 = ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power2_eq_iff_nonneg
% 0.30/0.47 thf(fact_228_le0,axiom,
% 0.30/0.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.30/0.47
% 0.30/0.47 % le0
% 0.30/0.47 thf(fact_229_bot__nat__0_Oextremum,axiom,
% 0.30/0.47 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.30/0.47
% 0.30/0.47 % bot_nat_0.extremum
% 0.30/0.47 thf(fact_230_d_I2_J,axiom,
% 0.30/0.47 ord_less_int @ zero_zero_int @ d ).
% 0.30/0.47
% 0.30/0.47 % d(2)
% 0.30/0.47 thf(fact_231_le__zero__eq,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.30/0.47 = ( N = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_zero_eq
% 0.30/0.47 thf(fact_232_of__nat__0,axiom,
% 0.30/0.47 ( ( semiri1382578993at_nat @ zero_zero_nat )
% 0.30/0.47 = zero_zero_nat ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_0
% 0.30/0.47 thf(fact_233_of__nat__0,axiom,
% 0.30/0.47 ( ( semiri2110766477t_real @ zero_zero_nat )
% 0.30/0.47 = zero_zero_real ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_0
% 0.30/0.47 thf(fact_234_of__nat__0,axiom,
% 0.30/0.47 ( ( semiri2019852685at_int @ zero_zero_nat )
% 0.30/0.47 = zero_zero_int ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_0
% 0.30/0.47 thf(fact_235_of__nat__0__eq__iff,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( zero_zero_nat
% 0.30/0.47 = ( semiri1382578993at_nat @ N ) )
% 0.30/0.47 = ( zero_zero_nat = N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_0_eq_iff
% 0.30/0.47 thf(fact_236_of__nat__0__eq__iff,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( zero_zero_real
% 0.30/0.47 = ( semiri2110766477t_real @ N ) )
% 0.30/0.47 = ( zero_zero_nat = N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_0_eq_iff
% 0.30/0.47 thf(fact_237_of__nat__0__eq__iff,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( zero_zero_int
% 0.30/0.47 = ( semiri2019852685at_int @ N ) )
% 0.30/0.47 = ( zero_zero_nat = N ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_0_eq_iff
% 0.30/0.47 thf(fact_238_of__nat__eq__0__iff,axiom,
% 0.30/0.47 ! [M: nat] :
% 0.30/0.47 ( ( ( semiri1382578993at_nat @ M )
% 0.30/0.47 = zero_zero_nat )
% 0.30/0.47 = ( M = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_eq_0_iff
% 0.30/0.47 thf(fact_239_of__nat__eq__0__iff,axiom,
% 0.30/0.47 ! [M: nat] :
% 0.30/0.47 ( ( ( semiri2110766477t_real @ M )
% 0.30/0.47 = zero_zero_real )
% 0.30/0.47 = ( M = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_eq_0_iff
% 0.30/0.47 thf(fact_240_of__nat__eq__0__iff,axiom,
% 0.30/0.47 ! [M: nat] :
% 0.30/0.47 ( ( ( semiri2019852685at_int @ M )
% 0.30/0.47 = zero_zero_int )
% 0.30/0.47 = ( M = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_eq_0_iff
% 0.30/0.47 thf(fact_241_power__Suc0__right,axiom,
% 0.30/0.47 ! [A: int] :
% 0.30/0.47 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.47 = A ) ).
% 0.30/0.47
% 0.30/0.47 % power_Suc0_right
% 0.30/0.47 thf(fact_242_power__Suc0__right,axiom,
% 0.30/0.47 ! [A: nat] :
% 0.30/0.47 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.47 = A ) ).
% 0.30/0.47
% 0.30/0.47 % power_Suc0_right
% 0.30/0.47 thf(fact_243_power__Suc0__right,axiom,
% 0.30/0.47 ! [A: real] :
% 0.30/0.47 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.47 = A ) ).
% 0.30/0.47
% 0.30/0.47 % power_Suc0_right
% 0.30/0.47 thf(fact_244_of__int__0,axiom,
% 0.30/0.47 ( ( ring_1_of_int_real @ zero_zero_int )
% 0.30/0.47 = zero_zero_real ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_0
% 0.30/0.47 thf(fact_245_of__int__0,axiom,
% 0.30/0.47 ( ( ring_1_of_int_int @ zero_zero_int )
% 0.30/0.47 = zero_zero_int ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_0
% 0.30/0.47 thf(fact_246_of__int__0__eq__iff,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( zero_zero_real
% 0.30/0.47 = ( ring_1_of_int_real @ Z ) )
% 0.30/0.47 = ( Z = zero_zero_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_0_eq_iff
% 0.30/0.47 thf(fact_247_of__int__0__eq__iff,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( zero_zero_int
% 0.30/0.47 = ( ring_1_of_int_int @ Z ) )
% 0.30/0.47 = ( Z = zero_zero_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_0_eq_iff
% 0.30/0.47 thf(fact_248_of__int__eq__0__iff,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ( ring_1_of_int_real @ Z )
% 0.30/0.47 = zero_zero_real )
% 0.30/0.47 = ( Z = zero_zero_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_eq_0_iff
% 0.30/0.47 thf(fact_249_of__int__eq__0__iff,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ( ring_1_of_int_int @ Z )
% 0.30/0.47 = zero_zero_int )
% 0.30/0.47 = ( Z = zero_zero_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_eq_0_iff
% 0.30/0.47 thf(fact_250_powr__eq__0__iff,axiom,
% 0.30/0.47 ! [W: real,Z: real] :
% 0.30/0.47 ( ( ( powr_real @ W @ Z )
% 0.30/0.47 = zero_zero_real )
% 0.30/0.47 = ( W = zero_zero_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % powr_eq_0_iff
% 0.30/0.47 thf(fact_251_powr__0,axiom,
% 0.30/0.47 ! [Z: real] :
% 0.30/0.47 ( ( powr_real @ zero_zero_real @ Z )
% 0.30/0.47 = zero_zero_real ) ).
% 0.30/0.47
% 0.30/0.47 % powr_0
% 0.30/0.47 thf(fact_252_ceiling__zero,axiom,
% 0.30/0.47 ( ( archim1371465213g_real @ zero_zero_real )
% 0.30/0.47 = zero_zero_int ) ).
% 0.30/0.47
% 0.30/0.47 % ceiling_zero
% 0.30/0.47 thf(fact_253_nat__le__0,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.30/0.47 => ( ( nat2 @ Z )
% 0.30/0.47 = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_le_0
% 0.30/0.47 thf(fact_254_nat__0__iff,axiom,
% 0.30/0.47 ! [I2: int] :
% 0.30/0.47 ( ( ( nat2 @ I2 )
% 0.30/0.47 = zero_zero_nat )
% 0.30/0.47 = ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_0_iff
% 0.30/0.47 thf(fact_255_power__Suc__0,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.47 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_Suc_0
% 0.30/0.47 thf(fact_256_nat__power__eq__Suc__0__iff,axiom,
% 0.30/0.47 ! [X: nat,M: nat] :
% 0.30/0.47 ( ( ( power_power_nat @ X @ M )
% 0.30/0.47 = ( suc @ zero_zero_nat ) )
% 0.30/0.47 = ( ( M = zero_zero_nat )
% 0.30/0.47 | ( X
% 0.30/0.47 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_power_eq_Suc_0_iff
% 0.30/0.47 thf(fact_257_nat__int,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( nat2 @ ( semiri2019852685at_int @ N ) )
% 0.30/0.47 = N ) ).
% 0.30/0.47
% 0.30/0.47 % nat_int
% 0.30/0.47 thf(fact_258_dbl__simps_I2_J,axiom,
% 0.30/0.47 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 0.30/0.47 = zero_zero_real ) ).
% 0.30/0.47
% 0.30/0.47 % dbl_simps(2)
% 0.30/0.47 thf(fact_259_dbl__simps_I2_J,axiom,
% 0.30/0.47 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 0.30/0.47 = zero_zero_int ) ).
% 0.30/0.47
% 0.30/0.47 % dbl_simps(2)
% 0.30/0.47 thf(fact_260_of__nat__le__0__iff,axiom,
% 0.30/0.47 ! [M: nat] :
% 0.30/0.47 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int )
% 0.30/0.47 = ( M = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_le_0_iff
% 0.30/0.47 thf(fact_261_of__nat__le__0__iff,axiom,
% 0.30/0.47 ! [M: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat )
% 0.30/0.47 = ( M = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_le_0_iff
% 0.30/0.47 thf(fact_262_of__nat__le__0__iff,axiom,
% 0.30/0.47 ! [M: nat] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( semiri2110766477t_real @ M ) @ zero_zero_real )
% 0.30/0.47 = ( M = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_le_0_iff
% 0.30/0.47 thf(fact_263_power__0__Suc,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 0.30/0.47 = zero_zero_int ) ).
% 0.30/0.47
% 0.30/0.47 % power_0_Suc
% 0.30/0.47 thf(fact_264_power__0__Suc,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.30/0.47 = zero_zero_nat ) ).
% 0.30/0.47
% 0.30/0.47 % power_0_Suc
% 0.30/0.47 thf(fact_265_power__0__Suc,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 0.30/0.47 = zero_zero_real ) ).
% 0.30/0.47
% 0.30/0.47 % power_0_Suc
% 0.30/0.47 thf(fact_266_power__zero__numeral,axiom,
% 0.30/0.47 ! [K: num] :
% 0.30/0.47 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 0.30/0.47 = zero_zero_int ) ).
% 0.30/0.47
% 0.30/0.47 % power_zero_numeral
% 0.30/0.47 thf(fact_267_power__zero__numeral,axiom,
% 0.30/0.47 ! [K: num] :
% 0.30/0.47 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.47 = zero_zero_nat ) ).
% 0.30/0.47
% 0.30/0.47 % power_zero_numeral
% 0.30/0.47 thf(fact_268_power__zero__numeral,axiom,
% 0.30/0.47 ! [K: num] :
% 0.30/0.47 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 0.30/0.47 = zero_zero_real ) ).
% 0.30/0.47
% 0.30/0.47 % power_zero_numeral
% 0.30/0.47 thf(fact_269_powr__zero__eq__one,axiom,
% 0.30/0.47 ! [X: real] :
% 0.30/0.47 ( ( ( X = zero_zero_real )
% 0.30/0.47 => ( ( powr_real @ X @ zero_zero_real )
% 0.30/0.47 = zero_zero_real ) )
% 0.30/0.47 & ( ( X != zero_zero_real )
% 0.30/0.47 => ( ( powr_real @ X @ zero_zero_real )
% 0.30/0.47 = one_one_real ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % powr_zero_eq_one
% 0.30/0.47 thf(fact_270_powr__nonneg__iff,axiom,
% 0.30/0.47 ! [A: real,X: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
% 0.30/0.47 = ( A = zero_zero_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % powr_nonneg_iff
% 0.30/0.47 thf(fact_271_nat__1,axiom,
% 0.30/0.47 ( ( nat2 @ one_one_int )
% 0.30/0.47 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_1
% 0.30/0.47 thf(fact_272_int__eq__iff__numeral,axiom,
% 0.30/0.47 ! [M: nat,V: num] :
% 0.30/0.47 ( ( ( semiri2019852685at_int @ M )
% 0.30/0.47 = ( numeral_numeral_int @ V ) )
% 0.30/0.47 = ( M
% 0.30/0.47 = ( numeral_numeral_nat @ V ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % int_eq_iff_numeral
% 0.30/0.47 thf(fact_273_log__one,axiom,
% 0.30/0.47 ! [A: real] :
% 0.30/0.47 ( ( log @ A @ one_one_real )
% 0.30/0.47 = zero_zero_real ) ).
% 0.30/0.47
% 0.30/0.47 % log_one
% 0.30/0.47 thf(fact_274_int__nat__eq,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.47 => ( ( semiri2019852685at_int @ ( nat2 @ Z ) )
% 0.30/0.47 = Z ) )
% 0.30/0.47 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.47 => ( ( semiri2019852685at_int @ ( nat2 @ Z ) )
% 0.30/0.47 = zero_zero_int ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % int_nat_eq
% 0.30/0.47 thf(fact_275_dbl__simps_I5_J,axiom,
% 0.30/0.47 ! [K: num] :
% 0.30/0.47 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 0.30/0.47 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dbl_simps(5)
% 0.30/0.47 thf(fact_276_dbl__simps_I5_J,axiom,
% 0.30/0.47 ! [K: num] :
% 0.30/0.47 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 0.30/0.47 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dbl_simps(5)
% 0.30/0.47 thf(fact_277_of__int__0__le__iff,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.47 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_0_le_iff
% 0.30/0.47 thf(fact_278_of__int__0__le__iff,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.47 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_0_le_iff
% 0.30/0.47 thf(fact_279_of__int__le__0__iff,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.30/0.47 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_le_0_iff
% 0.30/0.47 thf(fact_280_of__int__le__0__iff,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.30/0.47 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_le_0_iff
% 0.30/0.47 thf(fact_281_ceiling__le__zero,axiom,
% 0.30/0.47 ! [X: real] :
% 0.30/0.47 ( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ zero_zero_int )
% 0.30/0.47 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % ceiling_le_zero
% 0.30/0.47 thf(fact_282_powr__one,axiom,
% 0.30/0.47 ! [X: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.47 => ( ( powr_real @ X @ one_one_real )
% 0.30/0.47 = X ) ) ).
% 0.30/0.47
% 0.30/0.47 % powr_one
% 0.30/0.47 thf(fact_283_powr__one__gt__zero__iff,axiom,
% 0.30/0.47 ! [X: real] :
% 0.30/0.47 ( ( ( powr_real @ X @ one_one_real )
% 0.30/0.47 = X )
% 0.30/0.47 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % powr_one_gt_zero_iff
% 0.30/0.47 thf(fact_284_of__nat__nat,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.47 => ( ( semiri2110766477t_real @ ( nat2 @ Z ) )
% 0.30/0.47 = ( ring_1_of_int_real @ Z ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_nat
% 0.30/0.47 thf(fact_285_of__nat__nat,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.47 => ( ( semiri2019852685at_int @ ( nat2 @ Z ) )
% 0.30/0.47 = ( ring_1_of_int_int @ Z ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_nat
% 0.30/0.47 thf(fact_286_zero__eq__power2,axiom,
% 0.30/0.47 ! [A: int] :
% 0.30/0.47 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.47 = zero_zero_int )
% 0.30/0.47 = ( A = zero_zero_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_eq_power2
% 0.30/0.47 thf(fact_287_zero__eq__power2,axiom,
% 0.30/0.47 ! [A: nat] :
% 0.30/0.47 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.47 = zero_zero_nat )
% 0.30/0.47 = ( A = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_eq_power2
% 0.30/0.47 thf(fact_288_zero__eq__power2,axiom,
% 0.30/0.47 ! [A: real] :
% 0.30/0.47 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.47 = zero_zero_real )
% 0.30/0.47 = ( A = zero_zero_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_eq_power2
% 0.30/0.47 thf(fact_289_power2__less__eq__zero__iff,axiom,
% 0.30/0.47 ! [A: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.30/0.47 = ( A = zero_zero_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % power2_less_eq_zero_iff
% 0.30/0.47 thf(fact_290_power2__less__eq__zero__iff,axiom,
% 0.30/0.47 ! [A: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 0.30/0.47 = ( A = zero_zero_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % power2_less_eq_zero_iff
% 0.30/0.47 thf(fact_291_int__if,axiom,
% 0.30/0.47 ! [P: $o,A: nat,B: nat] :
% 0.30/0.47 ( ( P
% 0.30/0.47 => ( ( semiri2019852685at_int @ ( if_nat @ P @ A @ B ) )
% 0.30/0.47 = ( semiri2019852685at_int @ A ) ) )
% 0.30/0.47 & ( ~ P
% 0.30/0.47 => ( ( semiri2019852685at_int @ ( if_nat @ P @ A @ B ) )
% 0.30/0.47 = ( semiri2019852685at_int @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % int_if
% 0.30/0.47 thf(fact_292_nat__int__comparison_I1_J,axiom,
% 0.30/0.47 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.30/0.47 = ( ^ [A3: nat,B2: nat] :
% 0.30/0.47 ( ( semiri2019852685at_int @ A3 )
% 0.30/0.47 = ( semiri2019852685at_int @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_int_comparison(1)
% 0.30/0.47 thf(fact_293_int__ops_I1_J,axiom,
% 0.30/0.47 ( ( semiri2019852685at_int @ zero_zero_nat )
% 0.30/0.47 = zero_zero_int ) ).
% 0.30/0.47
% 0.30/0.47 % int_ops(1)
% 0.30/0.47 thf(fact_294_zero__le__imp__eq__int,axiom,
% 0.30/0.47 ! [K: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.47 => ? [N3: nat] :
% 0.30/0.47 ( K
% 0.30/0.47 = ( semiri2019852685at_int @ N3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_le_imp_eq_int
% 0.30/0.47 thf(fact_295_nonneg__int__cases,axiom,
% 0.30/0.47 ! [K: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.47 => ~ ! [N3: nat] :
% 0.30/0.47 ( K
% 0.30/0.47 != ( semiri2019852685at_int @ N3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nonneg_int_cases
% 0.30/0.47 thf(fact_296_int__int__eq,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ( semiri2019852685at_int @ M )
% 0.30/0.47 = ( semiri2019852685at_int @ N ) )
% 0.30/0.47 = ( M = N ) ) ).
% 0.30/0.47
% 0.30/0.47 % int_int_eq
% 0.30/0.47 thf(fact_297_nat__eq__iff2,axiom,
% 0.30/0.47 ! [M: nat,W: int] :
% 0.30/0.47 ( ( M
% 0.30/0.47 = ( nat2 @ W ) )
% 0.30/0.47 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.47 => ( W
% 0.30/0.47 = ( semiri2019852685at_int @ M ) ) )
% 0.30/0.47 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.47 => ( M = zero_zero_nat ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_eq_iff2
% 0.30/0.47 thf(fact_298_nat__eq__iff,axiom,
% 0.30/0.47 ! [W: int,M: nat] :
% 0.30/0.47 ( ( ( nat2 @ W )
% 0.30/0.47 = M )
% 0.30/0.47 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.47 => ( W
% 0.30/0.47 = ( semiri2019852685at_int @ M ) ) )
% 0.30/0.47 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.47 => ( M = zero_zero_nat ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_eq_iff
% 0.30/0.47 thf(fact_299_zero__reorient,axiom,
% 0.30/0.47 ! [X: real] :
% 0.30/0.47 ( ( zero_zero_real = X )
% 0.30/0.47 = ( X = zero_zero_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_reorient
% 0.30/0.47 thf(fact_300_zero__reorient,axiom,
% 0.30/0.47 ! [X: nat] :
% 0.30/0.47 ( ( zero_zero_nat = X )
% 0.30/0.47 = ( X = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_reorient
% 0.30/0.47 thf(fact_301_zero__reorient,axiom,
% 0.30/0.47 ! [X: int] :
% 0.30/0.47 ( ( zero_zero_int = X )
% 0.30/0.47 = ( X = zero_zero_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_reorient
% 0.30/0.47 thf(fact_302_zero__le,axiom,
% 0.30/0.47 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.30/0.47
% 0.30/0.47 % zero_le
% 0.30/0.47 thf(fact_303_nat__zero__as__int,axiom,
% 0.30/0.47 ( zero_zero_nat
% 0.30/0.47 = ( nat2 @ zero_zero_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_zero_as_int
% 0.30/0.47 thf(fact_304_power__0__left,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ( N = zero_zero_nat )
% 0.30/0.47 => ( ( power_power_int @ zero_zero_int @ N )
% 0.30/0.47 = one_one_int ) )
% 0.30/0.47 & ( ( N != zero_zero_nat )
% 0.30/0.47 => ( ( power_power_int @ zero_zero_int @ N )
% 0.30/0.47 = zero_zero_int ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_0_left
% 0.30/0.47 thf(fact_305_power__0__left,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ( N = zero_zero_nat )
% 0.30/0.47 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.30/0.47 = one_one_nat ) )
% 0.30/0.47 & ( ( N != zero_zero_nat )
% 0.30/0.47 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.30/0.47 = zero_zero_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_0_left
% 0.30/0.47 thf(fact_306_power__0__left,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ( N = zero_zero_nat )
% 0.30/0.47 => ( ( power_power_real @ zero_zero_real @ N )
% 0.30/0.47 = one_one_real ) )
% 0.30/0.47 & ( ( N != zero_zero_nat )
% 0.30/0.47 => ( ( power_power_real @ zero_zero_real @ N )
% 0.30/0.47 = zero_zero_real ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_0_left
% 0.30/0.47 thf(fact_307_nat__0__le,axiom,
% 0.30/0.47 ! [Z: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.47 => ( ( semiri2019852685at_int @ ( nat2 @ Z ) )
% 0.30/0.47 = Z ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_0_le
% 0.30/0.47 thf(fact_308_int__eq__iff,axiom,
% 0.30/0.47 ! [M: nat,Z: int] :
% 0.30/0.47 ( ( ( semiri2019852685at_int @ M )
% 0.30/0.47 = Z )
% 0.30/0.47 = ( ( M
% 0.30/0.47 = ( nat2 @ Z ) )
% 0.30/0.47 & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % int_eq_iff
% 0.30/0.47 thf(fact_309_int__ops_I2_J,axiom,
% 0.30/0.47 ( ( semiri2019852685at_int @ one_one_nat )
% 0.30/0.47 = one_one_int ) ).
% 0.30/0.47
% 0.30/0.47 % int_ops(2)
% 0.30/0.47 thf(fact_310_le__numeral__extra_I3_J,axiom,
% 0.30/0.47 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.30/0.47
% 0.30/0.47 % le_numeral_extra(3)
% 0.30/0.47 thf(fact_311_le__numeral__extra_I3_J,axiom,
% 0.30/0.47 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.30/0.47
% 0.30/0.47 % le_numeral_extra(3)
% 0.30/0.47 thf(fact_312_le__numeral__extra_I3_J,axiom,
% 0.30/0.47 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.30/0.47
% 0.30/0.47 % le_numeral_extra(3)
% 0.30/0.47 thf(fact_313_zero__neq__numeral,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( zero_zero_int
% 0.30/0.47 != ( numeral_numeral_int @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_neq_numeral
% 0.30/0.47 thf(fact_314_zero__neq__numeral,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( zero_zero_real
% 0.30/0.47 != ( numeral_numeral_real @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_neq_numeral
% 0.30/0.47 thf(fact_315_zero__neq__numeral,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( zero_zero_nat
% 0.30/0.47 != ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_neq_numeral
% 0.30/0.47 thf(fact_316_power__not__zero,axiom,
% 0.30/0.47 ! [A: int,N: nat] :
% 0.30/0.47 ( ( A != zero_zero_int )
% 0.30/0.47 => ( ( power_power_int @ A @ N )
% 0.30/0.47 != zero_zero_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_not_zero
% 0.30/0.47 thf(fact_317_power__not__zero,axiom,
% 0.30/0.47 ! [A: nat,N: nat] :
% 0.30/0.47 ( ( A != zero_zero_nat )
% 0.30/0.47 => ( ( power_power_nat @ A @ N )
% 0.30/0.47 != zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_not_zero
% 0.30/0.47 thf(fact_318_power__not__zero,axiom,
% 0.30/0.47 ! [A: real,N: nat] :
% 0.30/0.47 ( ( A != zero_zero_real )
% 0.30/0.47 => ( ( power_power_real @ A @ N )
% 0.30/0.47 != zero_zero_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_not_zero
% 0.30/0.47 thf(fact_319_exists__least__lemma,axiom,
% 0.30/0.47 ! [P: nat > $o] :
% 0.30/0.47 ( ~ ( P @ zero_zero_nat )
% 0.30/0.47 => ( ? [X_1: nat] : ( P @ X_1 )
% 0.30/0.47 => ? [N3: nat] :
% 0.30/0.47 ( ~ ( P @ N3 )
% 0.30/0.47 & ( P @ ( suc @ N3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % exists_least_lemma
% 0.30/0.47 thf(fact_320_nat_Odistinct_I1_J,axiom,
% 0.30/0.47 ! [X2: nat] :
% 0.30/0.47 ( zero_zero_nat
% 0.30/0.47 != ( suc @ X2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat.distinct(1)
% 0.30/0.47 thf(fact_321_old_Onat_Odistinct_I2_J,axiom,
% 0.30/0.47 ! [Nat2: nat] :
% 0.30/0.47 ( ( suc @ Nat2 )
% 0.30/0.47 != zero_zero_nat ) ).
% 0.30/0.47
% 0.30/0.47 % old.nat.distinct(2)
% 0.30/0.47 thf(fact_322_old_Onat_Odistinct_I1_J,axiom,
% 0.30/0.47 ! [Nat2: nat] :
% 0.30/0.47 ( zero_zero_nat
% 0.30/0.47 != ( suc @ Nat2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.nat.distinct(1)
% 0.30/0.47 thf(fact_323_nat_OdiscI,axiom,
% 0.30/0.47 ! [Nat: nat,X2: nat] :
% 0.30/0.47 ( ( Nat
% 0.30/0.47 = ( suc @ X2 ) )
% 0.30/0.47 => ( Nat != zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat.discI
% 0.30/0.47 thf(fact_324_nat__induct,axiom,
% 0.30/0.47 ! [P: nat > $o,N: nat] :
% 0.30/0.47 ( ( P @ zero_zero_nat )
% 0.30/0.47 => ( ! [N3: nat] :
% 0.30/0.47 ( ( P @ N3 )
% 0.30/0.47 => ( P @ ( suc @ N3 ) ) )
% 0.30/0.47 => ( P @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_induct
% 0.30/0.47 thf(fact_325_One__nat__def,axiom,
% 0.30/0.47 ( one_one_nat
% 0.30/0.47 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % One_nat_def
% 0.30/0.47 thf(fact_326_diff__induct,axiom,
% 0.30/0.47 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.30/0.47 ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
% 0.30/0.47 => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
% 0.30/0.47 => ( ! [X5: nat,Y3: nat] :
% 0.30/0.47 ( ( P @ X5 @ Y3 )
% 0.30/0.47 => ( P @ ( suc @ X5 ) @ ( suc @ Y3 ) ) )
% 0.30/0.47 => ( P @ M @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_induct
% 0.30/0.47 thf(fact_327_zero__induct,axiom,
% 0.30/0.47 ! [P: nat > $o,K: nat] :
% 0.30/0.47 ( ( P @ K )
% 0.30/0.47 => ( ! [N3: nat] :
% 0.30/0.47 ( ( P @ ( suc @ N3 ) )
% 0.30/0.47 => ( P @ N3 ) )
% 0.30/0.47 => ( P @ zero_zero_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_induct
% 0.30/0.47 thf(fact_328_Suc__neq__Zero,axiom,
% 0.30/0.47 ! [M: nat] :
% 0.30/0.47 ( ( suc @ M )
% 0.30/0.47 != zero_zero_nat ) ).
% 0.30/0.47
% 0.30/0.47 % Suc_neq_Zero
% 0.30/0.47 thf(fact_329_Zero__neq__Suc,axiom,
% 0.30/0.47 ! [M: nat] :
% 0.30/0.47 ( zero_zero_nat
% 0.30/0.47 != ( suc @ M ) ) ).
% 0.30/0.47
% 0.30/0.47 % Zero_neq_Suc
% 0.30/0.47 thf(fact_330_Zero__not__Suc,axiom,
% 0.30/0.47 ! [M: nat] :
% 0.30/0.47 ( zero_zero_nat
% 0.30/0.47 != ( suc @ M ) ) ).
% 0.30/0.47
% 0.30/0.47 % Zero_not_Suc
% 0.30/0.47 thf(fact_331_old_Onat_Oexhaust,axiom,
% 0.30/0.47 ! [Y: nat] :
% 0.30/0.47 ( ( Y != zero_zero_nat )
% 0.30/0.47 => ~ ! [Nat3: nat] :
% 0.30/0.47 ( Y
% 0.30/0.47 != ( suc @ Nat3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.nat.exhaust
% 0.30/0.47 thf(fact_332_old_Onat_Oinducts,axiom,
% 0.30/0.47 ! [P: nat > $o,Nat: nat] :
% 0.30/0.47 ( ( P @ zero_zero_nat )
% 0.30/0.47 => ( ! [Nat3: nat] :
% 0.30/0.47 ( ( P @ Nat3 )
% 0.30/0.47 => ( P @ ( suc @ Nat3 ) ) )
% 0.30/0.47 => ( P @ Nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % old.nat.inducts
% 0.30/0.47 thf(fact_333_not0__implies__Suc,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( N != zero_zero_nat )
% 0.30/0.47 => ? [M4: nat] :
% 0.30/0.47 ( N
% 0.30/0.47 = ( suc @ M4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not0_implies_Suc
% 0.30/0.47 thf(fact_334_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.30/0.47 ! [A: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.47 => ( A = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_nat_0.extremum_uniqueI
% 0.30/0.47 thf(fact_335_bot__nat__0_Oextremum__unique,axiom,
% 0.30/0.47 ! [A: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.47 = ( A = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_nat_0.extremum_unique
% 0.30/0.47 thf(fact_336_le__0__eq,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.30/0.47 = ( N = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_0_eq
% 0.30/0.47 thf(fact_337_less__eq__nat_Osimps_I1_J,axiom,
% 0.30/0.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_nat.simps(1)
% 0.30/0.47 thf(fact_338_less__eq__int__code_I1_J,axiom,
% 0.30/0.47 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.30/0.47
% 0.30/0.47 % less_eq_int_code(1)
% 0.30/0.47 thf(fact_339_le__nat__iff,axiom,
% 0.30/0.47 ! [K: int,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.47 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 0.30/0.47 = ( ord_less_eq_int @ ( semiri2019852685at_int @ N ) @ K ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_nat_iff
% 0.30/0.47 thf(fact_340_nat__int__comparison_I3_J,axiom,
% 0.30/0.47 ( ord_less_eq_nat
% 0.30/0.47 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri2019852685at_int @ A3 ) @ ( semiri2019852685at_int @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_int_comparison(3)
% 0.30/0.47 thf(fact_341_zle__int,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 0.30/0.47 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % zle_int
% 0.30/0.47 thf(fact_342_int__ops_I3_J,axiom,
% 0.30/0.47 ! [N: num] :
% 0.30/0.47 ( ( semiri2019852685at_int @ ( numeral_numeral_nat @ N ) )
% 0.30/0.47 = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % int_ops(3)
% 0.30/0.47 thf(fact_343_ex__nat,axiom,
% 0.30/0.47 ( ( ^ [P2: nat > $o] :
% 0.30/0.47 ? [X6: nat] : ( P2 @ X6 ) )
% 0.30/0.47 = ( ^ [P3: nat > $o] :
% 0.30/0.47 ? [X3: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.47 & ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ex_nat
% 0.30/0.47 thf(fact_344_all__nat,axiom,
% 0.30/0.47 ( ( ^ [P2: nat > $o] :
% 0.30/0.47 ! [X6: nat] : ( P2 @ X6 ) )
% 0.30/0.47 = ( ^ [P3: nat > $o] :
% 0.30/0.47 ! [X3: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.47 => ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % all_nat
% 0.30/0.47 thf(fact_345_eq__nat__nat__iff,axiom,
% 0.30/0.47 ! [Z: int,Z5: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
% 0.30/0.47 => ( ( ( nat2 @ Z )
% 0.30/0.47 = ( nat2 @ Z5 ) )
% 0.30/0.47 = ( Z = Z5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_nat_nat_iff
% 0.30/0.47 thf(fact_346_powr__mono2,axiom,
% 0.30/0.47 ! [A: real,X: real,Y: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.47 => ( ( ord_less_eq_real @ X @ Y )
% 0.30/0.47 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % powr_mono2
% 0.30/0.47 thf(fact_347_powr__ge__pzero,axiom,
% 0.30/0.47 ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % powr_ge_pzero
% 0.30/0.47 thf(fact_348_pow_Osimps_I1_J,axiom,
% 0.30/0.47 ! [X: num] :
% 0.30/0.47 ( ( pow @ X @ one )
% 0.30/0.47 = X ) ).
% 0.30/0.47
% 0.30/0.47 % pow.simps(1)
% 0.30/0.47 thf(fact_349_nat__le__iff,axiom,
% 0.30/0.47 ! [X: int,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
% 0.30/0.47 = ( ord_less_eq_int @ X @ ( semiri2019852685at_int @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_le_iff
% 0.30/0.47
% 0.30/0.47 % Helper facts (3)
% 0.30/0.47 thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
% 0.30/0.47 ! [P: $o] :
% 0.30/0.47 ( ( P = $true )
% 0.30/0.47 | ( P = $false ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( if_nat @ $false @ X @ Y )
% 0.30/0.47 = Y ) ).
% 0.30/0.47
% 0.30/0.47 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( if_nat @ $true @ X @ Y )
% 0.30/0.47 = X ) ).
% 0.30/0.47
% 0.30/0.47 % Conjectures (1)
% 0.30/0.47 thf(conj_0,conjecture,
% 0.30/0.47 ord_less_eq_int @ d @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 0.32/1.50 ---- Embedded in THF ---
% 0.32/1.50 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.32/1.50 %%% Generated on Tue Feb 24 08:26:04 EST 2026
% 0.32/1.50 %%% using '$$dhol' embedding, version 1.3.0.
% 0.32/1.50 %%% Logic specification used:
% 0.32/1.50 %%% thf(spec, logic, $$dhol).
% 0.32/1.50
% 0.32/1.50 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.Z5U7N5EVnJ/DTF2DTF17924.p
% See solution above
% 0.32/1.50 ------------------------
% 0.32/1.54 ---- Cleaned THF ---
% 0.32/1.54 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.32/1.54 set_real: $tType ).
% 0.32/1.54
% 0.32/1.54 thf(ty_n_t__Real__Oreal,type,
% 0.32/1.54 real: $tType ).
% 0.32/1.54
% 0.32/1.54 thf(ty_n_t__Num__Onum,type,
% 0.32/1.54 num: $tType ).
% 0.32/1.54
% 0.32/1.54 thf(ty_n_t__Nat__Onat,type,
% 0.32/1.54 nat: $tType ).
% 0.32/1.54
% 0.32/1.54 thf(ty_n_t__Int__Oint,type,
% 0.32/1.54 int: $tType ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
% 0.32/1.54 archim1371465213g_real: real > int ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
% 0.32/1.54 one_one_int: int ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.32/1.54 one_one_nat: nat ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 0.32/1.54 one_one_real: real ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 0.32/1.54 zero_zero_int: int ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.32/1.54 zero_zero_nat: nat ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.32/1.54 zero_zero_real: real ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_If_001t__Nat__Onat,type,
% 0.32/1.54 if_nat: $o > nat > nat > nat ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Int_Onat,type,
% 0.32/1.54 nat2: int > nat ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 0.32/1.54 ring_1_of_int_int: int > int ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 0.32/1.54 ring_1_of_int_real: int > real ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Nat_OSuc,type,
% 0.32/1.54 suc: nat > nat ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.32/1.54 semiri2019852685at_int: nat > int ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.32/1.54 semiri1382578993at_nat: nat > nat ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 0.32/1.54 semiri2110766477t_real: nat > real ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 0.32/1.54 neg_numeral_dbl_int: int > int ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 0.32/1.54 neg_numeral_dbl_real: real > real ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Num_Onum_OBit0,type,
% 0.32/1.54 bit0: num > num ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Num_Onum_OOne,type,
% 0.32/1.54 one: num ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 0.32/1.54 numeral_numeral_int: num > int ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 0.32/1.54 numeral_numeral_nat: num > nat ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 0.32/1.54 numeral_numeral_real: num > real ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Num_Opow,type,
% 0.32/1.54 pow: num > num > num ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.32/1.54 ord_less_int: int > int > $o ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.32/1.54 ord_less_eq_int: int > int > $o ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.32/1.54 ord_less_eq_nat: nat > nat > $o ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
% 0.32/1.54 ord_less_eq_num: num > num > $o ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.32/1.54 ord_less_eq_real: real > real > $o ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 0.32/1.54 power_power_int: int > nat > int ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 0.32/1.54 power_power_nat: nat > nat > nat ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
% 0.32/1.54 power_power_real: real > nat > real ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.32/1.54 collect_real: ( real > $o ) > set_real ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Transcendental_Olog,type,
% 0.32/1.54 log: real > real > real ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 0.32/1.54 powr_real: real > real > real ).
% 0.32/1.54
% 0.32/1.54 thf(sy_c_member_001t__Real__Oreal,type,
% 0.32/1.54 member_real: real > set_real > $o ).
% 0.32/1.54
% 0.32/1.54 thf(sy_v_d____,type,
% 0.32/1.54 d: int ).
% 0.32/1.54
% 0.32/1.54 thf(sy_v_n____,type,
% 0.32/1.54 n: nat ).
% 0.32/1.54
% 0.32/1.54 thf(fact_0_numeral__power__le__of__int__cancel__iff,axiom,
% 0.32/1.54 ! [X: num,N: nat,A: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.32/1.54 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_1_numeral__power__le__of__int__cancel__iff,axiom,
% 0.32/1.54 ! [X: num,N: nat,A: int] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.32/1.54 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_2_of__int__le__numeral__power__cancel__iff,axiom,
% 0.32/1.54 ! [A: int,X: num,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.32/1.54 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_3_of__int__le__numeral__power__cancel__iff,axiom,
% 0.32/1.54 ! [A: int,X: num,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.32/1.54 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_4_ceiling__numeral__power,axiom,
% 0.32/1.54 ! [X: num,N: nat] :
% 0.32/1.54 ( ( archim1371465213g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.32/1.54 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_5_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.32/1.54 ! [X: num,N: nat,Y: int] :
% 0.32/1.54 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.32/1.54 = ( ring_1_of_int_int @ Y ) )
% 0.32/1.54 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.32/1.54 = Y ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_6_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.32/1.54 ! [X: num,N: nat,Y: int] :
% 0.32/1.54 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 0.32/1.54 = ( ring_1_of_int_real @ Y ) )
% 0.32/1.54 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.32/1.54 = Y ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_7_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.32/1.54 ! [Y: int,X: num,N: nat] :
% 0.32/1.54 ( ( ( ring_1_of_int_int @ Y )
% 0.32/1.54 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.32/1.54 = ( Y
% 0.32/1.54 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_8_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.32/1.54 ! [Y: int,X: num,N: nat] :
% 0.32/1.54 ( ( ( ring_1_of_int_real @ Y )
% 0.32/1.54 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.32/1.54 = ( Y
% 0.32/1.54 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_9_of__int__le__of__int__power__cancel__iff,axiom,
% 0.32/1.54 ! [B: int,W: nat,X: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.32/1.54 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_10_of__int__le__of__int__power__cancel__iff,axiom,
% 0.32/1.54 ! [B: int,W: nat,X: int] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.32/1.54 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_11_of__int__power__le__of__int__cancel__iff,axiom,
% 0.32/1.54 ! [X: int,B: int,W: nat] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.32/1.54 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_12_of__int__power__le__of__int__cancel__iff,axiom,
% 0.32/1.54 ! [X: int,B: int,W: nat] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.32/1.54 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_13_ceiling__le__numeral,axiom,
% 0.32/1.54 ! [X: real,V: num] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.32/1.54 = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_14_of__int__le__numeral__iff,axiom,
% 0.32/1.54 ! [Z: int,N: num] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.32/1.54 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_15_of__int__le__numeral__iff,axiom,
% 0.32/1.54 ! [Z: int,N: num] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.32/1.54 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_16_of__int__numeral__le__iff,axiom,
% 0.32/1.54 ! [N: num,Z: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.32/1.54 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_17_of__int__numeral__le__iff,axiom,
% 0.32/1.54 ! [N: num,Z: int] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.32/1.54 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_18_of__int__power,axiom,
% 0.32/1.54 ! [Z: int,N: nat] :
% 0.32/1.54 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 0.32/1.54 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_19_of__int__power,axiom,
% 0.32/1.54 ! [Z: int,N: nat] :
% 0.32/1.54 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 0.32/1.54 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_20_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.32/1.54 ! [B: int,W: nat,X: int] :
% 0.32/1.54 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 0.32/1.54 = ( ring_1_of_int_int @ X ) )
% 0.32/1.54 = ( ( power_power_int @ B @ W )
% 0.32/1.54 = X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_21_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.32/1.54 ! [B: int,W: nat,X: int] :
% 0.32/1.54 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 0.32/1.54 = ( ring_1_of_int_real @ X ) )
% 0.32/1.54 = ( ( power_power_int @ B @ W )
% 0.32/1.54 = X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_22_of__int__eq__iff,axiom,
% 0.32/1.54 ! [W: int,Z: int] :
% 0.32/1.54 ( ( ( ring_1_of_int_real @ W )
% 0.32/1.54 = ( ring_1_of_int_real @ Z ) )
% 0.32/1.54 = ( W = Z ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_23_of__int__eq__iff,axiom,
% 0.32/1.54 ! [W: int,Z: int] :
% 0.32/1.54 ( ( ( ring_1_of_int_int @ W )
% 0.32/1.54 = ( ring_1_of_int_int @ Z ) )
% 0.32/1.54 = ( W = Z ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_24_nat__numeral,axiom,
% 0.32/1.54 ! [K: num] :
% 0.32/1.54 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 0.32/1.54 = ( numeral_numeral_nat @ K ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_25_ceiling__of__int,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( archim1371465213g_real @ ( ring_1_of_int_real @ Z ) )
% 0.32/1.54 = Z ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_26_of__int__ceiling__cancel,axiom,
% 0.32/1.54 ! [X: real] :
% 0.32/1.54 ( ( ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) )
% 0.32/1.54 = X )
% 0.32/1.54 = ( ? [N2: int] :
% 0.32/1.54 ( X
% 0.32/1.54 = ( ring_1_of_int_real @ N2 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_27_of__int__le__iff,axiom,
% 0.32/1.54 ! [W: int,Z: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.32/1.54 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_28_of__int__le__iff,axiom,
% 0.32/1.54 ! [W: int,Z: int] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.32/1.54 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_29_of__int__numeral,axiom,
% 0.32/1.54 ! [K: num] :
% 0.32/1.54 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 0.32/1.54 = ( numeral_numeral_int @ K ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_30_of__int__numeral,axiom,
% 0.32/1.54 ! [K: num] :
% 0.32/1.54 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 0.32/1.54 = ( numeral_numeral_real @ K ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_31_of__int__eq__numeral__iff,axiom,
% 0.32/1.54 ! [Z: int,N: num] :
% 0.32/1.54 ( ( ( ring_1_of_int_int @ Z )
% 0.32/1.54 = ( numeral_numeral_int @ N ) )
% 0.32/1.54 = ( Z
% 0.32/1.54 = ( numeral_numeral_int @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_32_of__int__eq__numeral__iff,axiom,
% 0.32/1.54 ! [Z: int,N: num] :
% 0.32/1.54 ( ( ( ring_1_of_int_real @ Z )
% 0.32/1.54 = ( numeral_numeral_real @ N ) )
% 0.32/1.54 = ( Z
% 0.32/1.54 = ( numeral_numeral_int @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_33_ceiling__numeral,axiom,
% 0.32/1.54 ! [V: num] :
% 0.32/1.54 ( ( archim1371465213g_real @ ( numeral_numeral_real @ V ) )
% 0.32/1.54 = ( numeral_numeral_int @ V ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_34_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.32/1.54 ! [X: int,B: int,W: nat] :
% 0.32/1.54 ( ( ( ring_1_of_int_int @ X )
% 0.32/1.54 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.32/1.54 = ( X
% 0.32/1.54 = ( power_power_int @ B @ W ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_35_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.32/1.54 ! [X: int,B: int,W: nat] :
% 0.32/1.54 ( ( ( ring_1_of_int_real @ X )
% 0.32/1.54 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.32/1.54 = ( X
% 0.32/1.54 = ( power_power_int @ B @ W ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_36_numeral__power__eq__nat__cancel__iff,axiom,
% 0.32/1.54 ! [X: num,N: nat,Y: int] :
% 0.32/1.54 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/1.54 = ( nat2 @ Y ) )
% 0.32/1.54 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.32/1.54 = Y ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_37_nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/1.54 ! [Y: int,X: num,N: nat] :
% 0.32/1.54 ( ( ( nat2 @ Y )
% 0.32/1.54 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.32/1.54 = ( Y
% 0.32/1.54 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_38_numeral__power__le__nat__cancel__iff,axiom,
% 0.32/1.54 ! [X: num,N: nat,A: int] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 0.32/1.54 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_39_nat__le__numeral__power__cancel__iff,axiom,
% 0.32/1.54 ! [A: int,X: num,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.32/1.54 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_40_nat__mono,axiom,
% 0.32/1.54 ! [X: int,Y: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ X @ Y )
% 0.32/1.54 => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_41_ex__le__of__int,axiom,
% 0.32/1.54 ! [X: real] :
% 0.32/1.54 ? [Z2: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_42_le__of__int__ceiling,axiom,
% 0.32/1.54 ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_43_ceiling__mono,axiom,
% 0.32/1.54 ! [Y: real,X: real] :
% 0.32/1.54 ( ( ord_less_eq_real @ Y @ X )
% 0.32/1.54 => ( ord_less_eq_int @ ( archim1371465213g_real @ Y ) @ ( archim1371465213g_real @ X ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_44_ceiling__le__iff,axiom,
% 0.32/1.54 ! [X: real,Z: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ Z )
% 0.32/1.54 = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_45_ceiling__le,axiom,
% 0.32/1.54 ! [X: real,A: int] :
% 0.32/1.54 ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
% 0.32/1.54 => ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ A ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_46__092_060open_062log_A2_A1_A_092_060le_062_Alog_A2_A_Ireal__of__int_Ad_J_092_060close_062,axiom,
% 0.32/1.54 ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ one_one_real ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_47_semiring__norm_I69_J,axiom,
% 0.32/1.54 ! [M: num] :
% 0.32/1.54 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_48_semiring__norm_I83_J,axiom,
% 0.32/1.54 ! [N: num] :
% 0.32/1.54 ( one
% 0.32/1.54 != ( bit0 @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_49_semiring__norm_I85_J,axiom,
% 0.32/1.54 ! [M: num] :
% 0.32/1.54 ( ( bit0 @ M )
% 0.32/1.54 != one ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_50_numeral__le__iff,axiom,
% 0.32/1.54 ! [M: num,N: num] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/1.54 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_51_numeral__le__iff,axiom,
% 0.32/1.54 ! [M: num,N: num] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/1.54 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_52_numeral__le__iff,axiom,
% 0.32/1.54 ! [M: num,N: num] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.32/1.54 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_53__092_060open_062real__of__int_Ad_A_061_A2_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_092_060close_062,axiom,
% 0.32/1.54 ( ( ring_1_of_int_real @ d )
% 0.32/1.54 = ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_54_n__def,axiom,
% 0.32/1.54 ( n
% 0.32/1.54 = ( suc @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_55_semiring__norm_I68_J,axiom,
% 0.32/1.54 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_56_semiring__norm_I71_J,axiom,
% 0.32/1.54 ! [M: num,N: num] :
% 0.32/1.54 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/1.54 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_57_semiring__norm_I87_J,axiom,
% 0.32/1.54 ! [M: num,N: num] :
% 0.32/1.54 ( ( ( bit0 @ M )
% 0.32/1.54 = ( bit0 @ N ) )
% 0.32/1.54 = ( M = N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_58_verit__eq__simplify_I8_J,axiom,
% 0.32/1.54 ! [X2: num,Y2: num] :
% 0.32/1.54 ( ( ( bit0 @ X2 )
% 0.32/1.54 = ( bit0 @ Y2 ) )
% 0.32/1.54 = ( X2 = Y2 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_59_numeral__eq__iff,axiom,
% 0.32/1.54 ! [M: num,N: num] :
% 0.32/1.54 ( ( ( numeral_numeral_int @ M )
% 0.32/1.54 = ( numeral_numeral_int @ N ) )
% 0.32/1.54 = ( M = N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_60_numeral__eq__iff,axiom,
% 0.32/1.54 ! [M: num,N: num] :
% 0.32/1.54 ( ( ( numeral_numeral_real @ M )
% 0.32/1.54 = ( numeral_numeral_real @ N ) )
% 0.32/1.54 = ( M = N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_61_numeral__eq__iff,axiom,
% 0.32/1.54 ! [M: num,N: num] :
% 0.32/1.54 ( ( ( numeral_numeral_nat @ M )
% 0.32/1.54 = ( numeral_numeral_nat @ N ) )
% 0.32/1.54 = ( M = N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_62_of__int__eq__1__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ( ring_1_of_int_real @ Z )
% 0.32/1.54 = one_one_real )
% 0.32/1.54 = ( Z = one_one_int ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_63_of__int__eq__1__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ( ring_1_of_int_int @ Z )
% 0.32/1.54 = one_one_int )
% 0.32/1.54 = ( Z = one_one_int ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_64_mem__Collect__eq,axiom,
% 0.32/1.54 ! [A: real,P: real > $o] :
% 0.32/1.54 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.32/1.54 = ( P @ A ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_65_Collect__mem__eq,axiom,
% 0.32/1.54 ! [A2: set_real] :
% 0.32/1.54 ( ( collect_real
% 0.32/1.54 @ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
% 0.32/1.54 = A2 ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_66_of__int__1,axiom,
% 0.32/1.54 ( ( ring_1_of_int_real @ one_one_int )
% 0.32/1.54 = one_one_real ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_67_of__int__1,axiom,
% 0.32/1.54 ( ( ring_1_of_int_int @ one_one_int )
% 0.32/1.54 = one_one_int ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_68_ceiling__one,axiom,
% 0.32/1.54 ( ( archim1371465213g_real @ one_one_real )
% 0.32/1.54 = one_one_int ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_69_numeral__eq__one__iff,axiom,
% 0.32/1.54 ! [N: num] :
% 0.32/1.54 ( ( ( numeral_numeral_int @ N )
% 0.32/1.54 = one_one_int )
% 0.32/1.54 = ( N = one ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_70_numeral__eq__one__iff,axiom,
% 0.32/1.54 ! [N: num] :
% 0.32/1.54 ( ( ( numeral_numeral_real @ N )
% 0.32/1.54 = one_one_real )
% 0.32/1.54 = ( N = one ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_71_numeral__eq__one__iff,axiom,
% 0.32/1.54 ! [N: num] :
% 0.32/1.54 ( ( ( numeral_numeral_nat @ N )
% 0.32/1.54 = one_one_nat )
% 0.32/1.54 = ( N = one ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_72_one__eq__numeral__iff,axiom,
% 0.32/1.54 ! [N: num] :
% 0.32/1.54 ( ( one_one_int
% 0.32/1.54 = ( numeral_numeral_int @ N ) )
% 0.32/1.54 = ( one = N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_73_one__eq__numeral__iff,axiom,
% 0.32/1.54 ! [N: num] :
% 0.32/1.54 ( ( one_one_real
% 0.32/1.54 = ( numeral_numeral_real @ N ) )
% 0.32/1.54 = ( one = N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_74_one__eq__numeral__iff,axiom,
% 0.32/1.54 ! [N: num] :
% 0.32/1.54 ( ( one_one_nat
% 0.32/1.54 = ( numeral_numeral_nat @ N ) )
% 0.32/1.54 = ( one = N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_75_of__int__1__le__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.32/1.54 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_76_of__int__1__le__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.32/1.54 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_77_of__int__le__1__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.32/1.54 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_78_of__int__le__1__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.32/1.54 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_79_ceiling__le__one,axiom,
% 0.32/1.54 ! [X: real] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ one_one_int )
% 0.32/1.54 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_80_numeral__le__one__iff,axiom,
% 0.32/1.54 ! [N: num] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.32/1.54 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_81_numeral__le__one__iff,axiom,
% 0.32/1.54 ! [N: num] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.32/1.54 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_82_numeral__le__one__iff,axiom,
% 0.32/1.54 ! [N: num] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.32/1.54 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_83_complete__real,axiom,
% 0.32/1.54 ! [S: set_real] :
% 0.32/1.54 ( ? [X4: real] : ( member_real @ X4 @ S )
% 0.32/1.54 => ( ? [Z3: real] :
% 0.32/1.54 ! [X5: real] :
% 0.32/1.54 ( ( member_real @ X5 @ S )
% 0.32/1.54 => ( ord_less_eq_real @ X5 @ Z3 ) )
% 0.32/1.54 => ? [Y3: real] :
% 0.32/1.54 ( ! [X4: real] :
% 0.32/1.54 ( ( member_real @ X4 @ S )
% 0.32/1.54 => ( ord_less_eq_real @ X4 @ Y3 ) )
% 0.32/1.54 & ! [Z3: real] :
% 0.32/1.54 ( ! [X5: real] :
% 0.32/1.54 ( ( member_real @ X5 @ S )
% 0.32/1.54 => ( ord_less_eq_real @ X5 @ Z3 ) )
% 0.32/1.54 => ( ord_less_eq_real @ Y3 @ Z3 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_84_le__numeral__extra_I4_J,axiom,
% 0.32/1.54 ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.32/1.54
% 0.32/1.54 thf(fact_85_le__numeral__extra_I4_J,axiom,
% 0.32/1.54 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.32/1.54
% 0.32/1.54 thf(fact_86_le__numeral__extra_I4_J,axiom,
% 0.32/1.54 ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.32/1.54
% 0.32/1.54 thf(fact_87_one__le__numeral,axiom,
% 0.32/1.54 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_88_one__le__numeral,axiom,
% 0.32/1.54 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_89_one__le__numeral,axiom,
% 0.32/1.54 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_90_numeral__One,axiom,
% 0.32/1.54 ( ( numeral_numeral_int @ one )
% 0.32/1.54 = one_one_int ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_91_numeral__One,axiom,
% 0.32/1.54 ( ( numeral_numeral_real @ one )
% 0.32/1.54 = one_one_real ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_92_numeral__One,axiom,
% 0.32/1.54 ( ( numeral_numeral_nat @ one )
% 0.32/1.54 = one_one_nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_93_verit__la__disequality,axiom,
% 0.32/1.54 ! [A: int,B: int] :
% 0.32/1.54 ( ( A = B )
% 0.32/1.54 | ~ ( ord_less_eq_int @ A @ B )
% 0.32/1.54 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_94_verit__la__disequality,axiom,
% 0.32/1.54 ! [A: nat,B: nat] :
% 0.32/1.54 ( ( A = B )
% 0.32/1.54 | ~ ( ord_less_eq_nat @ A @ B )
% 0.32/1.54 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_95_verit__la__disequality,axiom,
% 0.32/1.54 ! [A: real,B: real] :
% 0.32/1.54 ( ( A = B )
% 0.32/1.54 | ~ ( ord_less_eq_real @ A @ B )
% 0.32/1.54 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_96_verit__la__disequality,axiom,
% 0.32/1.54 ! [A: num,B: num] :
% 0.32/1.54 ( ( A = B )
% 0.32/1.54 | ~ ( ord_less_eq_num @ A @ B )
% 0.32/1.54 | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_97_le__num__One__iff,axiom,
% 0.32/1.54 ! [X: num] :
% 0.32/1.54 ( ( ord_less_eq_num @ X @ one )
% 0.32/1.54 = ( X = one ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_98_two__realpow__ge__one,axiom,
% 0.32/1.54 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_99_verit__la__generic,axiom,
% 0.32/1.54 ! [A: int,X: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ A @ X )
% 0.32/1.54 | ( A = X )
% 0.32/1.54 | ( ord_less_eq_int @ X @ A ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_100_verit__eq__simplify_I10_J,axiom,
% 0.32/1.54 ! [X2: num] :
% 0.32/1.54 ( one
% 0.32/1.54 != ( bit0 @ X2 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_101_nat__numeral__as__int,axiom,
% 0.32/1.54 ( numeral_numeral_nat
% 0.32/1.54 = ( ^ [I: num] : ( nat2 @ ( numeral_numeral_int @ I ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_102__092_060open_0622_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_A_061_Areal__of__int_A_I2_A_094_Anat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062,axiom,
% 0.32/1.54 ( ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) )
% 0.32/1.54 = ( ring_1_of_int_real @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_103__092_060open_0622_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_A_092_060le_062_A2_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062,axiom,
% 0.32/1.54 ord_less_eq_real @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) @ ( powr_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_104_numeral__powr__numeral__real,axiom,
% 0.32/1.54 ! [M: num,N: num] :
% 0.32/1.54 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.32/1.54 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_105_powr__one__eq__one,axiom,
% 0.32/1.54 ! [A: real] :
% 0.32/1.54 ( ( powr_real @ one_one_real @ A )
% 0.32/1.54 = one_one_real ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_106_self__le__ge2__pow,axiom,
% 0.32/1.54 ! [K: nat,M: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.32/1.54 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_107_power2__nat__le__eq__le,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/1.54 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_108_power2__nat__le__imp__le,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.32/1.54 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_109_Suc__le__mono,axiom,
% 0.32/1.54 ! [N: nat,M: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.32/1.54 = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_110_power__one__right,axiom,
% 0.32/1.54 ! [A: int] :
% 0.32/1.54 ( ( power_power_int @ A @ one_one_nat )
% 0.32/1.54 = A ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_111_power__one__right,axiom,
% 0.32/1.54 ! [A: nat] :
% 0.32/1.54 ( ( power_power_nat @ A @ one_one_nat )
% 0.32/1.54 = A ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_112_power__one__right,axiom,
% 0.32/1.54 ! [A: real] :
% 0.32/1.54 ( ( power_power_real @ A @ one_one_nat )
% 0.32/1.54 = A ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_113_nat_Oinject,axiom,
% 0.32/1.54 ! [X2: nat,Y2: nat] :
% 0.32/1.54 ( ( ( suc @ X2 )
% 0.32/1.54 = ( suc @ Y2 ) )
% 0.32/1.54 = ( X2 = Y2 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_114_old_Onat_Oinject,axiom,
% 0.32/1.54 ! [Nat: nat,Nat2: nat] :
% 0.32/1.54 ( ( ( suc @ Nat )
% 0.32/1.54 = ( suc @ Nat2 ) )
% 0.32/1.54 = ( Nat = Nat2 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_115_of__nat__eq__iff,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ( semiri2110766477t_real @ M )
% 0.32/1.54 = ( semiri2110766477t_real @ N ) )
% 0.32/1.54 = ( M = N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_116_of__nat__eq__iff,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ( semiri2019852685at_int @ M )
% 0.32/1.54 = ( semiri2019852685at_int @ N ) )
% 0.32/1.54 = ( M = N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_117_power__one,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( power_power_int @ one_one_int @ N )
% 0.32/1.54 = one_one_int ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_118_power__one,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( power_power_nat @ one_one_nat @ N )
% 0.32/1.54 = one_one_nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_119_power__one,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( power_power_real @ one_one_real @ N )
% 0.32/1.54 = one_one_real ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_120_of__nat__eq__1__iff,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( ( semiri1382578993at_nat @ N )
% 0.32/1.54 = one_one_nat )
% 0.32/1.54 = ( N = one_one_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_121_of__nat__eq__1__iff,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( ( semiri2110766477t_real @ N )
% 0.32/1.54 = one_one_real )
% 0.32/1.54 = ( N = one_one_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_122_of__nat__eq__1__iff,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( ( semiri2019852685at_int @ N )
% 0.32/1.54 = one_one_int )
% 0.32/1.54 = ( N = one_one_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_123_of__nat__1__eq__iff,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( one_one_nat
% 0.32/1.54 = ( semiri1382578993at_nat @ N ) )
% 0.32/1.54 = ( N = one_one_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_124_of__nat__1__eq__iff,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( one_one_real
% 0.32/1.54 = ( semiri2110766477t_real @ N ) )
% 0.32/1.54 = ( N = one_one_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_125_of__nat__1__eq__iff,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( one_one_int
% 0.32/1.54 = ( semiri2019852685at_int @ N ) )
% 0.32/1.54 = ( N = one_one_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_126_of__nat__1,axiom,
% 0.32/1.54 ( ( semiri1382578993at_nat @ one_one_nat )
% 0.32/1.54 = one_one_nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_127_of__nat__1,axiom,
% 0.32/1.54 ( ( semiri2110766477t_real @ one_one_nat )
% 0.32/1.54 = one_one_real ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_128_of__nat__1,axiom,
% 0.32/1.54 ( ( semiri2019852685at_int @ one_one_nat )
% 0.32/1.54 = one_one_int ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_129_of__int__of__nat__eq,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( ring_1_of_int_real @ ( semiri2019852685at_int @ N ) )
% 0.32/1.54 = ( semiri2110766477t_real @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_130_of__int__of__nat__eq,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( ring_1_of_int_int @ ( semiri2019852685at_int @ N ) )
% 0.32/1.54 = ( semiri2019852685at_int @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_131_ceiling__of__nat,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( archim1371465213g_real @ ( semiri2110766477t_real @ N ) )
% 0.32/1.54 = ( semiri2019852685at_int @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_132_of__nat__le__iff,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 0.32/1.54 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_133_of__nat__le__iff,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 0.32/1.54 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_134_of__nat__le__iff,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) )
% 0.32/1.54 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_135_of__nat__numeral,axiom,
% 0.32/1.54 ! [N: num] :
% 0.32/1.54 ( ( semiri1382578993at_nat @ ( numeral_numeral_nat @ N ) )
% 0.32/1.54 = ( numeral_numeral_nat @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_136_of__nat__numeral,axiom,
% 0.32/1.54 ! [N: num] :
% 0.32/1.54 ( ( semiri2110766477t_real @ ( numeral_numeral_nat @ N ) )
% 0.32/1.54 = ( numeral_numeral_real @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_137_of__nat__numeral,axiom,
% 0.32/1.54 ! [N: num] :
% 0.32/1.54 ( ( semiri2019852685at_int @ ( numeral_numeral_nat @ N ) )
% 0.32/1.54 = ( numeral_numeral_int @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_138_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/1.54 ! [X: nat,B: nat,W: nat] :
% 0.32/1.54 ( ( ( semiri1382578993at_nat @ X )
% 0.32/1.54 = ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) )
% 0.32/1.54 = ( X
% 0.32/1.54 = ( power_power_nat @ B @ W ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_139_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/1.54 ! [X: nat,B: nat,W: nat] :
% 0.32/1.54 ( ( ( semiri2110766477t_real @ X )
% 0.32/1.54 = ( power_power_real @ ( semiri2110766477t_real @ B ) @ W ) )
% 0.32/1.54 = ( X
% 0.32/1.54 = ( power_power_nat @ B @ W ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_140_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/1.54 ! [X: nat,B: nat,W: nat] :
% 0.32/1.54 ( ( ( semiri2019852685at_int @ X )
% 0.32/1.54 = ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) )
% 0.32/1.54 = ( X
% 0.32/1.54 = ( power_power_nat @ B @ W ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_141_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/1.54 ! [B: nat,W: nat,X: nat] :
% 0.32/1.54 ( ( ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W )
% 0.32/1.54 = ( semiri1382578993at_nat @ X ) )
% 0.32/1.54 = ( ( power_power_nat @ B @ W )
% 0.32/1.54 = X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_142_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/1.54 ! [B: nat,W: nat,X: nat] :
% 0.32/1.54 ( ( ( power_power_real @ ( semiri2110766477t_real @ B ) @ W )
% 0.32/1.54 = ( semiri2110766477t_real @ X ) )
% 0.32/1.54 = ( ( power_power_nat @ B @ W )
% 0.32/1.54 = X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_143_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/1.54 ! [B: nat,W: nat,X: nat] :
% 0.32/1.54 ( ( ( power_power_int @ ( semiri2019852685at_int @ B ) @ W )
% 0.32/1.54 = ( semiri2019852685at_int @ X ) )
% 0.32/1.54 = ( ( power_power_nat @ B @ W )
% 0.32/1.54 = X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_144_of__nat__power,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( semiri1382578993at_nat @ ( power_power_nat @ M @ N ) )
% 0.32/1.54 = ( power_power_nat @ ( semiri1382578993at_nat @ M ) @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_145_of__nat__power,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( semiri2110766477t_real @ ( power_power_nat @ M @ N ) )
% 0.32/1.54 = ( power_power_real @ ( semiri2110766477t_real @ M ) @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_146_of__nat__power,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( semiri2019852685at_int @ ( power_power_nat @ M @ N ) )
% 0.32/1.54 = ( power_power_int @ ( semiri2019852685at_int @ M ) @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_147_Suc__1,axiom,
% 0.32/1.54 ( ( suc @ one_one_nat )
% 0.32/1.54 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_148_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.32/1.54 ! [X: nat,B: nat,W: nat] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ X ) @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) )
% 0.32/1.54 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_149_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.32/1.54 ! [X: nat,B: nat,W: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ X ) @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) )
% 0.32/1.54 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_150_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.32/1.54 ! [X: nat,B: nat,W: nat] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( semiri2110766477t_real @ X ) @ ( power_power_real @ ( semiri2110766477t_real @ B ) @ W ) )
% 0.32/1.54 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_151_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.32/1.54 ! [B: nat,W: nat,X: nat] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) @ ( semiri2019852685at_int @ X ) )
% 0.32/1.54 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_152_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.32/1.54 ! [B: nat,W: nat,X: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) @ ( semiri1382578993at_nat @ X ) )
% 0.32/1.54 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_153_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.32/1.54 ! [B: nat,W: nat,X: nat] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri2110766477t_real @ B ) @ W ) @ ( semiri2110766477t_real @ X ) )
% 0.32/1.54 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_154_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/1.54 ! [Y: nat,X: num,N: nat] :
% 0.32/1.54 ( ( ( semiri1382578993at_nat @ Y )
% 0.32/1.54 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.32/1.54 = ( Y
% 0.32/1.54 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_155_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/1.54 ! [Y: nat,X: num,N: nat] :
% 0.32/1.54 ( ( ( semiri2110766477t_real @ Y )
% 0.32/1.54 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.32/1.54 = ( Y
% 0.32/1.54 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_156_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/1.54 ! [Y: nat,X: num,N: nat] :
% 0.32/1.54 ( ( ( semiri2019852685at_int @ Y )
% 0.32/1.54 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.32/1.54 = ( Y
% 0.32/1.54 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_157_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.32/1.54 ! [X: num,N: nat,Y: nat] :
% 0.32/1.54 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/1.54 = ( semiri1382578993at_nat @ Y ) )
% 0.32/1.54 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/1.54 = Y ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_158_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.32/1.54 ! [X: num,N: nat,Y: nat] :
% 0.32/1.54 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 0.32/1.54 = ( semiri2110766477t_real @ Y ) )
% 0.32/1.54 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/1.54 = Y ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_159_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.32/1.54 ! [X: num,N: nat,Y: nat] :
% 0.32/1.54 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.32/1.54 = ( semiri2019852685at_int @ Y ) )
% 0.32/1.54 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/1.54 = Y ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_160_numeral__le__real__of__nat__iff,axiom,
% 0.32/1.54 ! [N: num,M: nat] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri2110766477t_real @ M ) )
% 0.32/1.54 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_161_nat__ceiling__le__eq,axiom,
% 0.32/1.54 ! [X: real,A: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( nat2 @ ( archim1371465213g_real @ X ) ) @ A )
% 0.32/1.54 = ( ord_less_eq_real @ X @ ( semiri2110766477t_real @ A ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_162_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.32/1.54 ! [X: nat,I2: num,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
% 0.32/1.54 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_163_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.32/1.54 ! [X: nat,I2: num,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
% 0.32/1.54 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_164_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.32/1.54 ! [X: nat,I2: num,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( semiri2110766477t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) )
% 0.32/1.54 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_165_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.32/1.54 ! [I2: num,N: nat,X: nat] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri2019852685at_int @ X ) )
% 0.32/1.54 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_166_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.32/1.54 ! [I2: num,N: nat,X: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1382578993at_nat @ X ) )
% 0.32/1.54 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_167_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.32/1.54 ! [I2: num,N: nat,X: nat] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) @ ( semiri2110766477t_real @ X ) )
% 0.32/1.54 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_168_nat__one__as__int,axiom,
% 0.32/1.54 ( one_one_nat
% 0.32/1.54 = ( nat2 @ one_one_int ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_169_real__arch__simple,axiom,
% 0.32/1.54 ! [X: real] :
% 0.32/1.54 ? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri2110766477t_real @ N3 ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_170_of__nat__mono,axiom,
% 0.32/1.54 ! [I2: nat,J: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ I2 @ J )
% 0.32/1.54 => ( ord_less_eq_int @ ( semiri2019852685at_int @ I2 ) @ ( semiri2019852685at_int @ J ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_171_of__nat__mono,axiom,
% 0.32/1.54 ! [I2: nat,J: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ I2 @ J )
% 0.32/1.54 => ( ord_less_eq_nat @ ( semiri1382578993at_nat @ I2 ) @ ( semiri1382578993at_nat @ J ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_172_of__nat__mono,axiom,
% 0.32/1.54 ! [I2: nat,J: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ I2 @ J )
% 0.32/1.54 => ( ord_less_eq_real @ ( semiri2110766477t_real @ I2 ) @ ( semiri2110766477t_real @ J ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_173_numerals_I1_J,axiom,
% 0.32/1.54 ( ( numeral_numeral_nat @ one )
% 0.32/1.54 = one_one_nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_174_of__nat__ceiling,axiom,
% 0.32/1.54 ! [R: real] : ( ord_less_eq_real @ R @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ R ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_175_Suc__inject,axiom,
% 0.32/1.54 ! [X: nat,Y: nat] :
% 0.32/1.54 ( ( ( suc @ X )
% 0.32/1.54 = ( suc @ Y ) )
% 0.32/1.54 => ( X = Y ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_176_n__not__Suc__n,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( N
% 0.32/1.54 != ( suc @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_177_Nat_Oex__has__greatest__nat,axiom,
% 0.32/1.54 ! [P: nat > $o,K: nat,B: nat] :
% 0.32/1.54 ( ( P @ K )
% 0.32/1.54 => ( ! [Y3: nat] :
% 0.32/1.54 ( ( P @ Y3 )
% 0.32/1.54 => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.32/1.54 => ? [X5: nat] :
% 0.32/1.54 ( ( P @ X5 )
% 0.32/1.54 & ! [Y4: nat] :
% 0.32/1.54 ( ( P @ Y4 )
% 0.32/1.54 => ( ord_less_eq_nat @ Y4 @ X5 ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_178_nat__le__linear,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ M @ N )
% 0.32/1.54 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_179_le__antisym,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ M @ N )
% 0.32/1.54 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/1.54 => ( M = N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_180_eq__imp__le,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( M = N )
% 0.32/1.54 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_181_le__trans,axiom,
% 0.32/1.54 ! [I2: nat,J: nat,K: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ I2 @ J )
% 0.32/1.54 => ( ( ord_less_eq_nat @ J @ K )
% 0.32/1.54 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_182_le__refl,axiom,
% 0.32/1.54 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_183_real__nat__ceiling__ge,axiom,
% 0.32/1.54 ! [X: real] : ( ord_less_eq_real @ X @ ( semiri2110766477t_real @ ( nat2 @ ( archim1371465213g_real @ X ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_184_powr__powr__swap,axiom,
% 0.32/1.54 ! [X: real,A: real,B: real] :
% 0.32/1.54 ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 0.32/1.54 = ( powr_real @ ( powr_real @ X @ B ) @ A ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_185_log2__of__power__eq,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( M
% 0.32/1.54 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/1.54 => ( ( semiri2110766477t_real @ N )
% 0.32/1.54 = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ M ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_186_le__log2__of__power,axiom,
% 0.32/1.54 ! [N: nat,M: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.32/1.54 => ( ord_less_eq_real @ ( semiri2110766477t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri2110766477t_real @ M ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_187_transitive__stepwise__le,axiom,
% 0.32/1.54 ! [M: nat,N: nat,R2: nat > nat > $o] :
% 0.32/1.54 ( ( ord_less_eq_nat @ M @ N )
% 0.32/1.54 => ( ! [X5: nat] : ( R2 @ X5 @ X5 )
% 0.32/1.54 => ( ! [X5: nat,Y3: nat,Z2: nat] :
% 0.32/1.54 ( ( R2 @ X5 @ Y3 )
% 0.32/1.54 => ( ( R2 @ Y3 @ Z2 )
% 0.32/1.54 => ( R2 @ X5 @ Z2 ) ) )
% 0.32/1.54 => ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
% 0.32/1.54 => ( R2 @ M @ N ) ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_188_nat__induct__at__least,axiom,
% 0.32/1.54 ! [M: nat,N: nat,P: nat > $o] :
% 0.32/1.54 ( ( ord_less_eq_nat @ M @ N )
% 0.32/1.54 => ( ( P @ M )
% 0.32/1.54 => ( ! [N3: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ M @ N3 )
% 0.32/1.54 => ( ( P @ N3 )
% 0.32/1.54 => ( P @ ( suc @ N3 ) ) ) )
% 0.32/1.54 => ( P @ N ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_189_full__nat__induct,axiom,
% 0.32/1.54 ! [P: nat > $o,N: nat] :
% 0.32/1.54 ( ! [N3: nat] :
% 0.32/1.54 ( ! [M2: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
% 0.32/1.54 => ( P @ M2 ) )
% 0.32/1.54 => ( P @ N3 ) )
% 0.32/1.54 => ( P @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_190_not__less__eq__eq,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.32/1.54 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_191_Suc__n__not__le__n,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_192_le__Suc__eq,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/1.54 = ( ( ord_less_eq_nat @ M @ N )
% 0.32/1.54 | ( M
% 0.32/1.54 = ( suc @ N ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_193_Suc__le__D,axiom,
% 0.32/1.54 ! [N: nat,M3: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
% 0.32/1.54 => ? [M4: nat] :
% 0.32/1.54 ( M3
% 0.32/1.54 = ( suc @ M4 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_194_le__SucI,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ M @ N )
% 0.32/1.54 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_195_le__SucE,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/1.54 => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.32/1.54 => ( M
% 0.32/1.54 = ( suc @ N ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_196_Suc__leD,axiom,
% 0.32/1.54 ! [M: nat,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.32/1.54 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_197_one__le__power,axiom,
% 0.32/1.54 ! [A: int,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.32/1.54 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_198_one__le__power,axiom,
% 0.32/1.54 ! [A: nat,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.32/1.54 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_199_one__le__power,axiom,
% 0.32/1.54 ! [A: real,N: nat] :
% 0.32/1.54 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.32/1.54 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_200_lift__Suc__antimono__le,axiom,
% 0.32/1.54 ! [F: nat > int,N: nat,N4: nat] :
% 0.32/1.54 ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.32/1.54 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/1.54 => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_201_lift__Suc__antimono__le,axiom,
% 0.32/1.54 ! [F: nat > nat,N: nat,N4: nat] :
% 0.32/1.54 ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.32/1.54 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/1.54 => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_202_lift__Suc__antimono__le,axiom,
% 0.32/1.54 ! [F: nat > real,N: nat,N4: nat] :
% 0.32/1.54 ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.32/1.54 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/1.54 => ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_203_lift__Suc__antimono__le,axiom,
% 0.32/1.54 ! [F: nat > num,N: nat,N4: nat] :
% 0.32/1.54 ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.32/1.54 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/1.54 => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_204_lift__Suc__mono__le,axiom,
% 0.32/1.54 ! [F: nat > int,N: nat,N4: nat] :
% 0.32/1.54 ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.32/1.54 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/1.54 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_205_lift__Suc__mono__le,axiom,
% 0.32/1.54 ! [F: nat > nat,N: nat,N4: nat] :
% 0.32/1.54 ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.32/1.54 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/1.54 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_206_lift__Suc__mono__le,axiom,
% 0.32/1.54 ! [F: nat > real,N: nat,N4: nat] :
% 0.32/1.54 ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.32/1.54 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/1.54 => ( ord_less_eq_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_207_lift__Suc__mono__le,axiom,
% 0.32/1.54 ! [F: nat > num,N: nat,N4: nat] :
% 0.32/1.54 ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.32/1.54 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/1.54 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_208_powr__mono,axiom,
% 0.32/1.54 ! [A: real,B: real,X: real] :
% 0.32/1.54 ( ( ord_less_eq_real @ A @ B )
% 0.32/1.54 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.32/1.54 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_209_power__increasing,axiom,
% 0.32/1.54 ! [N: nat,N5: nat,A: int] :
% 0.32/1.54 ( ( ord_less_eq_nat @ N @ N5 )
% 0.32/1.54 => ( ( ord_less_eq_int @ one_one_int @ A )
% 0.32/1.54 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_210_power__increasing,axiom,
% 0.32/1.54 ! [N: nat,N5: nat,A: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ N @ N5 )
% 0.32/1.54 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.32/1.54 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_211_power__increasing,axiom,
% 0.32/1.54 ! [N: nat,N5: nat,A: real] :
% 0.32/1.54 ( ( ord_less_eq_nat @ N @ N5 )
% 0.32/1.54 => ( ( ord_less_eq_real @ one_one_real @ A )
% 0.32/1.54 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_212_one__power2,axiom,
% 0.32/1.54 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/1.54 = one_one_int ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_213_one__power2,axiom,
% 0.32/1.54 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/1.54 = one_one_nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_214_one__power2,axiom,
% 0.32/1.54 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/1.54 = one_one_real ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_215_dbl__simps_I3_J,axiom,
% 0.32/1.54 ( ( neg_numeral_dbl_int @ one_one_int )
% 0.32/1.54 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_216_dbl__simps_I3_J,axiom,
% 0.32/1.54 ( ( neg_numeral_dbl_real @ one_one_real )
% 0.32/1.54 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_217_power__numeral,axiom,
% 0.32/1.54 ! [K: num,L: num] :
% 0.32/1.54 ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.32/1.54 = ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_218_power__numeral,axiom,
% 0.32/1.54 ! [K: num,L: num] :
% 0.32/1.54 ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.32/1.54 = ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_219_power__numeral,axiom,
% 0.32/1.54 ! [K: num,L: num] :
% 0.32/1.54 ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.32/1.54 = ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_220_order__refl,axiom,
% 0.32/1.54 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_221_order__refl,axiom,
% 0.32/1.54 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_222_order__refl,axiom,
% 0.32/1.54 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_223_order__refl,axiom,
% 0.32/1.54 ! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_224_powr__numeral,axiom,
% 0.32/1.54 ! [X: real,N: num] :
% 0.32/1.54 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/1.54 => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
% 0.32/1.54 = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_225_power2__eq__iff__nonneg,axiom,
% 0.32/1.54 ! [X: int,Y: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/1.54 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.32/1.54 => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/1.54 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/1.54 = ( X = Y ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_226_power2__eq__iff__nonneg,axiom,
% 0.32/1.54 ! [X: nat,Y: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.32/1.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.32/1.54 => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/1.54 = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/1.54 = ( X = Y ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_227_power2__eq__iff__nonneg,axiom,
% 0.32/1.54 ! [X: real,Y: real] :
% 0.32/1.54 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/1.54 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/1.54 => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/1.54 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/1.54 = ( X = Y ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_228_le0,axiom,
% 0.32/1.54 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_229_bot__nat__0_Oextremum,axiom,
% 0.32/1.54 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_230_d_I2_J,axiom,
% 0.32/1.54 ord_less_int @ zero_zero_int @ d ).
% 0.32/1.54
% 0.32/1.54 thf(fact_231_le__zero__eq,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.32/1.54 = ( N = zero_zero_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_232_of__nat__0,axiom,
% 0.32/1.54 ( ( semiri1382578993at_nat @ zero_zero_nat )
% 0.32/1.54 = zero_zero_nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_233_of__nat__0,axiom,
% 0.32/1.54 ( ( semiri2110766477t_real @ zero_zero_nat )
% 0.32/1.54 = zero_zero_real ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_234_of__nat__0,axiom,
% 0.32/1.54 ( ( semiri2019852685at_int @ zero_zero_nat )
% 0.32/1.54 = zero_zero_int ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_235_of__nat__0__eq__iff,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( zero_zero_nat
% 0.32/1.54 = ( semiri1382578993at_nat @ N ) )
% 0.32/1.54 = ( zero_zero_nat = N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_236_of__nat__0__eq__iff,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( zero_zero_real
% 0.32/1.54 = ( semiri2110766477t_real @ N ) )
% 0.32/1.54 = ( zero_zero_nat = N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_237_of__nat__0__eq__iff,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( zero_zero_int
% 0.32/1.54 = ( semiri2019852685at_int @ N ) )
% 0.32/1.54 = ( zero_zero_nat = N ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_238_of__nat__eq__0__iff,axiom,
% 0.32/1.54 ! [M: nat] :
% 0.32/1.54 ( ( ( semiri1382578993at_nat @ M )
% 0.32/1.54 = zero_zero_nat )
% 0.32/1.54 = ( M = zero_zero_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_239_of__nat__eq__0__iff,axiom,
% 0.32/1.54 ! [M: nat] :
% 0.32/1.54 ( ( ( semiri2110766477t_real @ M )
% 0.32/1.54 = zero_zero_real )
% 0.32/1.54 = ( M = zero_zero_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_240_of__nat__eq__0__iff,axiom,
% 0.32/1.54 ! [M: nat] :
% 0.32/1.54 ( ( ( semiri2019852685at_int @ M )
% 0.32/1.54 = zero_zero_int )
% 0.32/1.54 = ( M = zero_zero_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_241_power__Suc0__right,axiom,
% 0.32/1.54 ! [A: int] :
% 0.32/1.54 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 0.32/1.54 = A ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_242_power__Suc0__right,axiom,
% 0.32/1.54 ! [A: nat] :
% 0.32/1.54 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.32/1.54 = A ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_243_power__Suc0__right,axiom,
% 0.32/1.54 ! [A: real] :
% 0.32/1.54 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 0.32/1.54 = A ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_244_of__int__0,axiom,
% 0.32/1.54 ( ( ring_1_of_int_real @ zero_zero_int )
% 0.32/1.54 = zero_zero_real ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_245_of__int__0,axiom,
% 0.32/1.54 ( ( ring_1_of_int_int @ zero_zero_int )
% 0.32/1.54 = zero_zero_int ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_246_of__int__0__eq__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( zero_zero_real
% 0.32/1.54 = ( ring_1_of_int_real @ Z ) )
% 0.32/1.54 = ( Z = zero_zero_int ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_247_of__int__0__eq__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( zero_zero_int
% 0.32/1.54 = ( ring_1_of_int_int @ Z ) )
% 0.32/1.54 = ( Z = zero_zero_int ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_248_of__int__eq__0__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ( ring_1_of_int_real @ Z )
% 0.32/1.54 = zero_zero_real )
% 0.32/1.54 = ( Z = zero_zero_int ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_249_of__int__eq__0__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ( ring_1_of_int_int @ Z )
% 0.32/1.54 = zero_zero_int )
% 0.32/1.54 = ( Z = zero_zero_int ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_250_powr__eq__0__iff,axiom,
% 0.32/1.54 ! [W: real,Z: real] :
% 0.32/1.54 ( ( ( powr_real @ W @ Z )
% 0.32/1.54 = zero_zero_real )
% 0.32/1.54 = ( W = zero_zero_real ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_251_powr__0,axiom,
% 0.32/1.54 ! [Z: real] :
% 0.32/1.54 ( ( powr_real @ zero_zero_real @ Z )
% 0.32/1.54 = zero_zero_real ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_252_ceiling__zero,axiom,
% 0.32/1.54 ( ( archim1371465213g_real @ zero_zero_real )
% 0.32/1.54 = zero_zero_int ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_253_nat__le__0,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.32/1.54 => ( ( nat2 @ Z )
% 0.32/1.54 = zero_zero_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_254_nat__0__iff,axiom,
% 0.32/1.54 ! [I2: int] :
% 0.32/1.54 ( ( ( nat2 @ I2 )
% 0.32/1.54 = zero_zero_nat )
% 0.32/1.54 = ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_255_power__Suc__0,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/1.54 = ( suc @ zero_zero_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_256_nat__power__eq__Suc__0__iff,axiom,
% 0.32/1.54 ! [X: nat,M: nat] :
% 0.32/1.54 ( ( ( power_power_nat @ X @ M )
% 0.32/1.54 = ( suc @ zero_zero_nat ) )
% 0.32/1.54 = ( ( M = zero_zero_nat )
% 0.32/1.54 | ( X
% 0.32/1.54 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_257_nat__int,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( nat2 @ ( semiri2019852685at_int @ N ) )
% 0.32/1.54 = N ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_258_dbl__simps_I2_J,axiom,
% 0.32/1.54 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 0.32/1.54 = zero_zero_real ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_259_dbl__simps_I2_J,axiom,
% 0.32/1.54 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 0.32/1.54 = zero_zero_int ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_260_of__nat__le__0__iff,axiom,
% 0.32/1.54 ! [M: nat] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int )
% 0.32/1.54 = ( M = zero_zero_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_261_of__nat__le__0__iff,axiom,
% 0.32/1.54 ! [M: nat] :
% 0.32/1.54 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat )
% 0.32/1.54 = ( M = zero_zero_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_262_of__nat__le__0__iff,axiom,
% 0.32/1.54 ! [M: nat] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( semiri2110766477t_real @ M ) @ zero_zero_real )
% 0.32/1.54 = ( M = zero_zero_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_263_power__0__Suc,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 0.32/1.54 = zero_zero_int ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_264_power__0__Suc,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.32/1.54 = zero_zero_nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_265_power__0__Suc,axiom,
% 0.32/1.54 ! [N: nat] :
% 0.32/1.54 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 0.32/1.54 = zero_zero_real ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_266_power__zero__numeral,axiom,
% 0.32/1.54 ! [K: num] :
% 0.32/1.54 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 0.32/1.54 = zero_zero_int ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_267_power__zero__numeral,axiom,
% 0.32/1.54 ! [K: num] :
% 0.32/1.54 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 0.32/1.54 = zero_zero_nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_268_power__zero__numeral,axiom,
% 0.32/1.54 ! [K: num] :
% 0.32/1.54 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 0.32/1.54 = zero_zero_real ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_269_powr__zero__eq__one,axiom,
% 0.32/1.54 ! [X: real] :
% 0.32/1.54 ( ( ( X = zero_zero_real )
% 0.32/1.54 => ( ( powr_real @ X @ zero_zero_real )
% 0.32/1.54 = zero_zero_real ) )
% 0.32/1.54 & ( ( X != zero_zero_real )
% 0.32/1.54 => ( ( powr_real @ X @ zero_zero_real )
% 0.32/1.54 = one_one_real ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_270_powr__nonneg__iff,axiom,
% 0.32/1.54 ! [A: real,X: real] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
% 0.32/1.54 = ( A = zero_zero_real ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_271_nat__1,axiom,
% 0.32/1.54 ( ( nat2 @ one_one_int )
% 0.32/1.54 = ( suc @ zero_zero_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_272_int__eq__iff__numeral,axiom,
% 0.32/1.54 ! [M: nat,V: num] :
% 0.32/1.54 ( ( ( semiri2019852685at_int @ M )
% 0.32/1.54 = ( numeral_numeral_int @ V ) )
% 0.32/1.54 = ( M
% 0.32/1.54 = ( numeral_numeral_nat @ V ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_273_log__one,axiom,
% 0.32/1.54 ! [A: real] :
% 0.32/1.54 ( ( log @ A @ one_one_real )
% 0.32/1.54 = zero_zero_real ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_274_int__nat__eq,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/1.54 => ( ( semiri2019852685at_int @ ( nat2 @ Z ) )
% 0.32/1.54 = Z ) )
% 0.32/1.54 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/1.54 => ( ( semiri2019852685at_int @ ( nat2 @ Z ) )
% 0.32/1.54 = zero_zero_int ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_275_dbl__simps_I5_J,axiom,
% 0.32/1.54 ! [K: num] :
% 0.32/1.54 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 0.32/1.54 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_276_dbl__simps_I5_J,axiom,
% 0.32/1.54 ! [K: num] :
% 0.32/1.54 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 0.32/1.54 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_277_of__int__0__le__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.32/1.54 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_278_of__int__0__le__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.32/1.54 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_279_of__int__le__0__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.32/1.54 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_280_of__int__le__0__iff,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.32/1.54 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_281_ceiling__le__zero,axiom,
% 0.32/1.54 ! [X: real] :
% 0.32/1.54 ( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ zero_zero_int )
% 0.32/1.54 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_282_powr__one,axiom,
% 0.32/1.54 ! [X: real] :
% 0.32/1.54 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/1.54 => ( ( powr_real @ X @ one_one_real )
% 0.32/1.54 = X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_283_powr__one__gt__zero__iff,axiom,
% 0.32/1.54 ! [X: real] :
% 0.32/1.54 ( ( ( powr_real @ X @ one_one_real )
% 0.32/1.54 = X )
% 0.32/1.54 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_284_of__nat__nat,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/1.54 => ( ( semiri2110766477t_real @ ( nat2 @ Z ) )
% 0.32/1.54 = ( ring_1_of_int_real @ Z ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_285_of__nat__nat,axiom,
% 0.32/1.54 ! [Z: int] :
% 0.32/1.54 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/1.54 => ( ( semiri2019852685at_int @ ( nat2 @ Z ) )
% 0.32/1.54 = ( ring_1_of_int_int @ Z ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_286_zero__eq__power2,axiom,
% 0.32/1.54 ! [A: int] :
% 0.32/1.54 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/1.54 = zero_zero_int )
% 0.32/1.54 = ( A = zero_zero_int ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_287_zero__eq__power2,axiom,
% 0.32/1.54 ! [A: nat] :
% 0.32/1.54 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/1.54 = zero_zero_nat )
% 0.32/1.54 = ( A = zero_zero_nat ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(fact_288_zero__eq__power2,axiom,
% 0.32/1.54 ! [A: real] :
% 0.32/1.54 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/1.55 = zero_zero_real )
% 0.32/1.55 = ( A = zero_zero_real ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_289_power2__less__eq__zero__iff,axiom,
% 0.32/1.55 ! [A: int] :
% 0.32/1.55 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.32/1.55 = ( A = zero_zero_int ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_290_power2__less__eq__zero__iff,axiom,
% 0.32/1.55 ! [A: real] :
% 0.32/1.55 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 0.32/1.55 = ( A = zero_zero_real ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_291_int__if,axiom,
% 0.32/1.55 ! [P: $o,A: nat,B: nat] :
% 0.32/1.55 ( ( P
% 0.32/1.55 => ( ( semiri2019852685at_int @ ( if_nat @ P @ A @ B ) )
% 0.32/1.55 = ( semiri2019852685at_int @ A ) ) )
% 0.32/1.55 & ( ~ P
% 0.32/1.55 => ( ( semiri2019852685at_int @ ( if_nat @ P @ A @ B ) )
% 0.32/1.55 = ( semiri2019852685at_int @ B ) ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_292_nat__int__comparison_I1_J,axiom,
% 0.32/1.55 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.32/1.55 = ( ^ [A3: nat,B2: nat] :
% 0.32/1.55 ( ( semiri2019852685at_int @ A3 )
% 0.32/1.55 = ( semiri2019852685at_int @ B2 ) ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_293_int__ops_I1_J,axiom,
% 0.32/1.55 ( ( semiri2019852685at_int @ zero_zero_nat )
% 0.32/1.55 = zero_zero_int ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_294_zero__le__imp__eq__int,axiom,
% 0.32/1.55 ! [K: int] :
% 0.32/1.55 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/1.55 => ? [N3: nat] :
% 0.32/1.55 ( K
% 0.32/1.55 = ( semiri2019852685at_int @ N3 ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_295_nonneg__int__cases,axiom,
% 0.32/1.55 ! [K: int] :
% 0.32/1.55 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/1.55 => ~ ! [N3: nat] :
% 0.32/1.55 ( K
% 0.32/1.55 != ( semiri2019852685at_int @ N3 ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_296_int__int__eq,axiom,
% 0.32/1.55 ! [M: nat,N: nat] :
% 0.32/1.55 ( ( ( semiri2019852685at_int @ M )
% 0.32/1.55 = ( semiri2019852685at_int @ N ) )
% 0.32/1.55 = ( M = N ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_297_nat__eq__iff2,axiom,
% 0.32/1.55 ! [M: nat,W: int] :
% 0.32/1.55 ( ( M
% 0.32/1.55 = ( nat2 @ W ) )
% 0.32/1.55 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.32/1.55 => ( W
% 0.32/1.55 = ( semiri2019852685at_int @ M ) ) )
% 0.32/1.55 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.32/1.55 => ( M = zero_zero_nat ) ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_298_nat__eq__iff,axiom,
% 0.32/1.55 ! [W: int,M: nat] :
% 0.32/1.55 ( ( ( nat2 @ W )
% 0.32/1.55 = M )
% 0.32/1.55 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.32/1.55 => ( W
% 0.32/1.55 = ( semiri2019852685at_int @ M ) ) )
% 0.32/1.55 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.32/1.55 => ( M = zero_zero_nat ) ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_299_zero__reorient,axiom,
% 0.32/1.55 ! [X: real] :
% 0.32/1.55 ( ( zero_zero_real = X )
% 0.32/1.55 = ( X = zero_zero_real ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_300_zero__reorient,axiom,
% 0.32/1.55 ! [X: nat] :
% 0.32/1.55 ( ( zero_zero_nat = X )
% 0.32/1.55 = ( X = zero_zero_nat ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_301_zero__reorient,axiom,
% 0.32/1.55 ! [X: int] :
% 0.32/1.55 ( ( zero_zero_int = X )
% 0.32/1.55 = ( X = zero_zero_int ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_302_zero__le,axiom,
% 0.32/1.55 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_303_nat__zero__as__int,axiom,
% 0.32/1.55 ( zero_zero_nat
% 0.32/1.55 = ( nat2 @ zero_zero_int ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_304_power__0__left,axiom,
% 0.32/1.55 ! [N: nat] :
% 0.32/1.55 ( ( ( N = zero_zero_nat )
% 0.32/1.55 => ( ( power_power_int @ zero_zero_int @ N )
% 0.32/1.55 = one_one_int ) )
% 0.32/1.55 & ( ( N != zero_zero_nat )
% 0.32/1.55 => ( ( power_power_int @ zero_zero_int @ N )
% 0.32/1.55 = zero_zero_int ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_305_power__0__left,axiom,
% 0.32/1.55 ! [N: nat] :
% 0.32/1.55 ( ( ( N = zero_zero_nat )
% 0.32/1.55 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.32/1.55 = one_one_nat ) )
% 0.32/1.55 & ( ( N != zero_zero_nat )
% 0.32/1.55 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.32/1.55 = zero_zero_nat ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_306_power__0__left,axiom,
% 0.32/1.55 ! [N: nat] :
% 0.32/1.55 ( ( ( N = zero_zero_nat )
% 0.32/1.55 => ( ( power_power_real @ zero_zero_real @ N )
% 0.32/1.55 = one_one_real ) )
% 0.32/1.55 & ( ( N != zero_zero_nat )
% 0.32/1.55 => ( ( power_power_real @ zero_zero_real @ N )
% 0.32/1.55 = zero_zero_real ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_307_nat__0__le,axiom,
% 0.32/1.55 ! [Z: int] :
% 0.32/1.55 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/1.55 => ( ( semiri2019852685at_int @ ( nat2 @ Z ) )
% 0.32/1.55 = Z ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_308_int__eq__iff,axiom,
% 0.32/1.55 ! [M: nat,Z: int] :
% 0.32/1.55 ( ( ( semiri2019852685at_int @ M )
% 0.32/1.55 = Z )
% 0.32/1.55 = ( ( M
% 0.32/1.55 = ( nat2 @ Z ) )
% 0.32/1.55 & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_309_int__ops_I2_J,axiom,
% 0.32/1.55 ( ( semiri2019852685at_int @ one_one_nat )
% 0.32/1.55 = one_one_int ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_310_le__numeral__extra_I3_J,axiom,
% 0.32/1.55 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.32/1.55
% 0.32/1.55 thf(fact_311_le__numeral__extra_I3_J,axiom,
% 0.32/1.55 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.32/1.55
% 0.32/1.55 thf(fact_312_le__numeral__extra_I3_J,axiom,
% 0.32/1.55 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.32/1.55
% 0.32/1.55 thf(fact_313_zero__neq__numeral,axiom,
% 0.32/1.55 ! [N: num] :
% 0.32/1.55 ( zero_zero_int
% 0.32/1.55 != ( numeral_numeral_int @ N ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_314_zero__neq__numeral,axiom,
% 0.32/1.55 ! [N: num] :
% 0.32/1.55 ( zero_zero_real
% 0.32/1.55 != ( numeral_numeral_real @ N ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_315_zero__neq__numeral,axiom,
% 0.32/1.55 ! [N: num] :
% 0.32/1.55 ( zero_zero_nat
% 0.32/1.55 != ( numeral_numeral_nat @ N ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_316_power__not__zero,axiom,
% 0.32/1.55 ! [A: int,N: nat] :
% 0.32/1.55 ( ( A != zero_zero_int )
% 0.32/1.55 => ( ( power_power_int @ A @ N )
% 0.32/1.55 != zero_zero_int ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_317_power__not__zero,axiom,
% 0.32/1.55 ! [A: nat,N: nat] :
% 0.32/1.55 ( ( A != zero_zero_nat )
% 0.32/1.55 => ( ( power_power_nat @ A @ N )
% 0.32/1.55 != zero_zero_nat ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_318_power__not__zero,axiom,
% 0.32/1.55 ! [A: real,N: nat] :
% 0.32/1.55 ( ( A != zero_zero_real )
% 0.32/1.55 => ( ( power_power_real @ A @ N )
% 0.32/1.55 != zero_zero_real ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_319_exists__least__lemma,axiom,
% 0.32/1.55 ! [P: nat > $o] :
% 0.32/1.55 ( ~ ( P @ zero_zero_nat )
% 0.32/1.55 => ( ? [X_1: nat] : ( P @ X_1 )
% 0.32/1.55 => ? [N3: nat] :
% 0.32/1.55 ( ~ ( P @ N3 )
% 0.32/1.55 & ( P @ ( suc @ N3 ) ) ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_320_nat_Odistinct_I1_J,axiom,
% 0.32/1.55 ! [X2: nat] :
% 0.32/1.55 ( zero_zero_nat
% 0.32/1.55 != ( suc @ X2 ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_321_old_Onat_Odistinct_I2_J,axiom,
% 0.32/1.55 ! [Nat2: nat] :
% 0.32/1.55 ( ( suc @ Nat2 )
% 0.32/1.55 != zero_zero_nat ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_322_old_Onat_Odistinct_I1_J,axiom,
% 0.32/1.55 ! [Nat2: nat] :
% 0.32/1.55 ( zero_zero_nat
% 0.32/1.55 != ( suc @ Nat2 ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_323_nat_OdiscI,axiom,
% 0.32/1.55 ! [Nat: nat,X2: nat] :
% 0.32/1.55 ( ( Nat
% 0.32/1.55 = ( suc @ X2 ) )
% 0.32/1.55 => ( Nat != zero_zero_nat ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_324_nat__induct,axiom,
% 0.32/1.55 ! [P: nat > $o,N: nat] :
% 0.32/1.55 ( ( P @ zero_zero_nat )
% 0.32/1.55 => ( ! [N3: nat] :
% 0.32/1.55 ( ( P @ N3 )
% 0.32/1.55 => ( P @ ( suc @ N3 ) ) )
% 0.32/1.55 => ( P @ N ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_325_One__nat__def,axiom,
% 0.32/1.55 ( one_one_nat
% 0.32/1.55 = ( suc @ zero_zero_nat ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_326_diff__induct,axiom,
% 0.32/1.55 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.32/1.55 ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
% 0.32/1.55 => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
% 0.32/1.55 => ( ! [X5: nat,Y3: nat] :
% 0.32/1.55 ( ( P @ X5 @ Y3 )
% 0.32/1.55 => ( P @ ( suc @ X5 ) @ ( suc @ Y3 ) ) )
% 0.32/1.55 => ( P @ M @ N ) ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_327_zero__induct,axiom,
% 0.32/1.55 ! [P: nat > $o,K: nat] :
% 0.32/1.55 ( ( P @ K )
% 0.32/1.55 => ( ! [N3: nat] :
% 0.32/1.55 ( ( P @ ( suc @ N3 ) )
% 0.32/1.55 => ( P @ N3 ) )
% 0.32/1.55 => ( P @ zero_zero_nat ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_328_Suc__neq__Zero,axiom,
% 0.32/1.55 ! [M: nat] :
% 0.32/1.55 ( ( suc @ M )
% 0.32/1.55 != zero_zero_nat ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_329_Zero__neq__Suc,axiom,
% 0.32/1.55 ! [M: nat] :
% 0.32/1.55 ( zero_zero_nat
% 0.32/1.55 != ( suc @ M ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_330_Zero__not__Suc,axiom,
% 0.32/1.55 ! [M: nat] :
% 0.32/1.55 ( zero_zero_nat
% 0.32/1.55 != ( suc @ M ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_331_old_Onat_Oexhaust,axiom,
% 0.32/1.55 ! [Y: nat] :
% 0.32/1.55 ( ( Y != zero_zero_nat )
% 0.32/1.55 => ~ ! [Nat3: nat] :
% 0.32/1.55 ( Y
% 0.32/1.55 != ( suc @ Nat3 ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_332_old_Onat_Oinducts,axiom,
% 0.32/1.55 ! [P: nat > $o,Nat: nat] :
% 0.32/1.55 ( ( P @ zero_zero_nat )
% 0.32/1.55 => ( ! [Nat3: nat] :
% 0.32/1.55 ( ( P @ Nat3 )
% 0.32/1.55 => ( P @ ( suc @ Nat3 ) ) )
% 0.32/1.55 => ( P @ Nat ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_333_not0__implies__Suc,axiom,
% 0.32/1.55 ! [N: nat] :
% 0.32/1.55 ( ( N != zero_zero_nat )
% 0.32/1.55 => ? [M4: nat] :
% 0.32/1.55 ( N
% 0.32/1.55 = ( suc @ M4 ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_334_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.32/1.55 ! [A: nat] :
% 0.32/1.55 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/1.55 => ( A = zero_zero_nat ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_335_bot__nat__0_Oextremum__unique,axiom,
% 0.32/1.55 ! [A: nat] :
% 0.32/1.55 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/1.55 = ( A = zero_zero_nat ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_336_le__0__eq,axiom,
% 0.32/1.55 ! [N: nat] :
% 0.32/1.55 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.32/1.55 = ( N = zero_zero_nat ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_337_less__eq__nat_Osimps_I1_J,axiom,
% 0.32/1.55 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_338_less__eq__int__code_I1_J,axiom,
% 0.32/1.55 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.32/1.55
% 0.32/1.55 thf(fact_339_le__nat__iff,axiom,
% 0.32/1.55 ! [K: int,N: nat] :
% 0.32/1.55 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/1.55 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 0.32/1.55 = ( ord_less_eq_int @ ( semiri2019852685at_int @ N ) @ K ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_340_nat__int__comparison_I3_J,axiom,
% 0.32/1.55 ( ord_less_eq_nat
% 0.32/1.55 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri2019852685at_int @ A3 ) @ ( semiri2019852685at_int @ B2 ) ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_341_zle__int,axiom,
% 0.32/1.55 ! [M: nat,N: nat] :
% 0.32/1.55 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 0.32/1.55 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_342_int__ops_I3_J,axiom,
% 0.32/1.55 ! [N: num] :
% 0.32/1.55 ( ( semiri2019852685at_int @ ( numeral_numeral_nat @ N ) )
% 0.32/1.55 = ( numeral_numeral_int @ N ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_343_ex__nat,axiom,
% 0.32/1.55 ( ( ^ [P2: nat > $o] :
% 0.32/1.55 ? [X6: nat] : ( P2 @ X6 ) )
% 0.32/1.55 = ( ^ [P3: nat > $o] :
% 0.32/1.55 ? [X3: int] :
% 0.32/1.55 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.32/1.55 & ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_344_all__nat,axiom,
% 0.32/1.55 ( ( ^ [P2: nat > $o] :
% 0.32/1.55 ! [X6: nat] : ( P2 @ X6 ) )
% 0.32/1.55 = ( ^ [P3: nat > $o] :
% 0.32/1.55 ! [X3: int] :
% 0.32/1.55 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.32/1.55 => ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_345_eq__nat__nat__iff,axiom,
% 0.32/1.55 ! [Z: int,Z5: int] :
% 0.32/1.55 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/1.55 => ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
% 0.32/1.55 => ( ( ( nat2 @ Z )
% 0.32/1.55 = ( nat2 @ Z5 ) )
% 0.32/1.55 = ( Z = Z5 ) ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_346_powr__mono2,axiom,
% 0.32/1.55 ! [A: real,X: real,Y: real] :
% 0.32/1.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/1.55 => ( ( ord_less_eq_real @ X @ Y )
% 0.32/1.55 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_347_powr__ge__pzero,axiom,
% 0.32/1.55 ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_348_pow_Osimps_I1_J,axiom,
% 0.32/1.55 ! [X: num] :
% 0.32/1.55 ( ( pow @ X @ one )
% 0.32/1.55 = X ) ).
% 0.32/1.55
% 0.32/1.55 thf(fact_349_nat__le__iff,axiom,
% 0.32/1.55 ! [X: int,N: nat] :
% 0.32/1.55 ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
% 0.32/1.55 = ( ord_less_eq_int @ X @ ( semiri2019852685at_int @ N ) ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
% 0.32/1.55 ! [P: $o] :
% 0.32/1.55 ( ( P = $true )
% 0.32/1.55 | ( P = $false ) ) ).
% 0.32/1.55
% 0.32/1.55 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.32/1.55 ! [X: nat,Y: nat] :
% 0.32/1.55 ( ( if_nat @ $false @ X @ Y )
% 0.32/1.55 = Y ) ).
% 0.32/1.55
% 0.32/1.55 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.32/1.55 ! [X: nat,Y: nat] :
% 0.32/1.55 ( ( if_nat @ $true @ X @ Y )
% 0.32/1.55 = X ) ).
% 0.32/1.55
% 0.32/1.55 thf(conj_0,conjecture,
% 0.32/1.55 ord_less_eq_int @ d @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( nat2 @ ( archim1371465213g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ d ) ) ) ) ) ).
% 0.32/1.55 ------------------------
% 0.32/1.57 % (18211)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/2Mi)
% 0.32/1.57 % (18210)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/27Mi)
% 0.32/1.57 % (18211)Instruction limit reached!
% 0.32/1.57 % (18211)------------------------------
% 0.32/1.57 % (18211)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.57 % (18211)Termination reason: Unknown
% 0.32/1.57 % (18211)Termination phase: shuffling
% 0.32/1.57
% 0.32/1.57 % (18211)Memory used [KB]: 1279
% 0.32/1.57 % (18211)Time elapsed: 0.002 s
% 0.32/1.57 % (18211)Instructions burned: 2 (million)
% 0.32/1.57 % (18211)------------------------------
% 0.32/1.57 % (18211)------------------------------
% 0.32/1.58 % (18208)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/183Mi)
% 0.32/1.58 % (18209)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/4Mi)
% 0.32/1.58 % (18212)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/2Mi)
% 0.32/1.58 % (18213)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/275Mi)
% 0.32/1.58 % (18214)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/18Mi)
% 0.32/1.58 % (18210)Instruction limit reached!
% 0.32/1.58 % (18210)------------------------------
% 0.32/1.58 % (18210)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.58 % (18210)Termination reason: Unknown
% 0.32/1.58 % (18210)Termination phase: Preprocessing 1
% 0.32/1.58
% 0.32/1.58 % (18210)Memory used [KB]: 1535
% 0.32/1.58 % (18210)Time elapsed: 0.009 s
% 0.32/1.58 % (18210)Instructions burned: 27 (million)
% 0.32/1.58 % (18210)------------------------------
% 0.32/1.58 % (18210)------------------------------
% 0.32/1.58 % (18212)Instruction limit reached!
% 0.32/1.58 % (18212)------------------------------
% 0.32/1.58 % (18212)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.58 % (18212)Termination reason: Unknown
% 0.32/1.58 % (18212)Termination phase: shuffling
% 0.32/1.58
% 0.32/1.58 % (18212)Memory used [KB]: 1279
% 0.32/1.58 % (18212)Time elapsed: 0.004 s
% 0.32/1.58 % (18212)Instructions burned: 3 (million)
% 0.32/1.58 % (18212)------------------------------
% 0.32/1.58 % (18212)------------------------------
% 0.32/1.58 % (18209)Instruction limit reached!
% 0.32/1.58 % (18209)------------------------------
% 0.32/1.58 % (18209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.58 % (18209)Termination reason: Unknown
% 0.32/1.58 % (18209)Termination phase: shuffling
% 0.32/1.58
% 0.32/1.58 % (18209)Memory used [KB]: 1407
% 0.32/1.58 % (18209)Time elapsed: 0.005 s
% 0.32/1.58 % (18209)Instructions burned: 5 (million)
% 0.32/1.58 % (18209)------------------------------
% 0.32/1.58 % (18209)------------------------------
% 0.32/1.58 % (18215)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/3Mi)
% 0.32/1.58 % (18215)Instruction limit reached!
% 0.32/1.58 % (18215)------------------------------
% 0.32/1.58 % (18215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.58 % (18215)Termination reason: Unknown
% 0.32/1.58 % (18215)Termination phase: shuffling
% 0.32/1.58
% 0.32/1.58 % (18215)Memory used [KB]: 1279
% 0.32/1.58 % (18215)Time elapsed: 0.002 s
% 0.32/1.58 % (18215)Instructions burned: 3 (million)
% 0.32/1.58 % (18215)------------------------------
% 0.32/1.58 % (18215)------------------------------
% 0.32/1.59 % (18216)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/37Mi)
% 0.32/1.59 % (18214)Instruction limit reached!
% 0.32/1.59 % (18214)------------------------------
% 0.32/1.59 % (18214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.59 % (18214)Termination reason: Unknown
% 0.32/1.59 % (18214)Termination phase: Property scanning
% 0.32/1.59
% 0.32/1.59 % (18214)Memory used [KB]: 1535
% 0.32/1.59 % (18214)Time elapsed: 0.013 s
% 0.32/1.59 % (18214)Instructions burned: 18 (million)
% 0.32/1.59 % (18214)------------------------------
% 0.32/1.59 % (18214)------------------------------
% 0.32/1.59 % (18219)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/1041Mi)
% 2.70/1.60 % (18217)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/15Mi)
% 2.70/1.60 % (18218)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/3Mi)
% 2.70/1.60 % (18216)Instruction limit reached!
% 2.70/1.60 % (18216)------------------------------
% 2.70/1.60 % (18216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (18216)Termination reason: Unknown
% 2.70/1.60 % (18216)Termination phase: Property scanning
% 2.70/1.60
% 2.70/1.60 % (18216)Memory used [KB]: 2046
% 2.70/1.60 % (18216)Time elapsed: 0.012 s
% 2.70/1.60 % (18216)Instructions burned: 40 (million)
% 2.70/1.60 % (18216)------------------------------
% 2.70/1.60 % (18216)------------------------------
% 2.70/1.60 % (18218)Instruction limit reached!
% 2.70/1.60 % (18218)------------------------------
% 2.70/1.60 % (18218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (18218)Termination reason: Unknown
% 2.70/1.60 % (18218)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (18218)Memory used [KB]: 1279
% 2.70/1.60 % (18218)Time elapsed: 0.004 s
% 2.70/1.60 % (18218)Instructions burned: 3 (million)
% 2.70/1.60 % (18218)------------------------------
% 2.70/1.60 % (18218)------------------------------
% 2.70/1.61 % (18220)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/7Mi)
% 2.70/1.61 % (18217)Instruction limit reached!
% 2.70/1.61 % (18217)------------------------------
% 2.70/1.61 % (18217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.61 % (18217)Termination reason: Unknown
% 2.70/1.61 % (18217)Termination phase: Property scanning
% 2.70/1.61
% 2.70/1.61 % (18217)Memory used [KB]: 1535
% 2.70/1.61 % (18217)Time elapsed: 0.012 s
% 2.70/1.61 % (18217)Instructions burned: 16 (million)
% 2.70/1.61 % (18217)------------------------------
% 2.70/1.61 % (18217)------------------------------
% 2.70/1.61 % (18221)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/16Mi)
% 2.70/1.61 % (18220)Instruction limit reached!
% 2.70/1.61 % (18220)------------------------------
% 2.70/1.61 % (18220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.61 % (18220)Termination reason: Unknown
% 2.70/1.61 % (18220)Termination phase: shuffling
% 2.70/1.61
% 2.70/1.61 % (18220)Memory used [KB]: 1407
% 2.70/1.61 % (18220)Time elapsed: 0.006 s
% 2.70/1.61 % (18220)Instructions burned: 7 (million)
% 2.70/1.61 % (18220)------------------------------
% 2.70/1.61 % (18220)------------------------------
% 2.70/1.61 % (18221)Instruction limit reached!
% 2.70/1.61 % (18221)------------------------------
% 2.70/1.61 % (18221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.61 % (18221)Termination reason: Unknown
% 2.70/1.61 % (18221)Termination phase: Property scanning
% 2.70/1.61
% 2.70/1.61 % (18221)Memory used [KB]: 1535
% 2.70/1.61 % (18221)Time elapsed: 0.006 s
% 2.70/1.61 % (18221)Instructions burned: 16 (million)
% 2.70/1.61 % (18221)------------------------------
% 2.70/1.61 % (18221)------------------------------
% 2.70/1.62 % (18222)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/3Mi)
% 2.70/1.62 % (18213)Refutation not found, incomplete strategy
% 2.70/1.62 % (18213)------------------------------
% 2.70/1.62 % (18213)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.62 % (18213)Termination reason: Refutation not found, incomplete strategy
% 2.70/1.62
% 2.70/1.62
% 2.70/1.62 % (18213)Memory used [KB]: 6908
% 2.70/1.62 % (18213)Time elapsed: 0.043 s
% 2.70/1.62 % (18213)Instructions burned: 83 (million)
% 2.70/1.62 % (18213)------------------------------
% 2.70/1.62 % (18213)------------------------------
% 2.70/1.62 % (18222)Instruction limit reached!
% 2.70/1.62 % (18222)------------------------------
% 2.70/1.62 % (18222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.62 % (18222)Termination reason: Unknown
% 2.70/1.62 % (18222)Termination phase: shuffling
% 2.70/1.62
% 2.70/1.62 % (18222)Memory used [KB]: 1279
% 2.70/1.62 % (18222)Time elapsed: 0.004 s
% 2.70/1.62 % (18222)Instructions burned: 3 (million)
% 2.70/1.62 % (18222)------------------------------
% 2.70/1.62 % (18222)------------------------------
% 2.70/1.62 % (18225)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/3Mi)
% 2.70/1.62 % (18225)Instruction limit reached!
% 2.70/1.62 % (18225)------------------------------
% 2.70/1.62 % (18225)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.62 % (18225)Termination reason: Unknown
% 2.70/1.62 % (18225)Termination phase: shuffling
% 2.70/1.62
% 2.70/1.62 % (18225)Memory used [KB]: 1279
% 2.70/1.62 % (18225)Time elapsed: 0.002 s
% 2.70/1.62 % (18225)Instructions burned: 3 (million)
% 2.70/1.62 % (18225)------------------------------
% 2.70/1.62 % (18225)------------------------------
% 2.70/1.62 % (18223)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/3Mi)
% 2.70/1.63 % (18223)Instruction limit reached!
% 2.70/1.63 % (18223)------------------------------
% 2.70/1.63 % (18223)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.63 % (18223)Termination reason: Unknown
% 2.70/1.63 % (18223)Termination phase: shuffling
% 2.70/1.63
% 2.70/1.63 % (18223)Memory used [KB]: 1279
% 2.70/1.63 % (18223)Time elapsed: 0.003 s
% 2.70/1.63 % (18223)Instructions burned: 3 (million)
% 2.70/1.63 % (18223)------------------------------
% 2.70/1.63 % (18223)------------------------------
% 2.70/1.63 % (18224)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/7Mi)
% 2.70/1.63 % (18227)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/18Mi)
% 2.70/1.63 % (18224)Instruction limit reached!
% 2.70/1.63 % (18224)------------------------------
% 2.70/1.63 % (18224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.63 % (18224)Termination reason: Unknown
% 2.70/1.63 % (18224)Termination phase: shuffling
% 2.70/1.63
% 2.70/1.63 % (18224)Memory used [KB]: 1407
% 2.70/1.63 % (18224)Time elapsed: 0.005 s
% 2.70/1.63 % (18224)Instructions burned: 7 (million)
% 2.70/1.63 % (18224)------------------------------
% 2.70/1.63 % (18224)------------------------------
% 2.70/1.63 % (18228)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/710Mi)
% 2.70/1.64 % (18227)Instruction limit reached!
% 2.70/1.64 % (18227)------------------------------
% 2.70/1.64 % (18227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.64 % (18227)Termination reason: Unknown
% 2.70/1.64 % (18227)Termination phase: Property scanning
% 2.70/1.64
% 2.70/1.64 % (18227)Memory used [KB]: 1535
% 2.70/1.64 % (18227)Time elapsed: 0.007 s
% 2.70/1.64 % (18227)Instructions burned: 20 (million)
% 2.70/1.64 % (18227)------------------------------
% 2.70/1.64 % (18227)------------------------------
% 2.70/1.64 % (18229)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/6Mi)
% 2.70/1.64 % (18226)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/4Mi)
% 2.70/1.64 % (18226)Instruction limit reached!
% 2.70/1.64 % (18226)------------------------------
% 2.70/1.64 % (18226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.64 % (18226)Termination reason: Unknown
% 2.70/1.64 % (18226)Termination phase: shuffling
% 2.70/1.64
% 2.70/1.64 % (18226)Memory used [KB]: 1279
% 2.70/1.64 % (18226)Time elapsed: 0.004 s
% 2.70/1.64 % (18226)Instructions burned: 4 (million)
% 2.70/1.64 % (18226)------------------------------
% 2.70/1.64 % (18226)------------------------------
% 2.70/1.64 % (18229)Instruction limit reached!
% 2.70/1.64 % (18229)------------------------------
% 2.70/1.64 % (18229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.64 % (18229)Termination reason: Unknown
% 2.70/1.64 % (18229)Termination phase: shuffling
% 2.70/1.64
% 2.70/1.64 % (18229)Memory used [KB]: 1407
% 2.70/1.64 % (18229)Time elapsed: 0.004 s
% 2.70/1.64 % (18229)Instructions burned: 7 (million)
% 2.70/1.64 % (18229)------------------------------
% 2.70/1.64 % (18229)------------------------------
% 2.70/1.64 % (18230)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/902Mi)
% 2.70/1.65 % (18231)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/21Mi)
% 2.70/1.65 % (18233)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/6Mi)
% 2.70/1.65 % (18208)Instruction limit reached!
% 2.70/1.65 % (18208)------------------------------
% 2.70/1.65 % (18208)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.65 % (18208)Termination reason: Unknown
% 2.70/1.65 % (18208)Termination phase: Saturation
% 2.70/1.65
% 2.70/1.65 % (18208)Memory used [KB]: 6908
% 2.70/1.65 % (18208)Time elapsed: 0.075 s
% 2.70/1.65 % (18208)Instructions burned: 184 (million)
% 2.70/1.65 % (18208)------------------------------
% 2.70/1.65 % (18208)------------------------------
% 2.70/1.65 % (18231)Instruction limit reached!
% 2.70/1.65 % (18231)------------------------------
% 2.70/1.65 % (18231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.65 % (18231)Termination reason: Unknown
% 2.70/1.65 % (18231)Termination phase: Property scanning
% 2.70/1.65
% 2.70/1.65 % (18231)Memory used [KB]: 1535
% 2.70/1.65 % (18231)Time elapsed: 0.008 s
% 2.70/1.65 % (18231)Instructions burned: 21 (million)
% 2.70/1.65 % (18231)------------------------------
% 2.70/1.65 % (18231)------------------------------
% 2.70/1.65 % (18233)Instruction limit reached!
% 2.70/1.65 % (18233)------------------------------
% 2.70/1.65 % (18233)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.65 % (18233)Termination reason: Unknown
% 2.70/1.65 % (18233)Termination phase: shuffling
% 2.70/1.65
% 2.70/1.65 % (18233)Memory used [KB]: 1407
% 2.70/1.65 % (18233)Time elapsed: 0.004 s
% 2.70/1.65 % (18233)Instructions burned: 7 (million)
% 2.70/1.65 % (18233)------------------------------
% 2.70/1.65 % (18233)------------------------------
% 2.70/1.66 % (18232)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/5Mi)
% 2.70/1.66 % (18232)Instruction limit reached!
% 2.70/1.66 % (18232)------------------------------
% 2.70/1.66 % (18232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.66 % (18232)Termination reason: Unknown
% 2.70/1.66 % (18232)Termination phase: shuffling
% 2.70/1.66
% 2.70/1.66 % (18232)Memory used [KB]: 1407
% 2.70/1.66 % (18232)Time elapsed: 0.005 s
% 2.70/1.66 % (18232)Instructions burned: 6 (million)
% 2.70/1.66 % (18232)------------------------------
% 2.70/1.66 % (18232)------------------------------
% 2.70/1.66 % (18234)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/377Mi)
% 2.70/1.66 % (18236)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/19Mi)
% 2.70/1.66 % (18235)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_17924 for (2999ds/779Mi)
% 2.70/1.67 % (18236)Instruction limit reached!
% 2.70/1.67 % (18236)------------------------------
% 2.70/1.67 % (18236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.67 % (18236)Termination reason: Unknown
% 2.70/1.67 % (18236)Termination phase: shuffling
% 2.70/1.67
% 2.70/1.67 % (18236)Memory used [KB]: 1791
% 2.70/1.67 % (18236)Time elapsed: 0.007 s
% 2.70/1.67 % (18236)Instructions burned: 22 (million)
% 2.70/1.67 % (18236)------------------------------
% 2.70/1.67 % (18236)------------------------------
% 2.70/1.67 % (18230)Refutation not found, incomplete strategy
% 2.70/1.67 % (18230)------------------------------
% 2.70/1.67 % (18230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.67 % (18230)Termination reason: Refutation not found, incomplete strategy
% 2.70/1.67
% 2.70/1.67
% 2.70/1.67 % (18230)Memory used [KB]: 6908
% 2.70/1.67 % (18230)Time elapsed: 0.032 s
% 2.70/1.67 % (18230)Instructions burned: 91 (million)
% 2.70/1.67 % (18230)------------------------------
% 2.70/1.67 % (18230)------------------------------
% 2.70/1.68 % (18238)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_17924 for (2998ds/17Mi)
% 2.70/1.68 % (18237)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_17924 for (2998ds/879Mi)
% 2.70/1.68 % (18239)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_17924 for (2998ds/3Mi)
% 2.70/1.68 % (18238)Instruction limit reached!
% 2.70/1.68 % (18238)------------------------------
% 2.70/1.68 % (18238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.68 % (18238)Termination reason: Unknown
% 2.70/1.68 % (18238)Termination phase: Property scanning
% 2.70/1.68
% 2.70/1.68 % (18238)Memory used [KB]: 1535
% 2.70/1.68 % (18238)Time elapsed: 0.006 s
% 2.70/1.68 % (18238)Instructions burned: 17 (million)
% 2.70/1.68 % (18238)------------------------------
% 2.70/1.68 % (18238)------------------------------
% 2.70/1.68 % (18239)Instruction limit reached!
% 2.70/1.68 % (18239)------------------------------
% 2.70/1.68 % (18239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.68 % (18239)Termination reason: Unknown
% 2.70/1.68 % (18239)Termination phase: shuffling
% 2.70/1.68
% 2.70/1.68 % (18239)Memory used [KB]: 1279
% 2.70/1.68 % (18239)Time elapsed: 0.003 s
% 2.70/1.68 % (18239)Instructions burned: 4 (million)
% 2.70/1.68 % (18239)------------------------------
% 2.70/1.68 % (18239)------------------------------
% 2.70/1.69 % (18240)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_17924 for (2998ds/30Mi)
% 2.70/1.69 % (18241)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_17924 for (2998ds/127Mi)
% 2.70/1.70 % (18228)First to succeed.
% 2.70/1.70 % (18228)Refutation found. Thanks to Tanya!
% 2.70/1.70 % SZS status Theorem for DTF2THF_17924
% 2.70/1.70 % SZS output start Proof for DTF2THF_17924
% 2.70/1.70 thf(type_def_5, type, real: $tType).
% 2.70/1.70 thf(type_def_6, type, int: $tType).
% 2.70/1.70 thf(type_def_8, type, nat: $tType).
% 2.70/1.70 thf(type_def_9, type, num: $tType).
% 2.70/1.70 thf(type_def_10, type, set_real: $tType).
% 2.70/1.70 thf(func_def_0, type, set_real: $tType).
% 2.70/1.70 thf(func_def_1, type, real: $tType).
% 2.70/1.70 thf(func_def_2, type, num: $tType).
% 2.70/1.70 thf(func_def_3, type, nat: $tType).
% 2.70/1.70 thf(func_def_4, type, int: $tType).
% 2.70/1.70 thf(func_def_5, type, archim1371465213g_real: real > int).
% 2.70/1.70 thf(func_def_6, type, one_one_int: int).
% 2.70/1.70 thf(func_def_7, type, one_one_nat: nat).
% 2.70/1.70 thf(func_def_8, type, one_one_real: real).
% 2.70/1.70 thf(func_def_9, type, zero_zero_int: int).
% 2.70/1.70 thf(func_def_10, type, zero_zero_nat: nat).
% 2.70/1.70 thf(func_def_11, type, zero_zero_real: real).
% 2.70/1.70 thf(func_def_12, type, if_nat: $o > nat > nat > nat).
% 2.70/1.70 thf(func_def_13, type, nat2: int > nat).
% 2.70/1.70 thf(func_def_14, type, ring_1_of_int_int: int > int).
% 2.70/1.70 thf(func_def_15, type, ring_1_of_int_real: int > real).
% 2.70/1.70 thf(func_def_16, type, suc: nat > nat).
% 2.70/1.70 thf(func_def_17, type, semiri2019852685at_int: nat > int).
% 2.70/1.70 thf(func_def_18, type, semiri1382578993at_nat: nat > nat).
% 2.70/1.70 thf(func_def_19, type, semiri2110766477t_real: nat > real).
% 2.70/1.70 thf(func_def_20, type, neg_numeral_dbl_int: int > int).
% 2.70/1.70 thf(func_def_21, type, neg_numeral_dbl_real: real > real).
% 2.70/1.70 thf(func_def_22, type, bit0: num > num).
% 2.70/1.70 thf(func_def_23, type, one: num).
% 2.70/1.70 thf(func_def_24, type, numeral_numeral_int: num > int).
% 2.70/1.70 thf(func_def_25, type, numeral_numeral_nat: num > nat).
% 2.70/1.70 thf(func_def_26, type, numeral_numeral_real: num > real).
% 2.70/1.70 thf(func_def_27, type, pow: num > num > num).
% 2.70/1.70 thf(func_def_28, type, ord_less_int: int > int > $o).
% 2.70/1.70 thf(func_def_29, type, ord_less_eq_int: int > int > $o).
% 2.70/1.70 thf(func_def_30, type, ord_less_eq_nat: nat > nat > $o).
% 2.70/1.70 thf(func_def_31, type, ord_less_eq_num: num > num > $o).
% 2.70/1.70 thf(func_def_32, type, ord_less_eq_real: real > real > $o).
% 2.70/1.70 thf(func_def_33, type, power_power_int: int > nat > int).
% 2.70/1.70 thf(func_def_34, type, power_power_nat: nat > nat > nat).
% 2.70/1.70 thf(func_def_35, type, power_power_real: real > nat > real).
% 2.70/1.70 thf(func_def_36, type, collect_real: (real > $o) > set_real).
% 2.70/1.70 thf(func_def_37, type, log: real > real > real).
% 2.70/1.70 thf(func_def_38, type, powr_real: real > real > real).
% 2.70/1.70 thf(func_def_39, type, member_real: real > set_real > $o).
% 2.70/1.70 thf(func_def_40, type, d: int).
% 2.70/1.70 thf(func_def_41, type, n: nat).
% 2.70/1.70 thf(func_def_53, type, sK0: real > int).
% 2.70/1.70 thf(func_def_54, type, sK1: real > int).
% 2.70/1.70 thf(f1411,plain,(
% 2.70/1.70 $false),
% 2.70/1.70 inference(subsumption_resolution,[status(thm)],[f1410,f888])).
% 2.70/1.70 thf(f888,plain,(
% 2.70/1.70 ((ord_less_eq_int @ d @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d)))))) != $true)),
% 2.70/1.70 inference(cnf_transformation,[status(thm)],[f738])).
% 2.70/1.70 thf(f738,plain,(
% 2.70/1.70 ((ord_less_eq_int @ d @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d)))))) != $true)),
% 2.70/1.70 inference(flattening,[status(thm)],[f609])).
% 2.70/1.70 thf(f609,plain,(
% 2.70/1.70 ~((ord_less_eq_int @ d @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d)))))) = $true)),
% 2.70/1.70 inference(fool_elimination,[status(thm)],[f608])).
% 2.70/1.70 thf(f608,plain,(
% 2.70/1.70 ~(ord_less_eq_int @ d @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))))))),
% 2.70/1.70 inference(rectify,[status(thm)],[f355])).
% 2.70/1.70 thf(f355,negated_conjecture,(
% 2.70/1.70 ~(ord_less_eq_int @ d @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))))))),
% 2.70/1.70 inference(negated_conjecture,[status(cth)],[f354])).
% 2.70/1.70 thf(f354,conjecture,(
% 2.70/1.70 (ord_less_eq_int @ d @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))))))),
% 2.70/1.70 file('/export/starexec/sandbox/tmp/tmp.Z5U7N5EVnJ/DTF2THF_17924.p',conj_0)).
% 2.70/1.70 thf(f1410,plain,(
% 2.70/1.70 ((ord_less_eq_int @ d @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d)))))) = $true)),
% 2.70/1.70 inference(backward_demodulation,[status(thm)],[f1376,f1399])).
% 2.70/1.70 thf(f1399,plain,(
% 2.70/1.70 ((power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))))) = (archim1371465213g_real @ (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))))))))),
% 2.70/1.70 inference(superposition,[status(thm)],[f864,f855])).
% 2.70/1.70 thf(f855,plain,(
% 2.70/1.70 ((powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d)))))) = (ring_1_of_int_real @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d)))))))),
% 2.70/1.70 inference(cnf_transformation,[status(thm)],[f103])).
% 2.70/1.70 thf(f103,axiom,(
% 2.70/1.70 ((powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d)))))) = (ring_1_of_int_real @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d)))))))),
% 2.70/1.70 file('/export/starexec/sandbox/tmp/tmp.Z5U7N5EVnJ/DTF2THF_17924.p',fact_102__092_060open_0622_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_A_061_Areal__of__int_A_I2_A_094_Anat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062)).
% 2.70/1.70 thf(f864,plain,(
% 2.70/1.70 ( ! [X0 : int] : (((archim1371465213g_real @ (ring_1_of_int_real @ X0)) = X0)) )),
% 2.70/1.70 inference(cnf_transformation,[status(thm)],[f742])).
% 2.70/1.70 thf(f742,plain,(
% 2.70/1.70 ! [X0 : int] : ((archim1371465213g_real @ (ring_1_of_int_real @ X0)) = X0)),
% 2.70/1.70 inference(rectify,[status(thm)],[f26])).
% 2.70/1.70 thf(f26,axiom,(
% 2.70/1.70 ! [X7 : int] : ((archim1371465213g_real @ (ring_1_of_int_real @ X7)) = X7)),
% 2.70/1.70 file('/export/starexec/sandbox/tmp/tmp.Z5U7N5EVnJ/DTF2THF_17924.p',fact_25_ceiling__of__int)).
% 2.70/1.70 thf(f1376,plain,(
% 2.70/1.70 ($true = (ord_less_eq_int @ d @ (archim1371465213g_real @ (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d)))))))))),
% 2.70/1.70 inference(forward_demodulation,[status(thm)],[f1375,f864])).
% 2.70/1.70 thf(f1375,plain,(
% 2.70/1.70 ($true = (ord_less_eq_int @ (archim1371465213g_real @ (ring_1_of_int_real @ d)) @ (archim1371465213g_real @ (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d)))))))))),
% 2.70/1.70 inference(trivial_inequality_removal,[status(thm)],[f1374])).
% 2.70/1.70 thf(f1374,plain,(
% 2.70/1.70 ($true != $true) | ($true = (ord_less_eq_int @ (archim1371465213g_real @ (ring_1_of_int_real @ d)) @ (archim1371465213g_real @ (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d)))))))))),
% 2.70/1.70 inference(superposition,[status(thm)],[f875,f1019])).
% 2.70/1.70 thf(f1019,plain,(
% 2.70/1.70 ($true = (ord_less_eq_real @ (ring_1_of_int_real @ d) @ (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))))))))),
% 2.70/1.70 inference(backward_demodulation,[status(thm)],[f869,f878])).
% 2.70/1.70 thf(f878,plain,(
% 2.70/1.70 ((ring_1_of_int_real @ d) = (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))))),
% 2.70/1.70 inference(cnf_transformation,[status(thm)],[f54])).
% 2.70/1.70 thf(f54,axiom,(
% 2.70/1.70 ((ring_1_of_int_real @ d) = (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))))),
% 2.70/1.70 file('/export/starexec/sandbox/tmp/tmp.Z5U7N5EVnJ/DTF2THF_17924.p',fact_53__092_060open_062real__of__int_Ad_A_061_A2_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_092_060close_062)).
% 2.70/1.70 thf(f869,plain,(
% 2.70/1.70 ((ord_less_eq_real @ (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))) @ (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))))))) = $true)),
% 2.70/1.70 inference(cnf_transformation,[status(thm)],[f534])).
% 2.70/1.70 thf(f534,plain,(
% 2.70/1.70 ((ord_less_eq_real @ (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))) @ (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))))))) = $true)),
% 2.70/1.70 inference(fool_elimination,[status(thm)],[f533])).
% 2.70/1.70 thf(f533,plain,(
% 2.70/1.70 (ord_less_eq_real @ (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))) @ (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d)))))))),
% 2.70/1.70 inference(rectify,[status(thm)],[f104])).
% 2.70/1.70 thf(f104,axiom,(
% 2.70/1.70 (ord_less_eq_real @ (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d))) @ (powr_real @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ (nat2 @ (archim1371465213g_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (ring_1_of_int_real @ d)))))))),
% 2.70/1.70 file('/export/starexec/sandbox/tmp/tmp.Z5U7N5EVnJ/DTF2THF_17924.p',fact_103__092_060open_0622_Apowr_Alog_A2_A_Ireal__of__int_Ad_J_A_092_060le_062_A2_Apowr_Areal_A_Inat_A_092_060lceil_062log_A2_A_Ireal__of__int_Ad_J_092_060rceil_062_J_092_060close_062)).
% 2.70/1.70 thf(f875,plain,(
% 2.70/1.70 ( ! [X0 : real,X1 : real] : (($true != (ord_less_eq_real @ X0 @ X1)) | ($true = (ord_less_eq_int @ (archim1371465213g_real @ X0) @ (archim1371465213g_real @ X1)))) )),
% 2.70/1.70 inference(cnf_transformation,[status(thm)],[f783])).
% 2.70/1.70 thf(f783,plain,(
% 2.70/1.70 ! [X0 : real,X1 : real] : (($true = (ord_less_eq_int @ (archim1371465213g_real @ X0) @ (archim1371465213g_real @ X1))) | ($true != (ord_less_eq_real @ X0 @ X1)))),
% 2.70/1.70 inference(ennf_transformation,[status(thm)],[f392])).
% 2.70/1.70 thf(f392,plain,(
% 2.70/1.70 ! [X0 : real,X1 : real] : (($true = (ord_less_eq_real @ X0 @ X1)) => ($true = (ord_less_eq_int @ (archim1371465213g_real @ X0) @ (archim1371465213g_real @ X1))))),
% 2.70/1.70 inference(fool_elimination,[status(thm)],[f391])).
% 2.70/1.70 thf(f391,plain,(
% 2.70/1.70 ! [X0 : real,X1 : real] : ((ord_less_eq_real @ X0 @ X1) => (ord_less_eq_int @ (archim1371465213g_real @ X0) @ (archim1371465213g_real @ X1)))),
% 2.70/1.70 inference(rectify,[status(thm)],[f44])).
% 2.70/1.70 thf(f44,axiom,(
% 2.70/1.70 ! [X3 : real,X0 : real] : ((ord_less_eq_real @ X3 @ X0) => (ord_less_eq_int @ (archim1371465213g_real @ X3) @ (archim1371465213g_real @ X0)))),
% 2.70/1.70 file('/export/starexec/sandbox/tmp/tmp.Z5U7N5EVnJ/DTF2THF_17924.p',fact_43_ceiling__mono)).
% 2.70/1.70 % SZS output end Proof for DTF2THF_17924
% 2.70/1.70 % (18228)------------------------------
% 2.70/1.70 % (18228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.70 % (18228)Termination reason: Refutation
% 2.70/1.70
% 2.70/1.70 % (18228)Memory used [KB]: 6780
% 2.70/1.70 % (18228)Time elapsed: 0.066 s
% 2.70/1.70 % (18228)Instructions burned: 194 (million)
% 2.70/1.70 % (18228)------------------------------
% 2.70/1.70 % (18228)------------------------------
% 2.70/1.70 % (18207)Success in time 0.142 s
%------------------------------------------------------------------------------