%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP095^1 : 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 : n028.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 08:55:30 AM UTC 2026
% Result : Theorem 5.62s 2.20s
% Output : Refutation 5.62s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 8
% Syntax : Number of formulae : 39 ( 27 unt; 0 typ; 0 def)
% Number of atoms : 142 ( 56 equ; 0 cnn)
% Maximal formula atoms : 4 ( 3 avg)
% Number of connectives : 512 ( 33 ~; 21 |; 0 &; 448 @)
% ( 0 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 100 ( 97 usr; 28 con; 0-3 aty)
% Number of variables : 35 ( 0 ^; 32 !; 3 ?; 35 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
int: $tType ).
thf(type_def_7,type,
poly_real: $tType ).
thf(type_def_8,type,
real: $tType ).
thf(type_def_9,type,
nat: $tType ).
thf(type_def_10,type,
poly_int: $tType ).
thf(type_def_11,type,
poly_nat: $tType ).
thf(type_def_12,type,
poly_poly_real: $tType ).
thf(type_def_13,type,
set_int: $tType ).
thf(type_def_14,type,
set_poly_real: $tType ).
thf(type_def_15,type,
set_real: $tType ).
thf(func_def_0,type,
poly_poly_real: $tType ).
thf(func_def_1,type,
set_poly_real: $tType ).
thf(func_def_2,type,
poly_real: $tType ).
thf(func_def_3,type,
poly_nat: $tType ).
thf(func_def_4,type,
poly_int: $tType ).
thf(func_def_5,type,
set_real: $tType ).
thf(func_def_6,type,
set_int: $tType ).
thf(func_def_7,type,
real: $tType ).
thf(func_def_8,type,
nat: $tType ).
thf(func_def_9,type,
int: $tType ).
thf(func_def_10,type,
abs_abs_int: int > int ).
thf(func_def_11,type,
abs_abs_poly_real: poly_real > poly_real ).
thf(func_def_12,type,
abs_abs_real: real > real ).
thf(func_def_13,type,
minus_minus_real: real > real > real ).
thf(func_def_14,type,
one_one_int: int ).
thf(func_def_15,type,
one_one_nat: nat ).
thf(func_def_16,type,
one_one_poly_int: poly_int ).
thf(func_def_17,type,
one_one_poly_nat: poly_nat ).
thf(func_def_18,type,
one_one_poly_real: poly_real ).
thf(func_def_19,type,
one_one_real: real ).
thf(func_def_20,type,
zero_zero_int: int ).
thf(func_def_21,type,
zero_zero_nat: nat ).
thf(func_def_22,type,
zero_zero_poly_int: poly_int ).
thf(func_def_23,type,
zero_zero_poly_nat: poly_nat ).
thf(func_def_24,type,
zero_z1423781445y_real: poly_poly_real ).
thf(func_def_25,type,
zero_zero_poly_real: poly_real ).
thf(func_def_26,type,
zero_zero_real: real ).
thf(func_def_27,type,
ring_1_Ints_int: set_int ).
thf(func_def_28,type,
ring_1690226883y_real: set_poly_real ).
thf(func_def_29,type,
ring_1_Ints_real: set_real ).
thf(func_def_30,type,
ring_1_of_int_int: int > int ).
thf(func_def_31,type,
ring_12102921859ly_int: int > poly_int ).
thf(func_def_32,type,
ring_11511526659y_real: int > poly_real ).
thf(func_def_33,type,
ring_1_of_int_real: int > real ).
thf(func_def_34,type,
ord_less_int: int > int > $o ).
thf(func_def_35,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_36,type,
ord_less_poly_real: poly_real > poly_real > $o ).
thf(func_def_37,type,
ord_less_real: real > real > $o ).
thf(func_def_38,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_39,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_40,type,
ord_le1180086932y_real: poly_real > poly_real > $o ).
thf(func_def_41,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_42,type,
algebraic_real: real > $o ).
thf(func_def_43,type,
degree_int: poly_int > nat ).
thf(func_def_44,type,
degree_nat: poly_nat > nat ).
thf(func_def_45,type,
degree_poly_real: poly_poly_real > nat ).
thf(func_def_46,type,
degree_real: poly_real > nat ).
thf(func_def_47,type,
poly_int2: poly_int > int > int ).
thf(func_def_48,type,
poly_nat2: poly_nat > nat > nat ).
thf(func_def_49,type,
poly_poly_real2: poly_poly_real > poly_real > poly_real ).
thf(func_def_50,type,
poly_real2: poly_real > real > real ).
thf(func_def_51,type,
coeff_int: poly_int > nat > int ).
thf(func_def_52,type,
coeff_nat: poly_nat > nat > nat ).
thf(func_def_53,type,
coeff_poly_real: poly_poly_real > nat > poly_real ).
thf(func_def_54,type,
coeff_real: poly_real > nat > real ).
thf(func_def_55,type,
power_power_int: int > nat > int ).
thf(func_def_56,type,
power_power_nat: nat > nat > nat ).
thf(func_def_57,type,
power_power_poly_int: poly_int > nat > poly_int ).
thf(func_def_58,type,
power_power_poly_nat: poly_nat > nat > poly_nat ).
thf(func_def_59,type,
power_2108872382y_real: poly_real > nat > poly_real ).
thf(func_def_60,type,
power_power_real: real > nat > real ).
thf(func_def_61,type,
field_1537545994s_real: set_real ).
thf(func_def_62,type,
divide_divide_int: int > int > int ).
thf(func_def_63,type,
divide_divide_nat: nat > nat > nat ).
thf(func_def_64,type,
divide1727078534y_real: poly_real > poly_real > poly_real ).
thf(func_def_65,type,
divide_divide_real: real > real > real ).
thf(func_def_66,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_67,type,
member_int: int > set_int > $o ).
thf(func_def_68,type,
member_poly_real: poly_real > set_poly_real > $o ).
thf(func_def_69,type,
member_real: real > set_real > $o ).
thf(func_def_70,type,
a: real ).
thf(func_def_71,type,
m: real ).
thf(func_def_72,type,
a2: int ).
thf(func_def_73,type,
b: int ).
thf(func_def_74,type,
n: nat ).
thf(func_def_75,type,
p: poly_real ).
thf(func_def_76,type,
roots: set_real ).
thf(func_def_78,type,
x: real ).
thf(func_def_93,type,
sK0:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_94,type,
sK1:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_95,type,
sK2: poly_real > poly_real > nat ).
thf(func_def_96,type,
sK3: poly_real > nat ).
thf(func_def_97,type,
sK4: real > poly_real > real > real ).
thf(func_def_98,type,
sK5: poly_poly_real > nat > nat ).
thf(func_def_99,type,
sK6: nat > poly_nat > nat ).
thf(func_def_100,type,
sK7: real > nat > int ).
thf(func_def_101,type,
sK8: real > nat > int ).
thf(func_def_102,type,
sK9: real > poly_real ).
thf(func_def_103,type,
sK10: poly_real > nat > nat ).
thf(func_def_104,type,
sK11: real > int ).
thf(func_def_105,type,
sK12: nat > poly_real > nat ).
thf(func_def_106,type,
sK13: real > poly_real > real > real ).
thf(func_def_107,type,
sK14: real > real ).
thf(func_def_108,type,
sK15: poly_real ).
thf(func_def_109,type,
sK16: nat > poly_int > nat ).
thf(func_def_110,type,
sK17: poly_real > real ).
thf(func_def_111,type,
sK18: nat > poly_int > nat ).
thf(func_def_112,type,
sK19: ( real > $o ) > int ).
thf(func_def_113,type,
sK20: real > real ).
thf(func_def_114,type,
sK21: poly_poly_real > nat > nat ).
thf(func_def_115,type,
sK22: ( int > $o ) > int ).
thf(func_def_116,type,
sK23: int > int ).
thf(func_def_117,type,
sK24: real > int ).
thf(func_def_118,type,
sK25: real > int ).
thf(func_def_119,type,
sK26: poly_real > nat ).
thf(func_def_120,type,
sK27: poly_nat > nat > nat ).
thf(func_def_122,type,
ph29:
!>[X0: $tType] : X0 ).
thf(f2062,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2061,f1486]) ).
thf(f1486,plain,
! [X0: nat] :
( ( member_real @ ( coeff_real @ p @ X0 ) @ ring_1_Ints_real )
= $true ),
inference(cnf_transformation,[status(thm)],[f403]) ).
thf(f403,plain,
! [X0: nat] :
( ( member_real @ ( coeff_real @ p @ X0 ) @ ring_1_Ints_real )
= $true ),
inference(fool_elimination,[status(thm)],[f402]) ).
thf(f402,plain,
! [X0: nat] : ( member_real @ ( coeff_real @ p @ X0 ) @ ring_1_Ints_real ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X0: nat] : ( member_real @ ( coeff_real @ p @ X0 ) @ ring_1_Ints_real ),
file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',fact_2_p_I1_J) ).
thf(f2061,plain,
( $true
!= ( member_real @ ( coeff_real @ p @ ( sK26 @ p ) ) @ ring_1_Ints_real ) ),
inference(subsumption_resolution,[status(thm)],[f2060,f1536]) ).
thf(f1536,plain,
( ( ord_less_int @ zero_zero_int @ b )
= $true ),
inference(cnf_transformation,[status(thm)],[f1024]) ).
thf(f1024,plain,
( ( ord_less_int @ zero_zero_int @ b )
= $true ),
inference(fool_elimination,[status(thm)],[f1023]) ).
thf(f1023,plain,
ord_less_int @ zero_zero_int @ b,
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
ord_less_int @ zero_zero_int @ b,
file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',fact_3_b) ).
thf(f2060,plain,
( ( ( ord_less_int @ zero_zero_int @ b )
!= $true )
| ( $true
!= ( member_real @ ( coeff_real @ p @ ( sK26 @ p ) ) @ ring_1_Ints_real ) ) ),
inference(subsumption_resolution,[status(thm)],[f2045,f1804]) ).
thf(f1804,plain,
( ( poly_real2 @ p @ x )
!= ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1445,f1400]) ).
thf(f1400,plain,
( ( poly_real2 @ p @ x )
= zero_zero_real ),
inference(cnf_transformation,[status(thm)],[f776]) ).
thf(f776,plain,
( ( poly_real2 @ p @ x )
= zero_zero_real ),
inference(fool_elimination,[status(thm)],[f5]) ).
thf(f5,axiom,
( ( poly_real2 @ p @ x )
= zero_zero_real ),
file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',fact_4_p_I3_J) ).
thf(f1445,plain,
( zero_zero_real
!= ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ),
inference(cnf_transformation,[status(thm)],[f699]) ).
thf(f699,plain,
( zero_zero_real
!= ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ),
inference(fool_elimination,[status(thm)],[f6]) ).
thf(f6,axiom,
( zero_zero_real
!= ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',fact_5_no__root) ).
thf(f2045,plain,
( ( ( poly_real2 @ p @ x )
= ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) )
| ( ( ord_less_int @ zero_zero_int @ b )
!= $true )
| ( $true
!= ( member_real @ ( coeff_real @ p @ ( sK26 @ p ) ) @ ring_1_Ints_real ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2044]) ).
thf(f2044,plain,
( ( ( poly_real2 @ p @ x )
= ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) )
| ( $true != $true )
| ( $true
!= ( member_real @ ( coeff_real @ p @ ( sK26 @ p ) ) @ ring_1_Ints_real ) )
| ( ( ord_less_int @ zero_zero_int @ b )
!= $true ) ),
inference(superposition,[status(thm)],[f1951,f2007]) ).
thf(f2007,plain,
! [X2: int,X0: poly_real,X1: int] :
( ( $true
= ( ord_less_eq_real @ ( divide_divide_real @ ( coeff_real @ one_one_poly_real @ ( degree_real @ one_one_poly_real ) ) @ ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ ( degree_real @ X0 ) ) ) @ ( abs_abs_real @ ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) ) ) )
| ( ( poly_real2 @ p @ x )
= ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) )
| ( $true
!= ( member_real @ ( coeff_real @ X0 @ ( sK26 @ X0 ) ) @ ring_1_Ints_real ) )
| ( ( ord_less_int @ zero_zero_int @ X1 )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f1729,f1400,f1437]) ).
thf(f1437,plain,
( one_one_real
= ( coeff_real @ one_one_poly_real @ ( degree_real @ one_one_poly_real ) ) ),
inference(cnf_transformation,[status(thm)],[f418]) ).
thf(f418,plain,
( one_one_real
= ( coeff_real @ one_one_poly_real @ ( degree_real @ one_one_poly_real ) ) ),
inference(fool_elimination,[status(thm)],[f69]) ).
thf(f69,axiom,
( one_one_real
= ( coeff_real @ one_one_poly_real @ ( degree_real @ one_one_poly_real ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',fact_68_lead__coeff__1) ).
thf(f1729,plain,
! [X2: int,X0: poly_real,X1: int] :
( ( ( ord_less_int @ zero_zero_int @ X1 )
!= $true )
| ( $true
!= ( member_real @ ( coeff_real @ X0 @ ( sK26 @ X0 ) ) @ ring_1_Ints_real ) )
| ( zero_zero_real
= ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) )
| ( $true
= ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ ( degree_real @ X0 ) ) ) @ ( abs_abs_real @ ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1375]) ).
thf(f1375,plain,
! [X0: poly_real,X1: int,X2: int] :
( ( ( ord_less_int @ zero_zero_int @ X1 )
!= $true )
| ( $true
!= ( member_real @ ( coeff_real @ X0 @ ( sK26 @ X0 ) ) @ ring_1_Ints_real ) )
| ( zero_zero_real
= ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) )
| ( $true
= ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ ( degree_real @ X0 ) ) ) @ ( abs_abs_real @ ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK26])],[f1309,f1374]) ).
thf(f1374,plain,
! [X0: poly_real] :
( ? [X3: nat] :
( $true
!= ( member_real @ ( coeff_real @ X0 @ X3 ) @ ring_1_Ints_real ) )
=> ( $true
!= ( member_real @ ( coeff_real @ X0 @ ( sK26 @ X0 ) ) @ ring_1_Ints_real ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1309,plain,
! [X0: poly_real,X1: int,X2: int] :
( ( ( ord_less_int @ zero_zero_int @ X1 )
!= $true )
| ? [X3: nat] :
( $true
!= ( member_real @ ( coeff_real @ X0 @ X3 ) @ ring_1_Ints_real ) )
| ( zero_zero_real
= ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) )
| ( $true
= ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ ( degree_real @ X0 ) ) ) @ ( abs_abs_real @ ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f1308]) ).
thf(f1308,plain,
! [X0: poly_real,X1: int,X2: int] :
( ( $true
= ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ ( degree_real @ X0 ) ) ) @ ( abs_abs_real @ ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) ) ) )
| ( zero_zero_real
= ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) )
| ( ( ord_less_int @ zero_zero_int @ X1 )
!= $true )
| ? [X3: nat] :
( $true
!= ( member_real @ ( coeff_real @ X0 @ X3 ) @ ring_1_Ints_real ) ) ),
inference(ennf_transformation,[status(thm)],[f890]) ).
thf(f890,plain,
! [X0: poly_real,X1: int,X2: int] :
( ! [X3: nat] :
( $true
= ( member_real @ ( coeff_real @ X0 @ X3 ) @ ring_1_Ints_real ) )
=> ( ( ( ord_less_int @ zero_zero_int @ X1 )
= $true )
=> ( ( zero_zero_real
!= ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) )
=> ( $true
= ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ ( degree_real @ X0 ) ) ) @ ( abs_abs_real @ ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f889]) ).
thf(f889,plain,
! [X0: poly_real,X1: int,X2: int] :
( ! [X3: nat] : ( member_real @ ( coeff_real @ X0 @ X3 ) @ ring_1_Ints_real )
=> ( ( ord_less_int @ zero_zero_int @ X1 )
=> ( ( zero_zero_real
!= ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ ( degree_real @ X0 ) ) ) @ ( abs_abs_real @ ( poly_real2 @ X0 @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f155]) ).
thf(f155,axiom,
! [X5: poly_real,X1: int,X12: int] :
( ! [X11: nat] : ( member_real @ ( coeff_real @ X5 @ X11 ) @ ring_1_Ints_real )
=> ( ( ord_less_int @ zero_zero_int @ X1 )
=> ( ( zero_zero_real
!= ( poly_real2 @ X5 @ ( divide_divide_real @ ( ring_1_of_int_real @ X12 ) @ ( ring_1_of_int_real @ X1 ) ) ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ ( degree_real @ X5 ) ) ) @ ( abs_abs_real @ ( poly_real2 @ X5 @ ( divide_divide_real @ ( ring_1_of_int_real @ X12 ) @ ( ring_1_of_int_real @ X1 ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',fact_154_int__poly__rat__no__root__ge) ).
thf(f1951,plain,
( $true
!= ( ord_less_eq_real @ ( divide_divide_real @ ( coeff_real @ one_one_poly_real @ ( degree_real @ one_one_poly_real ) ) @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1641,f1437]) ).
thf(f1641,plain,
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1071]) ).
thf(f1071,plain,
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) )
!= $true ),
inference(flattening,[status(thm)],[f863]) ).
thf(f863,plain,
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f862]) ).
thf(f862,plain,
~ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) ),
inference(rectify,[status(thm)],[f355]) ).
thf(f355,negated_conjecture,
~ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f354]) ).
thf(f354,conjecture,
ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP095^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n028.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 08:26:27 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
% 0.19/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : ITP095^1 : 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_189__5867098_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_189__5867098_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, 0.23 v8.1.0, 0.27 v7.5.0
% 0.30/0.46 % Syntax : Number of formulae : 433 ( 186 unt; 79 typ; 0 def)
% 0.30/0.46 % Number of atoms : 956 ( 415 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 6 ( 2 avg)
% 0.30/0.46 % Number of connectives : 2550 ( 66 ~; 22 |; 36 &;2130 @)
% 0.30/0.46 % ( 0 <=>; 296 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 14 ( 5 avg)
% 0.30/0.46 % Number of types : 11 ( 10 usr)
% 0.30/0.46 % Number of type conns : 77 ( 77 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 70 ( 69 usr; 26 con; 0-2 aty)
% 0.30/0.46 % Number of variables : 634 ( 11 ^; 612 !; 11 ?; 634 :)
% 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 15:40:52.492
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % Could-be-implicit typings (10)
% 0.30/0.46 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.30/0.46 poly_poly_real: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.30/0.46 set_poly_real: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 poly_real: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.30/0.46 poly_nat: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Polynomial__Opoly_It__Int__Oint_J,type,
% 0.30/0.46 poly_int: $tType ).
% 0.30/0.46
% 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__Set__Oset_It__Int__Oint_J,type,
% 0.30/0.46 set_int: $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__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 (69)
% 0.30/0.46 thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
% 0.30/0.46 abs_abs_int: int > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Oabs__class_Oabs_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 abs_abs_poly_real: poly_real > poly_real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 0.30/0.46 abs_abs_real: real > real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 0.30/0.46 minus_minus_real: real > real > real ).
% 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__Polynomial__Opoly_It__Int__Oint_J,type,
% 0.30/0.46 one_one_poly_int: poly_int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.30/0.46 one_one_poly_nat: poly_nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 one_one_poly_real: poly_real ).
% 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__Polynomial__Opoly_It__Int__Oint_J,type,
% 0.30/0.46 zero_zero_poly_int: poly_int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.30/0.46 zero_zero_poly_nat: poly_nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 0.30/0.46 zero_z1423781445y_real: poly_poly_real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 zero_zero_poly_real: poly_real ).
% 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_Int_Oring__1__class_OInts_001t__Int__Oint,type,
% 0.30/0.46 ring_1_Ints_int: set_int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Int_Oring__1__class_OInts_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 ring_1690226883y_real: set_poly_real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
% 0.30/0.46 ring_1_Ints_real: set_real ).
% 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__Polynomial__Opoly_It__Int__Oint_J,type,
% 0.30/0.46 ring_12102921859ly_int: int > poly_int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 ring_11511526659y_real: int > poly_real ).
% 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_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_001t__Nat__Onat,type,
% 0.30/0.46 ord_less_nat: nat > nat > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 ord_less_poly_real: poly_real > poly_real > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.30/0.46 ord_less_real: real > real > $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__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 ord_le1180086932y_real: poly_real > poly_real > $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_Polynomial_Oalgebraic_001t__Real__Oreal,type,
% 0.30/0.46 algebraic_real: real > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Polynomial_Odegree_001t__Int__Oint,type,
% 0.30/0.46 degree_int: poly_int > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Polynomial_Odegree_001t__Nat__Onat,type,
% 0.30/0.46 degree_nat: poly_nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 degree_poly_real: poly_poly_real > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Polynomial_Odegree_001t__Real__Oreal,type,
% 0.30/0.46 degree_real: poly_real > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Polynomial_Opoly_001t__Int__Oint,type,
% 0.30/0.46 poly_int2: poly_int > int > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Polynomial_Opoly_001t__Nat__Onat,type,
% 0.30/0.46 poly_nat2: poly_nat > nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 poly_poly_real2: poly_poly_real > poly_real > poly_real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Polynomial_Opoly_001t__Real__Oreal,type,
% 0.30/0.46 poly_real2: poly_real > real > real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Int__Oint,type,
% 0.30/0.46 coeff_int: poly_int > nat > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat,type,
% 0.30/0.46 coeff_nat: poly_nat > nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 coeff_poly_real: poly_poly_real > nat > poly_real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Real__Oreal,type,
% 0.30/0.46 coeff_real: poly_real > nat > real ).
% 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__Polynomial__Opoly_It__Int__Oint_J,type,
% 0.30/0.46 power_power_poly_int: poly_int > nat > poly_int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 0.30/0.46 power_power_poly_nat: poly_nat > nat > poly_nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 power_2108872382y_real: poly_real > nat > poly_real ).
% 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_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 0.30/0.46 field_1537545994s_real: set_real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 0.30/0.46 divide_divide_int: int > int > int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 0.30/0.46 divide_divide_nat: nat > nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Rings_Odivide__class_Odivide_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 divide1727078534y_real: poly_real > poly_real > poly_real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 0.30/0.46 divide_divide_real: real > real > 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_member_001t__Int__Oint,type,
% 0.30/0.46 member_int: int > set_int > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_member_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 0.30/0.46 member_poly_real: poly_real > set_poly_real > $o ).
% 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_A____,type,
% 0.30/0.46 a: real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_M____,type,
% 0.30/0.46 m: real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_a____,type,
% 0.30/0.46 a2: int ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_b____,type,
% 0.30/0.46 b: 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 thf(sy_v_p____,type,
% 0.30/0.46 p: poly_real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_roots____,type,
% 0.30/0.46 roots: set_real ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_thesis,type,
% 0.30/0.46 thesis: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_x,type,
% 0.30/0.46 x: real ).
% 0.30/0.46
% 0.30/0.46 % Relevant facts (353)
% 0.30/0.46 thf(fact_0_assms_I2_J,axiom,
% 0.30/0.46 algebraic_real @ x ).
% 0.30/0.46
% 0.30/0.46 % assms(2)
% 0.30/0.46 thf(fact_1_p_I2_J,axiom,
% 0.30/0.46 p != zero_zero_poly_real ).
% 0.30/0.46
% 0.30/0.46 % p(2)
% 0.30/0.46 thf(fact_2_p_I1_J,axiom,
% 0.30/0.46 ! [I: nat] : ( member_real @ ( coeff_real @ p @ I ) @ ring_1_Ints_real ) ).
% 0.30/0.46
% 0.30/0.46 % p(1)
% 0.30/0.46 thf(fact_3_b,axiom,
% 0.30/0.46 ord_less_int @ zero_zero_int @ b ).
% 0.30/0.46
% 0.30/0.46 % b
% 0.30/0.46 thf(fact_4_p_I3_J,axiom,
% 0.30/0.46 ( ( poly_real2 @ p @ x )
% 0.30/0.46 = zero_zero_real ) ).
% 0.30/0.46
% 0.30/0.46 % p(3)
% 0.30/0.46 thf(fact_5_no__root,axiom,
% 0.30/0.46 ( ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) )
% 0.30/0.46 != zero_zero_real ) ).
% 0.30/0.46
% 0.30/0.46 % no_root
% 0.30/0.46 thf(fact_6_n__def,axiom,
% 0.30/0.46 ( n
% 0.30/0.46 = ( degree_real @ p ) ) ).
% 0.30/0.46
% 0.30/0.46 % n_def
% 0.30/0.46 thf(fact_7_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_8_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_9_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_10_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_11_of__int__1__le__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.46 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_1_le_iff
% 0.30/0.46 thf(fact_12_of__int__1__le__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.46 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_1_le_iff
% 0.30/0.46 thf(fact_13_of__int__le__1__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.30/0.46 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_le_1_iff
% 0.30/0.46 thf(fact_14_of__int__le__1__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.30/0.46 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_le_1_iff
% 0.30/0.46 thf(fact_15_poly__power,axiom,
% 0.30/0.46 ! [P: poly_real,N: nat,X: real] :
% 0.30/0.46 ( ( poly_real2 @ ( power_2108872382y_real @ P @ N ) @ X )
% 0.30/0.46 = ( power_power_real @ ( poly_real2 @ P @ X ) @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % poly_power
% 0.30/0.46 thf(fact_16_poly__power,axiom,
% 0.30/0.46 ! [P: poly_nat,N: nat,X: nat] :
% 0.30/0.46 ( ( poly_nat2 @ ( power_power_poly_nat @ P @ N ) @ X )
% 0.30/0.46 = ( power_power_nat @ ( poly_nat2 @ P @ X ) @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % poly_power
% 0.30/0.46 thf(fact_17_poly__power,axiom,
% 0.30/0.46 ! [P: poly_int,N: nat,X: int] :
% 0.30/0.46 ( ( poly_int2 @ ( power_power_poly_int @ P @ N ) @ X )
% 0.30/0.46 = ( power_power_int @ ( poly_int2 @ P @ X ) @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % poly_power
% 0.30/0.46 thf(fact_18_poly__1,axiom,
% 0.30/0.46 ! [X: real] :
% 0.30/0.46 ( ( poly_real2 @ one_one_poly_real @ X )
% 0.30/0.46 = one_one_real ) ).
% 0.30/0.46
% 0.30/0.46 % poly_1
% 0.30/0.46 thf(fact_19_poly__1,axiom,
% 0.30/0.46 ! [X: int] :
% 0.30/0.46 ( ( poly_int2 @ one_one_poly_int @ X )
% 0.30/0.46 = one_one_int ) ).
% 0.30/0.46
% 0.30/0.46 % poly_1
% 0.30/0.46 thf(fact_20_poly__1,axiom,
% 0.30/0.46 ! [X: nat] :
% 0.30/0.46 ( ( poly_nat2 @ one_one_poly_nat @ X )
% 0.30/0.46 = one_one_nat ) ).
% 0.30/0.46
% 0.30/0.46 % poly_1
% 0.30/0.46 thf(fact_21_of__int__abs,axiom,
% 0.30/0.46 ! [X: int] :
% 0.30/0.46 ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
% 0.30/0.46 = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_abs
% 0.30/0.46 thf(fact_22_of__int__abs,axiom,
% 0.30/0.46 ! [X: int] :
% 0.30/0.46 ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
% 0.30/0.46 = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_abs
% 0.30/0.46 thf(fact_23_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_24_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_25_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_26_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_27_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_28_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_29_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_30_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_31_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_32_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_33_irrationsl,axiom,
% 0.30/0.46 ~ ( member_real @ x @ field_1537545994s_real ) ).
% 0.30/0.46
% 0.30/0.46 % irrationsl
% 0.30/0.46 thf(fact_34_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_35_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_36__092_060open_062_092_060And_062thesisa_O_A_I_092_060And_062p_O_A_092_060lbrakk_062_092_060And_062i_O_Acoeff_Ap_Ai_A_092_060in_062_A_092_060int_062_059_Ap_A_092_060noteq_062_A0_059_Apoly_Ap_Ax_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesisa_J_A_092_060Longrightarrow_062_Athesisa_092_060close_062,axiom,
% 0.30/0.46 ~ ! [P2: poly_real] :
% 0.30/0.46 ( ! [I2: nat] : ( member_real @ ( coeff_real @ P2 @ I2 ) @ ring_1_Ints_real )
% 0.30/0.46 => ( ( P2 != zero_zero_poly_real )
% 0.30/0.46 => ( ( poly_real2 @ P2 @ x )
% 0.30/0.46 != zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % \<open>\<And>thesisa. (\<And>p. \<lbrakk>\<And>i. coeff p i \<in> \<int>; p \<noteq> 0; poly p x = 0\<rbrakk> \<Longrightarrow> thesisa) \<Longrightarrow> thesisa\<close>
% 0.30/0.46 thf(fact_37__092_060open_062x_A_092_060notin_062_Aroots_092_060close_062,axiom,
% 0.30/0.46 ~ ( member_real @ x @ roots ) ).
% 0.30/0.46
% 0.30/0.46 % \<open>x \<notin> roots\<close>
% 0.30/0.46 thf(fact_38_of__int__eq__0__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ( ring_11511526659y_real @ Z )
% 0.30/0.46 = zero_zero_poly_real )
% 0.30/0.46 = ( Z = zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_eq_0_iff
% 0.30/0.46 thf(fact_39_of__int__eq__0__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ( ring_1_of_int_real @ Z )
% 0.30/0.46 = zero_zero_real )
% 0.30/0.46 = ( Z = zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_eq_0_iff
% 0.30/0.46 thf(fact_40_of__int__eq__0__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ( ring_1_of_int_int @ Z )
% 0.30/0.46 = zero_zero_int )
% 0.30/0.46 = ( Z = zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_eq_0_iff
% 0.30/0.46 thf(fact_41_of__int__0__eq__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( zero_zero_poly_real
% 0.30/0.46 = ( ring_11511526659y_real @ Z ) )
% 0.30/0.46 = ( Z = zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_0_eq_iff
% 0.30/0.46 thf(fact_42_of__int__0__eq__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( zero_zero_real
% 0.30/0.46 = ( ring_1_of_int_real @ Z ) )
% 0.30/0.46 = ( Z = zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_0_eq_iff
% 0.30/0.46 thf(fact_43_of__int__0__eq__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( zero_zero_int
% 0.30/0.46 = ( ring_1_of_int_int @ Z ) )
% 0.30/0.46 = ( Z = zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_0_eq_iff
% 0.30/0.46 thf(fact_44_of__int__0,axiom,
% 0.30/0.46 ( ( ring_11511526659y_real @ zero_zero_int )
% 0.30/0.46 = zero_zero_poly_real ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_0
% 0.30/0.46 thf(fact_45_of__int__0,axiom,
% 0.30/0.46 ( ( ring_1_of_int_real @ zero_zero_int )
% 0.30/0.46 = zero_zero_real ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_0
% 0.30/0.46 thf(fact_46_of__int__0,axiom,
% 0.30/0.46 ( ( ring_1_of_int_int @ zero_zero_int )
% 0.30/0.46 = zero_zero_int ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_0
% 0.30/0.46 thf(fact_47_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_48_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_49_of__int__less__iff,axiom,
% 0.30/0.46 ! [W: int,Z: int] :
% 0.30/0.46 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.46 = ( ord_less_int @ W @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_less_iff
% 0.30/0.46 thf(fact_50_of__int__less__iff,axiom,
% 0.30/0.46 ! [W: int,Z: int] :
% 0.30/0.46 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.46 = ( ord_less_int @ W @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_less_iff
% 0.30/0.46 thf(fact_51_coeff__0,axiom,
% 0.30/0.46 ! [N: nat] :
% 0.30/0.46 ( ( coeff_poly_real @ zero_z1423781445y_real @ N )
% 0.30/0.46 = zero_zero_poly_real ) ).
% 0.30/0.46
% 0.30/0.46 % coeff_0
% 0.30/0.46 thf(fact_52_coeff__0,axiom,
% 0.30/0.46 ! [N: nat] :
% 0.30/0.46 ( ( coeff_int @ zero_zero_poly_int @ N )
% 0.30/0.46 = zero_zero_int ) ).
% 0.30/0.46
% 0.30/0.46 % coeff_0
% 0.30/0.46 thf(fact_53_coeff__0,axiom,
% 0.30/0.46 ! [N: nat] :
% 0.30/0.46 ( ( coeff_nat @ zero_zero_poly_nat @ N )
% 0.30/0.46 = zero_zero_nat ) ).
% 0.30/0.46
% 0.30/0.46 % coeff_0
% 0.30/0.46 thf(fact_54_coeff__0,axiom,
% 0.30/0.46 ! [N: nat] :
% 0.30/0.46 ( ( coeff_real @ zero_zero_poly_real @ N )
% 0.30/0.46 = zero_zero_real ) ).
% 0.30/0.46
% 0.30/0.46 % coeff_0
% 0.30/0.46 thf(fact_55_poly__0,axiom,
% 0.30/0.46 ! [X: poly_real] :
% 0.30/0.46 ( ( poly_poly_real2 @ zero_z1423781445y_real @ X )
% 0.30/0.46 = zero_zero_poly_real ) ).
% 0.30/0.46
% 0.30/0.46 % poly_0
% 0.30/0.46 thf(fact_56_poly__0,axiom,
% 0.30/0.46 ! [X: int] :
% 0.30/0.46 ( ( poly_int2 @ zero_zero_poly_int @ X )
% 0.30/0.46 = zero_zero_int ) ).
% 0.30/0.46
% 0.30/0.46 % poly_0
% 0.30/0.46 thf(fact_57_poly__0,axiom,
% 0.30/0.46 ! [X: nat] :
% 0.30/0.46 ( ( poly_nat2 @ zero_zero_poly_nat @ X )
% 0.30/0.46 = zero_zero_nat ) ).
% 0.30/0.46
% 0.30/0.46 % poly_0
% 0.30/0.46 thf(fact_58_poly__0,axiom,
% 0.30/0.46 ! [X: real] :
% 0.30/0.46 ( ( poly_real2 @ zero_zero_poly_real @ X )
% 0.30/0.46 = zero_zero_real ) ).
% 0.30/0.46
% 0.30/0.46 % poly_0
% 0.30/0.46 thf(fact_59_lead__coeff__of__int,axiom,
% 0.30/0.46 ! [K: int] :
% 0.30/0.46 ( ( coeff_real @ ( ring_11511526659y_real @ K ) @ ( degree_real @ ( ring_11511526659y_real @ K ) ) )
% 0.30/0.46 = ( ring_1_of_int_real @ K ) ) ).
% 0.30/0.46
% 0.30/0.46 % lead_coeff_of_int
% 0.30/0.46 thf(fact_60_lead__coeff__of__int,axiom,
% 0.30/0.46 ! [K: int] :
% 0.30/0.46 ( ( coeff_int @ ( ring_12102921859ly_int @ K ) @ ( degree_int @ ( ring_12102921859ly_int @ K ) ) )
% 0.30/0.46 = ( ring_1_of_int_int @ K ) ) ).
% 0.30/0.46
% 0.30/0.46 % lead_coeff_of_int
% 0.30/0.46 thf(fact_61_leading__coeff__0__iff,axiom,
% 0.30/0.46 ! [P: poly_poly_real] :
% 0.30/0.46 ( ( ( coeff_poly_real @ P @ ( degree_poly_real @ P ) )
% 0.30/0.46 = zero_zero_poly_real )
% 0.30/0.46 = ( P = zero_z1423781445y_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % leading_coeff_0_iff
% 0.30/0.46 thf(fact_62_leading__coeff__0__iff,axiom,
% 0.30/0.46 ! [P: poly_int] :
% 0.30/0.46 ( ( ( coeff_int @ P @ ( degree_int @ P ) )
% 0.30/0.46 = zero_zero_int )
% 0.30/0.46 = ( P = zero_zero_poly_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % leading_coeff_0_iff
% 0.30/0.46 thf(fact_63_leading__coeff__0__iff,axiom,
% 0.30/0.46 ! [P: poly_real] :
% 0.30/0.46 ( ( ( coeff_real @ P @ ( degree_real @ P ) )
% 0.30/0.46 = zero_zero_real )
% 0.30/0.46 = ( P = zero_zero_poly_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % leading_coeff_0_iff
% 0.30/0.46 thf(fact_64_leading__coeff__0__iff,axiom,
% 0.30/0.46 ! [P: poly_nat] :
% 0.30/0.46 ( ( ( coeff_nat @ P @ ( degree_nat @ P ) )
% 0.30/0.46 = zero_zero_nat )
% 0.30/0.46 = ( P = zero_zero_poly_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % leading_coeff_0_iff
% 0.30/0.46 thf(fact_65_zabs__less__one__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 0.30/0.46 = ( Z = zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % zabs_less_one_iff
% 0.30/0.46 thf(fact_66_lead__coeff__1,axiom,
% 0.30/0.46 ( ( coeff_int @ one_one_poly_int @ ( degree_int @ one_one_poly_int ) )
% 0.30/0.46 = one_one_int ) ).
% 0.30/0.46
% 0.30/0.46 % lead_coeff_1
% 0.30/0.46 thf(fact_67_lead__coeff__1,axiom,
% 0.30/0.46 ( ( coeff_nat @ one_one_poly_nat @ ( degree_nat @ one_one_poly_nat ) )
% 0.30/0.46 = one_one_nat ) ).
% 0.30/0.46
% 0.30/0.46 % lead_coeff_1
% 0.30/0.46 thf(fact_68_lead__coeff__1,axiom,
% 0.30/0.46 ( ( coeff_real @ one_one_poly_real @ ( degree_real @ one_one_poly_real ) )
% 0.30/0.46 = one_one_real ) ).
% 0.30/0.46
% 0.30/0.46 % lead_coeff_1
% 0.30/0.46 thf(fact_69_of__int__le__0__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_le1180086932y_real @ ( ring_11511526659y_real @ Z ) @ zero_zero_poly_real )
% 0.30/0.46 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_le_0_iff
% 0.30/0.46 thf(fact_70_of__int__le__0__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.30/0.46 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_le_0_iff
% 0.30/0.46 thf(fact_71_of__int__le__0__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.30/0.46 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_le_0_iff
% 0.30/0.46 thf(fact_72_of__int__0__le__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_le1180086932y_real @ zero_zero_poly_real @ ( ring_11511526659y_real @ Z ) )
% 0.30/0.46 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_0_le_iff
% 0.30/0.46 thf(fact_73_of__int__0__le__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.46 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_0_le_iff
% 0.30/0.46 thf(fact_74_of__int__0__le__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.46 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_0_le_iff
% 0.30/0.46 thf(fact_75_of__int__less__0__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_poly_real @ ( ring_11511526659y_real @ Z ) @ zero_zero_poly_real )
% 0.30/0.46 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_less_0_iff
% 0.30/0.46 thf(fact_76_of__int__less__0__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.30/0.46 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_less_0_iff
% 0.30/0.46 thf(fact_77_of__int__less__0__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.30/0.46 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_less_0_iff
% 0.30/0.46 thf(fact_78_of__int__0__less__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_poly_real @ zero_zero_poly_real @ ( ring_11511526659y_real @ Z ) )
% 0.30/0.46 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_0_less_iff
% 0.30/0.46 thf(fact_79_of__int__0__less__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.46 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_0_less_iff
% 0.30/0.46 thf(fact_80_of__int__0__less__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.46 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_0_less_iff
% 0.30/0.46 thf(fact_81_of__int__less__1__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.30/0.46 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_less_1_iff
% 0.30/0.46 thf(fact_82_of__int__less__1__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.30/0.46 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_less_1_iff
% 0.30/0.46 thf(fact_83_of__int__1__less__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.46 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_1_less_iff
% 0.30/0.46 thf(fact_84_of__int__1__less__iff,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.46 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_1_less_iff
% 0.30/0.46 thf(fact_85_of__int__power__less__of__int__cancel__iff,axiom,
% 0.30/0.46 ! [X: int,B: int,W: nat] :
% 0.30/0.46 ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.30/0.46 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_power_less_of_int_cancel_iff
% 0.30/0.46 thf(fact_86_of__int__power__less__of__int__cancel__iff,axiom,
% 0.30/0.46 ! [X: int,B: int,W: nat] :
% 0.30/0.46 ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.30/0.46 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_power_less_of_int_cancel_iff
% 0.30/0.46 thf(fact_87_of__int__less__of__int__power__cancel__iff,axiom,
% 0.30/0.46 ! [B: int,W: nat,X: int] :
% 0.30/0.46 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.30/0.46 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_less_of_int_power_cancel_iff
% 0.30/0.46 thf(fact_88_of__int__less__of__int__power__cancel__iff,axiom,
% 0.30/0.46 ! [B: int,W: nat,X: int] :
% 0.30/0.46 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.30/0.46 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_less_of_int_power_cancel_iff
% 0.30/0.46 thf(fact_89_less__int__code_I1_J,axiom,
% 0.30/0.46 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.30/0.46
% 0.30/0.46 % less_int_code(1)
% 0.30/0.46 thf(fact_90_less__eq__int__code_I1_J,axiom,
% 0.30/0.46 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.30/0.46
% 0.30/0.46 % less_eq_int_code(1)
% 0.30/0.46 thf(fact_91_poly__eqI,axiom,
% 0.30/0.46 ! [P: poly_real,Q: poly_real] :
% 0.30/0.46 ( ! [N2: nat] :
% 0.30/0.46 ( ( coeff_real @ P @ N2 )
% 0.30/0.46 = ( coeff_real @ Q @ N2 ) )
% 0.30/0.46 => ( P = Q ) ) ).
% 0.30/0.46
% 0.30/0.46 % poly_eqI
% 0.30/0.46 thf(fact_92_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: real,P3: real > $o] :
% 0.30/0.46 ( ( member_real @ A @ ( collect_real @ P3 ) )
% 0.30/0.46 = ( P3 @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_93_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A2: set_real] :
% 0.30/0.46 ( ( collect_real
% 0.30/0.46 @ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
% 0.30/0.46 = A2 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_94_poly__eq__iff,axiom,
% 0.30/0.46 ( ( ^ [Y: poly_real,Z2: poly_real] : ( Y = Z2 ) )
% 0.30/0.46 = ( ^ [P4: poly_real,Q2: poly_real] :
% 0.30/0.46 ! [N3: nat] :
% 0.30/0.46 ( ( coeff_real @ P4 @ N3 )
% 0.30/0.46 = ( coeff_real @ Q2 @ N3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % poly_eq_iff
% 0.30/0.46 thf(fact_95_Ints__0,axiom,
% 0.30/0.46 member_poly_real @ zero_zero_poly_real @ ring_1690226883y_real ).
% 0.30/0.46
% 0.30/0.46 % Ints_0
% 0.30/0.46 thf(fact_96_Ints__0,axiom,
% 0.30/0.46 member_int @ zero_zero_int @ ring_1_Ints_int ).
% 0.30/0.46
% 0.30/0.46 % Ints_0
% 0.30/0.46 thf(fact_97_Ints__0,axiom,
% 0.30/0.46 member_real @ zero_zero_real @ ring_1_Ints_real ).
% 0.30/0.46
% 0.30/0.46 % Ints_0
% 0.30/0.46 thf(fact_98_zero__poly_Orep__eq,axiom,
% 0.30/0.46 ( ( coeff_poly_real @ zero_z1423781445y_real )
% 0.30/0.46 = ( ^ [Uu: nat] : zero_zero_poly_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_poly.rep_eq
% 0.30/0.46 thf(fact_99_zero__poly_Orep__eq,axiom,
% 0.30/0.46 ( ( coeff_int @ zero_zero_poly_int )
% 0.30/0.46 = ( ^ [Uu: nat] : zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_poly.rep_eq
% 0.30/0.46 thf(fact_100_zero__poly_Orep__eq,axiom,
% 0.30/0.46 ( ( coeff_nat @ zero_zero_poly_nat )
% 0.30/0.46 = ( ^ [Uu: nat] : zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_poly.rep_eq
% 0.30/0.46 thf(fact_101_zero__poly_Orep__eq,axiom,
% 0.30/0.46 ( ( coeff_real @ zero_zero_poly_real )
% 0.30/0.46 = ( ^ [Uu: nat] : zero_zero_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_poly.rep_eq
% 0.30/0.46 thf(fact_102_int__one__le__iff__zero__less,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ one_one_int @ Z )
% 0.30/0.46 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % int_one_le_iff_zero_less
% 0.30/0.46 thf(fact_103_coeff__inject,axiom,
% 0.30/0.46 ! [X: poly_real,Y2: poly_real] :
% 0.30/0.46 ( ( ( coeff_real @ X )
% 0.30/0.46 = ( coeff_real @ Y2 ) )
% 0.30/0.46 = ( X = Y2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % coeff_inject
% 0.30/0.46 thf(fact_104_le__degree,axiom,
% 0.30/0.46 ! [P: poly_poly_real,N: nat] :
% 0.30/0.46 ( ( ( coeff_poly_real @ P @ N )
% 0.30/0.46 != zero_zero_poly_real )
% 0.30/0.46 => ( ord_less_eq_nat @ N @ ( degree_poly_real @ P ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_degree
% 0.30/0.46 thf(fact_105_le__degree,axiom,
% 0.30/0.46 ! [P: poly_int,N: nat] :
% 0.30/0.46 ( ( ( coeff_int @ P @ N )
% 0.30/0.46 != zero_zero_int )
% 0.30/0.46 => ( ord_less_eq_nat @ N @ ( degree_int @ P ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_degree
% 0.30/0.46 thf(fact_106_le__degree,axiom,
% 0.30/0.46 ! [P: poly_real,N: nat] :
% 0.30/0.46 ( ( ( coeff_real @ P @ N )
% 0.30/0.46 != zero_zero_real )
% 0.30/0.46 => ( ord_less_eq_nat @ N @ ( degree_real @ P ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_degree
% 0.30/0.46 thf(fact_107_le__degree,axiom,
% 0.30/0.46 ! [P: poly_nat,N: nat] :
% 0.30/0.46 ( ( ( coeff_nat @ P @ N )
% 0.30/0.46 != zero_zero_nat )
% 0.30/0.46 => ( ord_less_eq_nat @ N @ ( degree_nat @ P ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_degree
% 0.30/0.46 thf(fact_108_poly__0__coeff__0,axiom,
% 0.30/0.46 ! [P: poly_poly_real] :
% 0.30/0.46 ( ( poly_poly_real2 @ P @ zero_zero_poly_real )
% 0.30/0.46 = ( coeff_poly_real @ P @ zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % poly_0_coeff_0
% 0.30/0.46 thf(fact_109_poly__0__coeff__0,axiom,
% 0.30/0.46 ! [P: poly_int] :
% 0.30/0.46 ( ( poly_int2 @ P @ zero_zero_int )
% 0.30/0.46 = ( coeff_int @ P @ zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % poly_0_coeff_0
% 0.30/0.46 thf(fact_110_poly__0__coeff__0,axiom,
% 0.30/0.46 ! [P: poly_real] :
% 0.30/0.46 ( ( poly_real2 @ P @ zero_zero_real )
% 0.30/0.46 = ( coeff_real @ P @ zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % poly_0_coeff_0
% 0.30/0.46 thf(fact_111_poly__0__coeff__0,axiom,
% 0.30/0.46 ! [P: poly_nat] :
% 0.30/0.46 ( ( poly_nat2 @ P @ zero_zero_nat )
% 0.30/0.46 = ( coeff_nat @ P @ zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % poly_0_coeff_0
% 0.30/0.46 thf(fact_112_algebraic__def,axiom,
% 0.30/0.46 ( algebraic_real
% 0.30/0.46 = ( ^ [X2: real] :
% 0.30/0.46 ? [P4: poly_real] :
% 0.30/0.46 ( ! [I3: nat] : ( member_real @ ( coeff_real @ P4 @ I3 ) @ ring_1_Ints_real )
% 0.30/0.46 & ( P4 != zero_zero_poly_real )
% 0.30/0.46 & ( ( poly_real2 @ P4 @ X2 )
% 0.30/0.46 = zero_zero_real ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % algebraic_def
% 0.30/0.46 thf(fact_113_algebraicI,axiom,
% 0.30/0.46 ! [P: poly_real,X: real] :
% 0.30/0.46 ( ! [I4: nat] : ( member_real @ ( coeff_real @ P @ I4 ) @ ring_1_Ints_real )
% 0.30/0.46 => ( ( P != zero_zero_poly_real )
% 0.30/0.46 => ( ( ( poly_real2 @ P @ X )
% 0.30/0.46 = zero_zero_real )
% 0.30/0.46 => ( algebraic_real @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % algebraicI
% 0.30/0.46 thf(fact_114_algebraicE,axiom,
% 0.30/0.46 ! [X: real] :
% 0.30/0.46 ( ( algebraic_real @ X )
% 0.30/0.46 => ~ ! [P2: poly_real] :
% 0.30/0.46 ( ! [I2: nat] : ( member_real @ ( coeff_real @ P2 @ I2 ) @ ring_1_Ints_real )
% 0.30/0.46 => ( ( P2 != zero_zero_poly_real )
% 0.30/0.46 => ( ( poly_real2 @ P2 @ X )
% 0.30/0.46 != zero_zero_real ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % algebraicE
% 0.30/0.46 thf(fact_115_of__int__pos,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.46 => ( ord_less_poly_real @ zero_zero_poly_real @ ( ring_11511526659y_real @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_pos
% 0.30/0.46 thf(fact_116_of__int__pos,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.46 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_pos
% 0.30/0.46 thf(fact_117_of__int__pos,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.46 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_pos
% 0.30/0.46 thf(fact_118_leading__coeff__neq__0,axiom,
% 0.30/0.46 ! [P: poly_poly_real] :
% 0.30/0.46 ( ( P != zero_z1423781445y_real )
% 0.30/0.46 => ( ( coeff_poly_real @ P @ ( degree_poly_real @ P ) )
% 0.30/0.46 != zero_zero_poly_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % leading_coeff_neq_0
% 0.30/0.46 thf(fact_119_leading__coeff__neq__0,axiom,
% 0.30/0.46 ! [P: poly_int] :
% 0.30/0.46 ( ( P != zero_zero_poly_int )
% 0.30/0.46 => ( ( coeff_int @ P @ ( degree_int @ P ) )
% 0.30/0.46 != zero_zero_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % leading_coeff_neq_0
% 0.30/0.46 thf(fact_120_leading__coeff__neq__0,axiom,
% 0.30/0.46 ! [P: poly_nat] :
% 0.30/0.46 ( ( P != zero_zero_poly_nat )
% 0.30/0.46 => ( ( coeff_nat @ P @ ( degree_nat @ P ) )
% 0.30/0.46 != zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % leading_coeff_neq_0
% 0.30/0.46 thf(fact_121_leading__coeff__neq__0,axiom,
% 0.30/0.46 ! [P: poly_real] :
% 0.30/0.46 ( ( P != zero_zero_poly_real )
% 0.30/0.46 => ( ( coeff_real @ P @ ( degree_real @ P ) )
% 0.30/0.46 != zero_zero_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % leading_coeff_neq_0
% 0.30/0.46 thf(fact_122_poly__all__0__iff__0,axiom,
% 0.30/0.46 ! [P: poly_poly_real] :
% 0.30/0.46 ( ( ! [X2: poly_real] :
% 0.30/0.46 ( ( poly_poly_real2 @ P @ X2 )
% 0.30/0.46 = zero_zero_poly_real ) )
% 0.30/0.46 = ( P = zero_z1423781445y_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % poly_all_0_iff_0
% 0.30/0.46 thf(fact_123_poly__all__0__iff__0,axiom,
% 0.30/0.46 ! [P: poly_int] :
% 0.30/0.46 ( ( ! [X2: int] :
% 0.30/0.46 ( ( poly_int2 @ P @ X2 )
% 0.30/0.46 = zero_zero_int ) )
% 0.30/0.46 = ( P = zero_zero_poly_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % poly_all_0_iff_0
% 0.30/0.46 thf(fact_124_poly__all__0__iff__0,axiom,
% 0.30/0.46 ! [P: poly_real] :
% 0.30/0.46 ( ( ! [X2: real] :
% 0.30/0.46 ( ( poly_real2 @ P @ X2 )
% 0.30/0.46 = zero_zero_real ) )
% 0.30/0.46 = ( P = zero_zero_poly_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % poly_all_0_iff_0
% 0.30/0.46 thf(fact_125_Ints__1,axiom,
% 0.30/0.46 member_int @ one_one_int @ ring_1_Ints_int ).
% 0.30/0.46
% 0.30/0.46 % Ints_1
% 0.30/0.46 thf(fact_126_Ints__1,axiom,
% 0.30/0.46 member_real @ one_one_real @ ring_1_Ints_real ).
% 0.30/0.46
% 0.30/0.46 % Ints_1
% 0.30/0.46 thf(fact_127_Ints__power,axiom,
% 0.30/0.46 ! [A: real,N: nat] :
% 0.30/0.46 ( ( member_real @ A @ ring_1_Ints_real )
% 0.30/0.46 => ( member_real @ ( power_power_real @ A @ N ) @ ring_1_Ints_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_power
% 0.30/0.46 thf(fact_128_Ints__power,axiom,
% 0.30/0.46 ! [A: int,N: nat] :
% 0.30/0.46 ( ( member_int @ A @ ring_1_Ints_int )
% 0.30/0.46 => ( member_int @ ( power_power_int @ A @ N ) @ ring_1_Ints_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_power
% 0.30/0.46 thf(fact_129_Ints__abs,axiom,
% 0.30/0.46 ! [A: int] :
% 0.30/0.46 ( ( member_int @ A @ ring_1_Ints_int )
% 0.30/0.46 => ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_abs
% 0.30/0.46 thf(fact_130_Ints__abs,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( member_real @ A @ ring_1_Ints_real )
% 0.30/0.46 => ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_abs
% 0.30/0.46 thf(fact_131_Ints__of__int,axiom,
% 0.30/0.46 ! [Z: int] : ( member_real @ ( ring_1_of_int_real @ Z ) @ ring_1_Ints_real ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_of_int
% 0.30/0.46 thf(fact_132_Ints__of__int,axiom,
% 0.30/0.46 ! [Z: int] : ( member_int @ ( ring_1_of_int_int @ Z ) @ ring_1_Ints_int ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_of_int
% 0.30/0.46 thf(fact_133_Ints__induct,axiom,
% 0.30/0.46 ! [Q: real,P3: real > $o] :
% 0.30/0.46 ( ( member_real @ Q @ ring_1_Ints_real )
% 0.30/0.46 => ( ! [Z3: int] : ( P3 @ ( ring_1_of_int_real @ Z3 ) )
% 0.30/0.46 => ( P3 @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_induct
% 0.30/0.46 thf(fact_134_Ints__induct,axiom,
% 0.30/0.46 ! [Q: int,P3: int > $o] :
% 0.30/0.46 ( ( member_int @ Q @ ring_1_Ints_int )
% 0.30/0.46 => ( ! [Z3: int] : ( P3 @ ( ring_1_of_int_int @ Z3 ) )
% 0.30/0.46 => ( P3 @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_induct
% 0.30/0.46 thf(fact_135_Ints__cases,axiom,
% 0.30/0.46 ! [Q: real] :
% 0.30/0.46 ( ( member_real @ Q @ ring_1_Ints_real )
% 0.30/0.46 => ~ ! [Z3: int] :
% 0.30/0.46 ( Q
% 0.30/0.46 != ( ring_1_of_int_real @ Z3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_cases
% 0.30/0.46 thf(fact_136_Ints__cases,axiom,
% 0.30/0.46 ! [Q: int] :
% 0.30/0.46 ( ( member_int @ Q @ ring_1_Ints_int )
% 0.30/0.46 => ~ ! [Z3: int] :
% 0.30/0.46 ( Q
% 0.30/0.46 != ( ring_1_of_int_int @ Z3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_cases
% 0.30/0.46 thf(fact_137_Ints__nonzero__abs__less1,axiom,
% 0.30/0.46 ! [X: poly_real] :
% 0.30/0.46 ( ( member_poly_real @ X @ ring_1690226883y_real )
% 0.30/0.46 => ( ( ord_less_poly_real @ ( abs_abs_poly_real @ X ) @ one_one_poly_real )
% 0.30/0.46 => ( X = zero_zero_poly_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_nonzero_abs_less1
% 0.30/0.46 thf(fact_138_Ints__nonzero__abs__less1,axiom,
% 0.30/0.46 ! [X: int] :
% 0.30/0.46 ( ( member_int @ X @ ring_1_Ints_int )
% 0.30/0.46 => ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
% 0.30/0.46 => ( X = zero_zero_int ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_nonzero_abs_less1
% 0.30/0.46 thf(fact_139_Ints__nonzero__abs__less1,axiom,
% 0.30/0.46 ! [X: real] :
% 0.30/0.46 ( ( member_real @ X @ ring_1_Ints_real )
% 0.30/0.46 => ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.30/0.46 => ( X = zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_nonzero_abs_less1
% 0.30/0.46 thf(fact_140_of__int__nonneg,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.46 => ( ord_le1180086932y_real @ zero_zero_poly_real @ ( ring_11511526659y_real @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_nonneg
% 0.30/0.46 thf(fact_141_of__int__nonneg,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_nonneg
% 0.30/0.46 thf(fact_142_of__int__nonneg,axiom,
% 0.30/0.46 ! [Z: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.46 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_nonneg
% 0.30/0.46 thf(fact_143_lead__coeff__power,axiom,
% 0.30/0.46 ! [P: poly_real,N: nat] :
% 0.30/0.46 ( ( coeff_real @ ( power_2108872382y_real @ P @ N ) @ ( degree_real @ ( power_2108872382y_real @ P @ N ) ) )
% 0.30/0.46 = ( power_power_real @ ( coeff_real @ P @ ( degree_real @ P ) ) @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % lead_coeff_power
% 0.30/0.46 thf(fact_144_lead__coeff__power,axiom,
% 0.30/0.46 ! [P: poly_nat,N: nat] :
% 0.30/0.46 ( ( coeff_nat @ ( power_power_poly_nat @ P @ N ) @ ( degree_nat @ ( power_power_poly_nat @ P @ N ) ) )
% 0.30/0.46 = ( power_power_nat @ ( coeff_nat @ P @ ( degree_nat @ P ) ) @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % lead_coeff_power
% 0.30/0.46 thf(fact_145_lead__coeff__power,axiom,
% 0.30/0.46 ! [P: poly_int,N: nat] :
% 0.30/0.46 ( ( coeff_int @ ( power_power_poly_int @ P @ N ) @ ( degree_int @ ( power_power_poly_int @ P @ N ) ) )
% 0.30/0.46 = ( power_power_int @ ( coeff_int @ P @ ( degree_int @ P ) ) @ N ) ) ).
% 0.30/0.46
% 0.30/0.46 % lead_coeff_power
% 0.30/0.46 thf(fact_146_Ints__nonzero__abs__ge1,axiom,
% 0.30/0.46 ! [X: poly_real] :
% 0.30/0.46 ( ( member_poly_real @ X @ ring_1690226883y_real )
% 0.30/0.46 => ( ( X != zero_zero_poly_real )
% 0.30/0.46 => ( ord_le1180086932y_real @ one_one_poly_real @ ( abs_abs_poly_real @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_nonzero_abs_ge1
% 0.30/0.46 thf(fact_147_Ints__nonzero__abs__ge1,axiom,
% 0.30/0.46 ! [X: real] :
% 0.30/0.46 ( ( member_real @ X @ ring_1_Ints_real )
% 0.30/0.46 => ( ( X != zero_zero_real )
% 0.30/0.46 => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_nonzero_abs_ge1
% 0.30/0.46 thf(fact_148_Ints__nonzero__abs__ge1,axiom,
% 0.30/0.46 ! [X: int] :
% 0.30/0.46 ( ( member_int @ X @ ring_1_Ints_int )
% 0.30/0.46 => ( ( X != zero_zero_int )
% 0.30/0.46 => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ints_nonzero_abs_ge1
% 0.30/0.46 thf(fact_149_poly__eq__poly__eq__iff,axiom,
% 0.30/0.46 ! [P: poly_real,Q: poly_real] :
% 0.30/0.46 ( ( ( poly_real2 @ P )
% 0.30/0.46 = ( poly_real2 @ Q ) )
% 0.30/0.46 = ( P = Q ) ) ).
% 0.30/0.46
% 0.30/0.46 % poly_eq_poly_eq_iff
% 0.30/0.46 thf(fact_150_of__int__lessD,axiom,
% 0.30/0.46 ! [N: int,X: int] :
% 0.30/0.46 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 0.30/0.46 => ( ( N = zero_zero_int )
% 0.30/0.46 | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_lessD
% 0.30/0.46 thf(fact_151_of__int__lessD,axiom,
% 0.30/0.46 ! [N: int,X: real] :
% 0.30/0.46 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 0.30/0.46 => ( ( N = zero_zero_int )
% 0.30/0.46 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_lessD
% 0.30/0.46 thf(fact_152_of__int__leD,axiom,
% 0.30/0.46 ! [N: int,X: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 0.30/0.46 => ( ( N = zero_zero_int )
% 0.30/0.46 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_leD
% 0.30/0.46 thf(fact_153_of__int__leD,axiom,
% 0.30/0.46 ! [N: int,X: int] :
% 0.30/0.46 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 0.30/0.46 => ( ( N = zero_zero_int )
% 0.30/0.46 | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % of_int_leD
% 0.30/0.46 thf(fact_154_int__poly__rat__no__root__ge,axiom,
% 0.30/0.46 ! [P: poly_real,B: int,A: int] :
% 0.30/0.46 ( ! [N2: nat] : ( member_real @ ( coeff_real @ P @ N2 ) @ ring_1_Ints_real )
% 0.30/0.46 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.46 => ( ( ( poly_real2 @ P @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
% 0.30/0.46 != zero_zero_real )
% 0.30/0.46 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ ( degree_real @ P ) ) ) @ ( abs_abs_real @ ( poly_real2 @ P @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % int_poly_rat_no_root_ge
% 0.30/0.46 thf(fact_155_power__decreasing__iff,axiom,
% 0.30/0.46 ! [B: poly_real,M: nat,N: nat] :
% 0.30/0.46 ( ( ord_less_poly_real @ zero_zero_poly_real @ B )
% 0.30/0.46 => ( ( ord_less_poly_real @ B @ one_one_poly_real )
% 0.30/0.46 => ( ( ord_le1180086932y_real @ ( power_2108872382y_real @ B @ M ) @ ( power_2108872382y_real @ B @ N ) )
% 0.30/0.46 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % power_decreasing_iff
% 0.30/0.46 thf(fact_156_power__decreasing__iff,axiom,
% 0.30/0.46 ! [B: real,M: nat,N: nat] :
% 0.30/0.46 ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.46 => ( ( ord_less_real @ B @ one_one_real )
% 0.30/0.46 => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.30/0.46 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % power_decreasing_iff
% 0.30/0.46 thf(fact_157_power__decreasing__iff,axiom,
% 0.30/0.46 ! [B: int,M: nat,N: nat] :
% 0.30/0.46 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.46 => ( ( ord_less_int @ B @ one_one_int )
% 0.30/0.46 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.30/0.46 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % power_decreasing_iff
% 0.30/0.46 thf(fact_158_power__decreasing__iff,axiom,
% 0.30/0.46 ! [B: nat,M: nat,N: nat] :
% 0.30/0.46 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.46 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.30/0.46 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.30/0.46 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % power_decreasing_iff
% 0.30/0.46 thf(fact_159_le__divide__eq__1__pos,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.46 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_divide_eq_1_pos
% 0.30/0.46 thf(fact_160_le__divide__eq__1__neg,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.46 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.46 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_divide_eq_1_neg
% 0.30/0.46 thf(fact_161_divide__le__eq__1__pos,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.46 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % divide_le_eq_1_pos
% 0.30/0.46 thf(fact_162_divide__le__eq__1__neg,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.46 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.46 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % divide_le_eq_1_neg
% 0.30/0.46 thf(fact_163__092_060open_062A_A_P_Areal__of__int_Ab_A_094_An_A_092_060le_062_AA_A_P_A1_092_060close_062,axiom,
% 0.30/0.46 ord_less_eq_real @ ( divide_divide_real @ a @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ n ) ) @ ( divide_divide_real @ a @ one_one_real ) ).
% 0.30/0.46
% 0.30/0.46 % \<open>A / real_of_int b ^ n \<le> A / 1\<close>
% 0.30/0.46 thf(fact_164_zero__less__power__abs__iff,axiom,
% 0.30/0.46 ! [A: poly_real,N: nat] :
% 0.30/0.46 ( ( ord_less_poly_real @ zero_zero_poly_real @ ( power_2108872382y_real @ ( abs_abs_poly_real @ A ) @ N ) )
% 0.30/0.46 = ( ( A != zero_zero_poly_real )
% 0.30/0.46 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_less_power_abs_iff
% 0.30/0.46 thf(fact_165_zero__less__power__abs__iff,axiom,
% 0.30/0.46 ! [A: int,N: nat] :
% 0.30/0.46 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 0.30/0.46 = ( ( A != zero_zero_int )
% 0.30/0.46 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_less_power_abs_iff
% 0.30/0.46 thf(fact_166_zero__less__power__abs__iff,axiom,
% 0.30/0.46 ! [A: real,N: nat] :
% 0.30/0.46 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.30/0.46 = ( ( A != zero_zero_real )
% 0.30/0.46 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_less_power_abs_iff
% 0.30/0.46 thf(fact_167_zero__le__divide__abs__iff,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 0.30/0.46 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.46 | ( B = zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_le_divide_abs_iff
% 0.30/0.46 thf(fact_168_divide__le__0__abs__iff,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 0.30/0.46 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.46 | ( B = zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % divide_le_0_abs_iff
% 0.30/0.46 thf(fact_169_A__pos,axiom,
% 0.30/0.46 ord_less_real @ zero_zero_real @ a ).
% 0.30/0.46
% 0.30/0.46 % A_pos
% 0.30/0.46 thf(fact_170_A__less_I1_J,axiom,
% 0.30/0.46 ord_less_real @ a @ one_one_real ).
% 0.30/0.46
% 0.30/0.46 % A_less(1)
% 0.30/0.46 thf(fact_171__092_060open_062A_A_092_060le_062_A1_092_060close_062,axiom,
% 0.30/0.46 ord_less_eq_real @ a @ one_one_real ).
% 0.30/0.46
% 0.30/0.46 % \<open>A \<le> 1\<close>
% 0.30/0.46 thf(fact_172_power__one__right,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( power_power_real @ A @ one_one_nat )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % power_one_right
% 0.30/0.46 thf(fact_173_power__one__right,axiom,
% 0.30/0.46 ! [A: nat] :
% 0.30/0.46 ( ( power_power_nat @ A @ one_one_nat )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % power_one_right
% 0.30/0.46 thf(fact_174_power__one__right,axiom,
% 0.30/0.46 ! [A: int] :
% 0.30/0.46 ( ( power_power_int @ A @ one_one_nat )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % power_one_right
% 0.30/0.46 thf(fact_175_nat__zero__less__power__iff,axiom,
% 0.30/0.46 ! [X: nat,N: nat] :
% 0.30/0.46 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 0.30/0.46 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.30/0.46 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nat_zero_less_power_iff
% 0.30/0.46 thf(fact_176_divide__eq__0__iff,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ( divide_divide_real @ A @ B )
% 0.30/0.46 = zero_zero_real )
% 0.30/0.46 = ( ( A = zero_zero_real )
% 0.30/0.46 | ( B = zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % divide_eq_0_iff
% 0.30/0.46 thf(fact_177_divide__cancel__left,axiom,
% 0.30/0.46 ! [C: real,A: real,B: real] :
% 0.30/0.46 ( ( ( divide_divide_real @ C @ A )
% 0.30/0.46 = ( divide_divide_real @ C @ B ) )
% 0.30/0.46 = ( ( C = zero_zero_real )
% 0.30/0.46 | ( A = B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % divide_cancel_left
% 0.30/0.46 thf(fact_178_divide__cancel__right,axiom,
% 0.30/0.46 ! [A: real,C: real,B: real] :
% 0.30/0.46 ( ( ( divide_divide_real @ A @ C )
% 0.30/0.46 = ( divide_divide_real @ B @ C ) )
% 0.30/0.46 = ( ( C = zero_zero_real )
% 0.30/0.46 | ( A = B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % divide_cancel_right
% 0.30/0.46 thf(fact_179_division__ring__divide__zero,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.30/0.46 = zero_zero_real ) ).
% 0.30/0.46
% 0.30/0.46 % division_ring_divide_zero
% 0.30/0.46 thf(fact_180_power__one,axiom,
% 0.30/0.46 ! [N: nat] :
% 0.30/0.46 ( ( power_power_real @ one_one_real @ N )
% 0.30/0.46 = one_one_real ) ).
% 0.30/0.46
% 0.30/0.46 % power_one
% 0.30/0.46 thf(fact_181_power__one,axiom,
% 0.30/0.46 ! [N: nat] :
% 0.30/0.46 ( ( power_power_nat @ one_one_nat @ N )
% 0.30/0.46 = one_one_nat ) ).
% 0.30/0.46
% 0.30/0.46 % power_one
% 0.30/0.46 thf(fact_182_power__one,axiom,
% 0.30/0.46 ! [N: nat] :
% 0.30/0.46 ( ( power_power_int @ one_one_int @ N )
% 0.30/0.46 = one_one_int ) ).
% 0.30/0.46
% 0.30/0.46 % power_one
% 0.30/0.46 thf(fact_183_abs__divide,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.46 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % abs_divide
% 0.30/0.46 thf(fact_184_divide__eq__1__iff,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ( divide_divide_real @ A @ B )
% 0.30/0.46 = one_one_real )
% 0.30/0.46 = ( ( B != zero_zero_real )
% 0.30/0.46 & ( A = B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % divide_eq_1_iff
% 0.30/0.46 thf(fact_185_one__eq__divide__iff,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( one_one_real
% 0.30/0.46 = ( divide_divide_real @ A @ B ) )
% 0.30/0.46 = ( ( B != zero_zero_real )
% 0.30/0.46 & ( A = B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % one_eq_divide_iff
% 0.30/0.46 thf(fact_186_divide__self,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( A != zero_zero_real )
% 0.30/0.46 => ( ( divide_divide_real @ A @ A )
% 0.30/0.46 = one_one_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % divide_self
% 0.30/0.46 thf(fact_187_divide__self__if,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( ( A = zero_zero_real )
% 0.30/0.46 => ( ( divide_divide_real @ A @ A )
% 0.30/0.47 = zero_zero_real ) )
% 0.30/0.47 & ( ( A != zero_zero_real )
% 0.30/0.47 => ( ( divide_divide_real @ A @ A )
% 0.30/0.47 = one_one_real ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_self_if
% 0.30/0.47 thf(fact_188_divide__eq__eq__1,axiom,
% 0.30/0.47 ! [B: real,A: real] :
% 0.30/0.47 ( ( ( divide_divide_real @ B @ A )
% 0.30/0.47 = one_one_real )
% 0.30/0.47 = ( ( A != zero_zero_real )
% 0.30/0.47 & ( A = B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_eq_eq_1
% 0.30/0.47 thf(fact_189_eq__divide__eq__1,axiom,
% 0.30/0.47 ! [B: real,A: real] :
% 0.30/0.47 ( ( one_one_real
% 0.30/0.47 = ( divide_divide_real @ B @ A ) )
% 0.30/0.47 = ( ( A != zero_zero_real )
% 0.30/0.47 & ( A = B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_divide_eq_1
% 0.30/0.47 thf(fact_190_one__divide__eq__0__iff,axiom,
% 0.30/0.47 ! [A: real] :
% 0.30/0.47 ( ( ( divide_divide_real @ one_one_real @ A )
% 0.30/0.47 = zero_zero_real )
% 0.30/0.47 = ( A = zero_zero_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_divide_eq_0_iff
% 0.30/0.47 thf(fact_191_zero__eq__1__divide__iff,axiom,
% 0.30/0.47 ! [A: real] :
% 0.30/0.47 ( ( zero_zero_real
% 0.30/0.47 = ( divide_divide_real @ one_one_real @ A ) )
% 0.30/0.47 = ( A = zero_zero_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_eq_1_divide_iff
% 0.30/0.47 thf(fact_192_power__inject__exp,axiom,
% 0.30/0.47 ! [A: int,M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.47 => ( ( ( power_power_int @ A @ M )
% 0.30/0.47 = ( power_power_int @ A @ N ) )
% 0.30/0.47 = ( M = N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_inject_exp
% 0.30/0.47 thf(fact_193_power__inject__exp,axiom,
% 0.30/0.47 ! [A: real,M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.47 => ( ( ( power_power_real @ A @ M )
% 0.30/0.47 = ( power_power_real @ A @ N ) )
% 0.30/0.47 = ( M = N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_inject_exp
% 0.30/0.47 thf(fact_194_power__inject__exp,axiom,
% 0.30/0.47 ! [A: nat,M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.47 => ( ( ( power_power_nat @ A @ M )
% 0.30/0.47 = ( power_power_nat @ A @ N ) )
% 0.30/0.47 = ( M = N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_inject_exp
% 0.30/0.47 thf(fact_195_power__strict__increasing__iff,axiom,
% 0.30/0.47 ! [B: int,X: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_int @ one_one_int @ B )
% 0.30/0.47 => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
% 0.30/0.47 = ( ord_less_nat @ X @ Y2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_strict_increasing_iff
% 0.30/0.47 thf(fact_196_power__strict__increasing__iff,axiom,
% 0.30/0.47 ! [B: real,X: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.47 => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y2 ) )
% 0.30/0.47 = ( ord_less_nat @ X @ Y2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_strict_increasing_iff
% 0.30/0.47 thf(fact_197_power__strict__increasing__iff,axiom,
% 0.30/0.47 ! [B: nat,X: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ one_one_nat @ B )
% 0.30/0.47 => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
% 0.30/0.47 = ( ord_less_nat @ X @ Y2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_strict_increasing_iff
% 0.30/0.47 thf(fact_198_power__eq__0__iff,axiom,
% 0.30/0.47 ! [A: poly_real,N: nat] :
% 0.30/0.47 ( ( ( power_2108872382y_real @ A @ N )
% 0.30/0.47 = zero_zero_poly_real )
% 0.30/0.47 = ( ( A = zero_zero_poly_real )
% 0.30/0.47 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_eq_0_iff
% 0.30/0.47 thf(fact_199_power__eq__0__iff,axiom,
% 0.30/0.47 ! [A: real,N: nat] :
% 0.30/0.47 ( ( ( power_power_real @ A @ N )
% 0.30/0.47 = zero_zero_real )
% 0.30/0.47 = ( ( A = zero_zero_real )
% 0.30/0.47 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_eq_0_iff
% 0.30/0.47 thf(fact_200_power__eq__0__iff,axiom,
% 0.30/0.47 ! [A: nat,N: nat] :
% 0.30/0.47 ( ( ( power_power_nat @ A @ N )
% 0.30/0.47 = zero_zero_nat )
% 0.30/0.47 = ( ( A = zero_zero_nat )
% 0.30/0.47 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_eq_0_iff
% 0.30/0.47 thf(fact_201_power__eq__0__iff,axiom,
% 0.30/0.47 ! [A: int,N: nat] :
% 0.30/0.47 ( ( ( power_power_int @ A @ N )
% 0.30/0.47 = zero_zero_int )
% 0.30/0.47 = ( ( A = zero_zero_int )
% 0.30/0.47 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_eq_0_iff
% 0.30/0.47 thf(fact_202_degree__0,axiom,
% 0.30/0.47 ( ( degree_real @ zero_zero_poly_real )
% 0.30/0.47 = zero_zero_nat ) ).
% 0.30/0.47
% 0.30/0.47 % degree_0
% 0.30/0.47 thf(fact_203_degree__1,axiom,
% 0.30/0.47 ( ( degree_real @ one_one_poly_real )
% 0.30/0.47 = zero_zero_nat ) ).
% 0.30/0.47
% 0.30/0.47 % degree_1
% 0.30/0.47 thf(fact_204_degree__of__int,axiom,
% 0.30/0.47 ! [K: int] :
% 0.30/0.47 ( ( degree_real @ ( ring_11511526659y_real @ K ) )
% 0.30/0.47 = zero_zero_nat ) ).
% 0.30/0.47
% 0.30/0.47 % degree_of_int
% 0.30/0.47 thf(fact_205_divide__le__0__1__iff,axiom,
% 0.30/0.47 ! [A: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.30/0.47 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_le_0_1_iff
% 0.30/0.47 thf(fact_206_zero__le__divide__1__iff,axiom,
% 0.30/0.47 ! [A: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.30/0.47 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_le_divide_1_iff
% 0.30/0.47 thf(fact_207_divide__less__0__1__iff,axiom,
% 0.30/0.47 ! [A: real] :
% 0.30/0.47 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.30/0.47 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_less_0_1_iff
% 0.30/0.47 thf(fact_208_divide__less__eq__1__neg,axiom,
% 0.30/0.47 ! [A: real,B: real] :
% 0.30/0.47 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.47 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.47 = ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_less_eq_1_neg
% 0.30/0.47 thf(fact_209_divide__less__eq__1__pos,axiom,
% 0.30/0.47 ! [A: real,B: real] :
% 0.30/0.47 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.47 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.47 = ( ord_less_real @ B @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_less_eq_1_pos
% 0.30/0.47 thf(fact_210_less__divide__eq__1__neg,axiom,
% 0.30/0.47 ! [A: real,B: real] :
% 0.30/0.47 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.47 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.47 = ( ord_less_real @ B @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_divide_eq_1_neg
% 0.30/0.47 thf(fact_211_less__divide__eq__1__pos,axiom,
% 0.30/0.47 ! [A: real,B: real] :
% 0.30/0.47 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.47 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.47 = ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_divide_eq_1_pos
% 0.30/0.47 thf(fact_212_zero__less__divide__1__iff,axiom,
% 0.30/0.47 ! [A: real] :
% 0.30/0.47 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.30/0.47 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_less_divide_1_iff
% 0.30/0.47 thf(fact_213_power__strict__decreasing__iff,axiom,
% 0.30/0.47 ! [B: poly_real,M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_poly_real @ zero_zero_poly_real @ B )
% 0.30/0.47 => ( ( ord_less_poly_real @ B @ one_one_poly_real )
% 0.30/0.47 => ( ( ord_less_poly_real @ ( power_2108872382y_real @ B @ M ) @ ( power_2108872382y_real @ B @ N ) )
% 0.30/0.47 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_strict_decreasing_iff
% 0.30/0.47 thf(fact_214_power__strict__decreasing__iff,axiom,
% 0.30/0.47 ! [B: int,M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.47 => ( ( ord_less_int @ B @ one_one_int )
% 0.30/0.47 => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.30/0.47 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_strict_decreasing_iff
% 0.30/0.47 thf(fact_215_power__strict__decreasing__iff,axiom,
% 0.30/0.47 ! [B: real,M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.47 => ( ( ord_less_real @ B @ one_one_real )
% 0.30/0.47 => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.30/0.47 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_strict_decreasing_iff
% 0.30/0.47 thf(fact_216_power__strict__decreasing__iff,axiom,
% 0.30/0.47 ! [B: nat,M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.47 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.30/0.47 => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.30/0.47 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_strict_decreasing_iff
% 0.30/0.47 thf(fact_217_power__mono__iff,axiom,
% 0.30/0.47 ! [A: poly_real,B: poly_real,N: nat] :
% 0.30/0.47 ( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
% 0.30/0.47 => ( ( ord_le1180086932y_real @ zero_zero_poly_real @ B )
% 0.30/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ( ord_le1180086932y_real @ ( power_2108872382y_real @ A @ N ) @ ( power_2108872382y_real @ B @ N ) )
% 0.30/0.47 = ( ord_le1180086932y_real @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_mono_iff
% 0.30/0.47 thf(fact_218_power__mono__iff,axiom,
% 0.30/0.47 ! [A: real,B: real,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.30/0.47 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_mono_iff
% 0.30/0.47 thf(fact_219_power__mono__iff,axiom,
% 0.30/0.47 ! [A: int,B: int,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.30/0.47 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_mono_iff
% 0.30/0.47 thf(fact_220_power__mono__iff,axiom,
% 0.30/0.47 ! [A: nat,B: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.30/0.47 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_mono_iff
% 0.30/0.47 thf(fact_221_power__increasing__iff,axiom,
% 0.30/0.47 ! [B: real,X: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.47 => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y2 ) )
% 0.30/0.47 = ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_increasing_iff
% 0.30/0.47 thf(fact_222_power__increasing__iff,axiom,
% 0.30/0.47 ! [B: int,X: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_int @ one_one_int @ B )
% 0.30/0.47 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
% 0.30/0.47 = ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_increasing_iff
% 0.30/0.47 thf(fact_223_power__increasing__iff,axiom,
% 0.30/0.47 ! [B: nat,X: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ one_one_nat @ B )
% 0.30/0.47 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
% 0.30/0.47 = ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_increasing_iff
% 0.30/0.47 thf(fact_224__092_060open_062_092_060not_062_AA_A_P_Areal__of__int_Ab_A_094_An_A_060_A_092_060bar_062x_A_N_Areal__of__int_Aa_A_P_Areal__of__int_Ab_092_060bar_062_092_060close_062,axiom,
% 0.30/0.47 ~ ( ord_less_real @ ( divide_divide_real @ a @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ n ) ) @ ( abs_abs_real @ ( minus_minus_real @ x @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>\<not> A / real_of_int b ^ n < \<bar>x - real_of_int a / real_of_int b\<bar>\<close>
% 0.30/0.47 thf(fact_225_nat__power__less__imp__less,axiom,
% 0.30/0.47 ! [I: nat,M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ I )
% 0.30/0.47 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.30/0.47 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_power_less_imp_less
% 0.30/0.47 thf(fact_226_div__poly__less,axiom,
% 0.30/0.47 ! [X: poly_real,Y2: poly_real] :
% 0.30/0.47 ( ( ord_less_nat @ ( degree_real @ X ) @ ( degree_real @ Y2 ) )
% 0.30/0.47 => ( ( divide1727078534y_real @ X @ Y2 )
% 0.30/0.47 = zero_zero_poly_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % div_poly_less
% 0.30/0.47 thf(fact_227_self__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_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % self_le_power
% 0.30/0.47 thf(fact_228_self__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_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % self_le_power
% 0.30/0.47 thf(fact_229_self__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_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % self_le_power
% 0.30/0.47 thf(fact_230_divide__poly__0,axiom,
% 0.30/0.47 ! [F: poly_real] :
% 0.30/0.47 ( ( divide1727078534y_real @ F @ zero_zero_poly_real )
% 0.30/0.47 = zero_zero_poly_real ) ).
% 0.30/0.47
% 0.30/0.47 % divide_poly_0
% 0.30/0.47 thf(fact_231_zero__power,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ( power_2108872382y_real @ zero_zero_poly_real @ N )
% 0.30/0.47 = zero_zero_poly_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_power
% 0.30/0.47 thf(fact_232_zero__power,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 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 % zero_power
% 0.30/0.47 thf(fact_233_zero__power,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 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 % zero_power
% 0.30/0.47 thf(fact_234_zero__power,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 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 % zero_power
% 0.30/0.47 thf(fact_235_power__0,axiom,
% 0.30/0.47 ! [A: real] :
% 0.30/0.47 ( ( power_power_real @ A @ zero_zero_nat )
% 0.30/0.47 = one_one_real ) ).
% 0.30/0.47
% 0.30/0.47 % power_0
% 0.30/0.47 thf(fact_236_power__0,axiom,
% 0.30/0.47 ! [A: nat] :
% 0.30/0.47 ( ( power_power_nat @ A @ zero_zero_nat )
% 0.30/0.47 = one_one_nat ) ).
% 0.30/0.47
% 0.30/0.47 % power_0
% 0.30/0.47 thf(fact_237_power__0,axiom,
% 0.30/0.47 ! [A: int] :
% 0.30/0.47 ( ( power_power_int @ A @ zero_zero_nat )
% 0.30/0.47 = one_one_int ) ).
% 0.30/0.47
% 0.30/0.47 % power_0
% 0.30/0.47 thf(fact_238_power__eq__iff__eq__base,axiom,
% 0.30/0.47 ! [N: nat,A: poly_real,B: poly_real] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
% 0.30/0.47 => ( ( ord_le1180086932y_real @ zero_zero_poly_real @ B )
% 0.30/0.47 => ( ( ( power_2108872382y_real @ A @ N )
% 0.30/0.47 = ( power_2108872382y_real @ B @ N ) )
% 0.30/0.47 = ( A = B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_eq_iff_eq_base
% 0.30/0.47 thf(fact_239_power__eq__iff__eq__base,axiom,
% 0.30/0.47 ! [N: nat,A: real,B: real] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.47 => ( ( ( power_power_real @ A @ N )
% 0.30/0.47 = ( power_power_real @ B @ N ) )
% 0.30/0.47 = ( A = B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_eq_iff_eq_base
% 0.30/0.47 thf(fact_240_power__eq__iff__eq__base,axiom,
% 0.30/0.47 ! [N: nat,A: int,B: int] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.47 => ( ( ( power_power_int @ A @ N )
% 0.30/0.47 = ( power_power_int @ B @ N ) )
% 0.30/0.47 = ( A = B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_eq_iff_eq_base
% 0.30/0.47 thf(fact_241_power__eq__iff__eq__base,axiom,
% 0.30/0.47 ! [N: nat,A: nat,B: nat] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.47 => ( ( ( power_power_nat @ A @ N )
% 0.30/0.47 = ( power_power_nat @ B @ N ) )
% 0.30/0.47 = ( A = B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_eq_iff_eq_base
% 0.30/0.47 thf(fact_242_power__eq__imp__eq__base,axiom,
% 0.30/0.47 ! [A: poly_real,N: nat,B: poly_real] :
% 0.30/0.47 ( ( ( power_2108872382y_real @ A @ N )
% 0.30/0.47 = ( power_2108872382y_real @ B @ N ) )
% 0.30/0.47 => ( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
% 0.30/0.47 => ( ( ord_le1180086932y_real @ zero_zero_poly_real @ B )
% 0.30/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( A = B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_eq_imp_eq_base
% 0.30/0.47 thf(fact_243_power__eq__imp__eq__base,axiom,
% 0.30/0.47 ! [A: real,N: nat,B: real] :
% 0.30/0.47 ( ( ( power_power_real @ A @ N )
% 0.30/0.47 = ( power_power_real @ B @ N ) )
% 0.30/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( A = B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_eq_imp_eq_base
% 0.30/0.47 thf(fact_244_power__eq__imp__eq__base,axiom,
% 0.30/0.47 ! [A: int,N: nat,B: int] :
% 0.30/0.47 ( ( ( power_power_int @ A @ N )
% 0.30/0.47 = ( power_power_int @ B @ N ) )
% 0.30/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( A = B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_eq_imp_eq_base
% 0.30/0.47 thf(fact_245_power__eq__imp__eq__base,axiom,
% 0.30/0.47 ! [A: nat,N: nat,B: nat] :
% 0.30/0.47 ( ( ( power_power_nat @ A @ N )
% 0.30/0.47 = ( power_power_nat @ B @ N ) )
% 0.30/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( A = B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_eq_imp_eq_base
% 0.30/0.47 thf(fact_246_one__less__power,axiom,
% 0.30/0.47 ! [A: int,N: nat] :
% 0.30/0.47 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_less_power
% 0.30/0.47 thf(fact_247_one__less__power,axiom,
% 0.30/0.47 ! [A: real,N: nat] :
% 0.30/0.47 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_less_power
% 0.30/0.47 thf(fact_248_one__less__power,axiom,
% 0.30/0.47 ! [A: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.47 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_less_power
% 0.30/0.47 thf(fact_249_poly__IVT__neg,axiom,
% 0.30/0.47 ! [A: real,B: real,P: poly_real] :
% 0.30/0.47 ( ( ord_less_real @ A @ B )
% 0.30/0.47 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ A ) )
% 0.30/0.47 => ( ( ord_less_real @ ( poly_real2 @ P @ B ) @ zero_zero_real )
% 0.30/0.47 => ? [X3: real] :
% 0.30/0.47 ( ( ord_less_real @ A @ X3 )
% 0.30/0.47 & ( ord_less_real @ X3 @ B )
% 0.30/0.47 & ( ( poly_real2 @ P @ X3 )
% 0.30/0.47 = zero_zero_real ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % poly_IVT_neg
% 0.30/0.47 thf(fact_250_poly__IVT__pos,axiom,
% 0.30/0.47 ! [A: real,B: real,P: poly_real] :
% 0.30/0.47 ( ( ord_less_real @ A @ B )
% 0.30/0.47 => ( ( ord_less_real @ ( poly_real2 @ P @ A ) @ zero_zero_real )
% 0.30/0.47 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ B ) )
% 0.30/0.47 => ? [X3: real] :
% 0.30/0.47 ( ( ord_less_real @ A @ X3 )
% 0.30/0.47 & ( ord_less_real @ X3 @ B )
% 0.30/0.47 & ( ( poly_real2 @ P @ X3 )
% 0.30/0.47 = zero_zero_real ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % poly_IVT_pos
% 0.30/0.47 thf(fact_251_linordered__field__no__ub,axiom,
% 0.30/0.47 ! [X4: real] :
% 0.30/0.47 ? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% 0.30/0.47
% 0.30/0.47 % linordered_field_no_ub
% 0.30/0.47 thf(fact_252_linordered__field__no__lb,axiom,
% 0.30/0.47 ! [X4: real] :
% 0.30/0.47 ? [Y3: real] : ( ord_less_real @ Y3 @ X4 ) ).
% 0.30/0.47
% 0.30/0.47 % linordered_field_no_lb
% 0.30/0.47 thf(fact_253_coeff__eq__0,axiom,
% 0.30/0.47 ! [P: poly_poly_real,N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ ( degree_poly_real @ P ) @ N )
% 0.30/0.47 => ( ( coeff_poly_real @ P @ N )
% 0.30/0.47 = zero_zero_poly_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % coeff_eq_0
% 0.30/0.47 thf(fact_254_coeff__eq__0,axiom,
% 0.30/0.47 ! [P: poly_int,N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ ( degree_int @ P ) @ N )
% 0.30/0.47 => ( ( coeff_int @ P @ N )
% 0.30/0.47 = zero_zero_int ) ) ).
% 0.30/0.47
% 0.30/0.47 % coeff_eq_0
% 0.30/0.47 thf(fact_255_coeff__eq__0,axiom,
% 0.30/0.47 ! [P: poly_real,N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ ( degree_real @ P ) @ N )
% 0.30/0.47 => ( ( coeff_real @ P @ N )
% 0.30/0.47 = zero_zero_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % coeff_eq_0
% 0.30/0.47 thf(fact_256_coeff__eq__0,axiom,
% 0.30/0.47 ! [P: poly_nat,N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ ( degree_nat @ P ) @ N )
% 0.30/0.47 => ( ( coeff_nat @ P @ N )
% 0.30/0.47 = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % coeff_eq_0
% 0.30/0.47 thf(fact_257_less__degree__imp,axiom,
% 0.30/0.47 ! [N: nat,P: poly_poly_real] :
% 0.30/0.47 ( ( ord_less_nat @ N @ ( degree_poly_real @ P ) )
% 0.30/0.47 => ? [I4: nat] :
% 0.30/0.47 ( ( ord_less_nat @ N @ I4 )
% 0.30/0.47 & ( ( coeff_poly_real @ P @ I4 )
% 0.30/0.47 != zero_zero_poly_real ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_degree_imp
% 0.30/0.47 thf(fact_258_less__degree__imp,axiom,
% 0.30/0.47 ! [N: nat,P: poly_int] :
% 0.30/0.47 ( ( ord_less_nat @ N @ ( degree_int @ P ) )
% 0.30/0.47 => ? [I4: nat] :
% 0.30/0.47 ( ( ord_less_nat @ N @ I4 )
% 0.30/0.47 & ( ( coeff_int @ P @ I4 )
% 0.30/0.47 != zero_zero_int ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_degree_imp
% 0.30/0.47 thf(fact_259_less__degree__imp,axiom,
% 0.30/0.47 ! [N: nat,P: poly_real] :
% 0.30/0.47 ( ( ord_less_nat @ N @ ( degree_real @ P ) )
% 0.30/0.47 => ? [I4: nat] :
% 0.30/0.47 ( ( ord_less_nat @ N @ I4 )
% 0.30/0.47 & ( ( coeff_real @ P @ I4 )
% 0.30/0.47 != zero_zero_real ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_degree_imp
% 0.30/0.47 thf(fact_260_less__degree__imp,axiom,
% 0.30/0.47 ! [N: nat,P: poly_nat] :
% 0.30/0.47 ( ( ord_less_nat @ N @ ( degree_nat @ P ) )
% 0.30/0.47 => ? [I4: nat] :
% 0.30/0.47 ( ( ord_less_nat @ N @ I4 )
% 0.30/0.47 & ( ( coeff_nat @ P @ I4 )
% 0.30/0.47 != zero_zero_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_degree_imp
% 0.30/0.47 thf(fact_261_degree__le,axiom,
% 0.30/0.47 ! [N: nat,P: poly_poly_real] :
% 0.30/0.47 ( ! [I4: nat] :
% 0.30/0.47 ( ( ord_less_nat @ N @ I4 )
% 0.30/0.47 => ( ( coeff_poly_real @ P @ I4 )
% 0.30/0.47 = zero_zero_poly_real ) )
% 0.30/0.47 => ( ord_less_eq_nat @ ( degree_poly_real @ P ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % degree_le
% 0.30/0.47 thf(fact_262_degree__le,axiom,
% 0.30/0.47 ! [N: nat,P: poly_int] :
% 0.30/0.47 ( ! [I4: nat] :
% 0.30/0.47 ( ( ord_less_nat @ N @ I4 )
% 0.30/0.47 => ( ( coeff_int @ P @ I4 )
% 0.30/0.47 = zero_zero_int ) )
% 0.30/0.47 => ( ord_less_eq_nat @ ( degree_int @ P ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % degree_le
% 0.30/0.47 thf(fact_263_degree__le,axiom,
% 0.30/0.47 ! [N: nat,P: poly_real] :
% 0.30/0.47 ( ! [I4: nat] :
% 0.30/0.47 ( ( ord_less_nat @ N @ I4 )
% 0.30/0.47 => ( ( coeff_real @ P @ I4 )
% 0.30/0.47 = zero_zero_real ) )
% 0.30/0.47 => ( ord_less_eq_nat @ ( degree_real @ P ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % degree_le
% 0.30/0.47 thf(fact_264_degree__le,axiom,
% 0.30/0.47 ! [N: nat,P: poly_nat] :
% 0.30/0.47 ( ! [I4: nat] :
% 0.30/0.47 ( ( ord_less_nat @ N @ I4 )
% 0.30/0.47 => ( ( coeff_nat @ P @ I4 )
% 0.30/0.47 = zero_zero_nat ) )
% 0.30/0.47 => ( ord_less_eq_nat @ ( degree_nat @ P ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % degree_le
% 0.30/0.47 thf(fact_265_Liouville__Numbers__Misc_ORats__cases_H,axiom,
% 0.30/0.47 ! [X: real] :
% 0.30/0.47 ( ( member_real @ X @ field_1537545994s_real )
% 0.30/0.47 => ~ ! [P2: int,Q3: int] :
% 0.30/0.47 ( ( ord_less_int @ zero_zero_int @ Q3 )
% 0.30/0.47 => ( X
% 0.30/0.47 != ( divide_divide_real @ ( ring_1_of_int_real @ P2 ) @ ( ring_1_of_int_real @ Q3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Liouville_Numbers_Misc.Rats_cases'
% 0.30/0.47 thf(fact_266_coeff__0__power,axiom,
% 0.30/0.47 ! [P: poly_real,N: nat] :
% 0.30/0.47 ( ( coeff_real @ ( power_2108872382y_real @ P @ N ) @ zero_zero_nat )
% 0.30/0.47 = ( power_power_real @ ( coeff_real @ P @ zero_zero_nat ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % coeff_0_power
% 0.30/0.47 thf(fact_267_coeff__0__power,axiom,
% 0.30/0.47 ! [P: poly_nat,N: nat] :
% 0.30/0.47 ( ( coeff_nat @ ( power_power_poly_nat @ P @ N ) @ zero_zero_nat )
% 0.30/0.47 = ( power_power_nat @ ( coeff_nat @ P @ zero_zero_nat ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % coeff_0_power
% 0.30/0.47 thf(fact_268_coeff__0__power,axiom,
% 0.30/0.47 ! [P: poly_int,N: nat] :
% 0.30/0.47 ( ( coeff_int @ ( power_power_poly_int @ P @ N ) @ zero_zero_nat )
% 0.30/0.47 = ( power_power_int @ ( coeff_int @ P @ zero_zero_nat ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % coeff_0_power
% 0.30/0.47 thf(fact_269_eq__zero__or__degree__less,axiom,
% 0.30/0.47 ! [P: poly_poly_real,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( degree_poly_real @ P ) @ N )
% 0.30/0.47 => ( ( ( coeff_poly_real @ P @ N )
% 0.30/0.47 = zero_zero_poly_real )
% 0.30/0.47 => ( ( P = zero_z1423781445y_real )
% 0.30/0.47 | ( ord_less_nat @ ( degree_poly_real @ P ) @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_zero_or_degree_less
% 0.30/0.47 thf(fact_270_eq__zero__or__degree__less,axiom,
% 0.30/0.47 ! [P: poly_int,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( degree_int @ P ) @ N )
% 0.30/0.47 => ( ( ( coeff_int @ P @ N )
% 0.30/0.47 = zero_zero_int )
% 0.30/0.47 => ( ( P = zero_zero_poly_int )
% 0.30/0.47 | ( ord_less_nat @ ( degree_int @ P ) @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_zero_or_degree_less
% 0.30/0.47 thf(fact_271_eq__zero__or__degree__less,axiom,
% 0.30/0.47 ! [P: poly_real,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( degree_real @ P ) @ N )
% 0.30/0.47 => ( ( ( coeff_real @ P @ N )
% 0.30/0.47 = zero_zero_real )
% 0.30/0.47 => ( ( P = zero_zero_poly_real )
% 0.30/0.47 | ( ord_less_nat @ ( degree_real @ P ) @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_zero_or_degree_less
% 0.30/0.47 thf(fact_272_eq__zero__or__degree__less,axiom,
% 0.30/0.47 ! [P: poly_nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( degree_nat @ P ) @ N )
% 0.30/0.47 => ( ( ( coeff_nat @ P @ N )
% 0.30/0.47 = zero_zero_nat )
% 0.30/0.47 => ( ( P = zero_zero_poly_nat )
% 0.30/0.47 | ( ord_less_nat @ ( degree_nat @ P ) @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_zero_or_degree_less
% 0.30/0.47 thf(fact_273_power__not__zero,axiom,
% 0.30/0.47 ! [A: poly_real,N: nat] :
% 0.30/0.47 ( ( A != zero_zero_poly_real )
% 0.30/0.47 => ( ( power_2108872382y_real @ A @ N )
% 0.30/0.47 != zero_zero_poly_real ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_not_zero
% 0.30/0.47 thf(fact_274_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_275_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_276_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_277_power__divide,axiom,
% 0.30/0.47 ! [A: real,B: real,N: nat] :
% 0.30/0.47 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 0.30/0.47 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_divide
% 0.30/0.47 thf(fact_278_poly__pinfty__gt__lc,axiom,
% 0.30/0.47 ! [P: poly_real] :
% 0.30/0.47 ( ( ord_less_real @ zero_zero_real @ ( coeff_real @ P @ ( degree_real @ P ) ) )
% 0.30/0.47 => ? [N2: real] :
% 0.30/0.47 ! [X4: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ N2 @ X4 )
% 0.30/0.47 => ( ord_less_eq_real @ ( coeff_real @ P @ ( degree_real @ P ) ) @ ( poly_real2 @ P @ X4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % poly_pinfty_gt_lc
% 0.30/0.47 thf(fact_279_power__abs,axiom,
% 0.30/0.47 ! [A: real,N: nat] :
% 0.30/0.47 ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 0.30/0.47 = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_abs
% 0.30/0.47 thf(fact_280_power__abs,axiom,
% 0.30/0.47 ! [A: int,N: nat] :
% 0.30/0.47 ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 0.30/0.47 = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_abs
% 0.30/0.47 thf(fact_281_algebraic__altdef,axiom,
% 0.30/0.47 ( algebraic_real
% 0.30/0.47 = ( ^ [X2: real] :
% 0.30/0.47 ? [P4: poly_real] :
% 0.30/0.47 ( ! [I3: nat] : ( member_real @ ( coeff_real @ P4 @ I3 ) @ field_1537545994s_real )
% 0.30/0.47 & ( P4 != zero_zero_poly_real )
% 0.30/0.47 & ( ( poly_real2 @ P4 @ X2 )
% 0.30/0.47 = zero_zero_real ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % algebraic_altdef
% 0.30/0.47 thf(fact_282_divide__le__0__iff,axiom,
% 0.30/0.47 ! [A: real,B: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.30/0.47 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.47 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.30/0.47 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.47 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_le_0_iff
% 0.30/0.47 thf(fact_283_divide__right__mono,axiom,
% 0.30/0.47 ! [A: real,B: real,C: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.47 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_right_mono
% 0.30/0.47 thf(fact_284_zero__le__divide__iff,axiom,
% 0.30/0.47 ! [A: real,B: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.47 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.47 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.30/0.47 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.47 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_le_divide_iff
% 0.30/0.47 thf(fact_285_divide__nonneg__nonneg,axiom,
% 0.30/0.47 ! [X: real,Y2: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
% 0.30/0.47 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_nonneg_nonneg
% 0.30/0.47 thf(fact_286_divide__nonneg__nonpos,axiom,
% 0.30/0.47 ! [X: real,Y2: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.47 => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
% 0.30/0.47 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_nonneg_nonpos
% 0.30/0.47 thf(fact_287_divide__nonpos__nonneg,axiom,
% 0.30/0.47 ! [X: real,Y2: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
% 0.30/0.47 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_nonpos_nonneg
% 0.30/0.47 thf(fact_288_divide__nonpos__nonpos,axiom,
% 0.30/0.47 ! [X: real,Y2: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/0.47 => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
% 0.30/0.47 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_nonpos_nonpos
% 0.30/0.47 thf(fact_289_divide__right__mono__neg,axiom,
% 0.30/0.47 ! [A: real,B: real,C: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.47 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.47 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_right_mono_neg
% 0.30/0.47 thf(fact_290_power__mono,axiom,
% 0.30/0.47 ! [A: poly_real,B: poly_real,N: nat] :
% 0.30/0.47 ( ( ord_le1180086932y_real @ A @ B )
% 0.30/0.47 => ( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
% 0.30/0.47 => ( ord_le1180086932y_real @ ( power_2108872382y_real @ A @ N ) @ ( power_2108872382y_real @ B @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_mono
% 0.30/0.47 thf(fact_291_power__mono,axiom,
% 0.30/0.47 ! [A: real,B: real,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.47 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_mono
% 0.30/0.47 thf(fact_292_power__mono,axiom,
% 0.30/0.47 ! [A: int,B: int,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.47 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_mono
% 0.30/0.47 thf(fact_293_power__mono,axiom,
% 0.30/0.47 ! [A: nat,B: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.47 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_mono
% 0.30/0.47 thf(fact_294_zero__le__power,axiom,
% 0.30/0.47 ! [A: poly_real,N: nat] :
% 0.30/0.47 ( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
% 0.30/0.47 => ( ord_le1180086932y_real @ zero_zero_poly_real @ ( power_2108872382y_real @ A @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_le_power
% 0.30/0.47 thf(fact_295_zero__le__power,axiom,
% 0.30/0.47 ! [A: real,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.47 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_le_power
% 0.30/0.47 thf(fact_296_zero__le__power,axiom,
% 0.30/0.47 ! [A: int,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.47 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_le_power
% 0.30/0.47 thf(fact_297_zero__le__power,axiom,
% 0.30/0.47 ! [A: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.47 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_le_power
% 0.30/0.47 thf(fact_298_divide__neg__neg,axiom,
% 0.30/0.47 ! [X: real,Y2: real] :
% 0.30/0.47 ( ( ord_less_real @ X @ zero_zero_real )
% 0.30/0.47 => ( ( ord_less_real @ Y2 @ zero_zero_real )
% 0.30/0.47 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_neg_neg
% 0.30/0.47 thf(fact_299_divide__neg__pos,axiom,
% 0.30/0.47 ! [X: real,Y2: real] :
% 0.30/0.47 ( ( ord_less_real @ X @ zero_zero_real )
% 0.30/0.47 => ( ( ord_less_real @ zero_zero_real @ Y2 )
% 0.30/0.47 => ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_neg_pos
% 0.30/0.47 thf(fact_300_divide__pos__neg,axiom,
% 0.30/0.47 ! [X: real,Y2: real] :
% 0.30/0.47 ( ( ord_less_real @ zero_zero_real @ X )
% 0.30/0.47 => ( ( ord_less_real @ Y2 @ zero_zero_real )
% 0.30/0.47 => ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_pos_neg
% 0.30/0.47 thf(fact_301_divide__pos__pos,axiom,
% 0.30/0.47 ! [X: real,Y2: real] :
% 0.30/0.47 ( ( ord_less_real @ zero_zero_real @ X )
% 0.30/0.47 => ( ( ord_less_real @ zero_zero_real @ Y2 )
% 0.30/0.47 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_pos_pos
% 0.30/0.47 thf(fact_302_divide__less__0__iff,axiom,
% 0.30/0.47 ! [A: real,B: real] :
% 0.30/0.47 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.30/0.47 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.47 & ( ord_less_real @ B @ zero_zero_real ) )
% 0.30/0.47 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.47 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_less_0_iff
% 0.30/0.47 thf(fact_303_divide__less__cancel,axiom,
% 0.30/0.47 ! [A: real,C: real,B: real] :
% 0.30/0.47 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.30/0.47 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.47 => ( ord_less_real @ A @ B ) )
% 0.30/0.47 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.47 => ( ord_less_real @ B @ A ) )
% 0.30/0.47 & ( C != zero_zero_real ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_less_cancel
% 0.30/0.47 thf(fact_304_zero__less__divide__iff,axiom,
% 0.30/0.47 ! [A: real,B: real] :
% 0.30/0.47 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.47 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.47 & ( ord_less_real @ zero_zero_real @ B ) )
% 0.30/0.47 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.47 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_less_divide_iff
% 0.30/0.47 thf(fact_305_divide__strict__right__mono,axiom,
% 0.30/0.47 ! [A: real,B: real,C: real] :
% 0.30/0.47 ( ( ord_less_real @ A @ B )
% 0.30/0.47 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.47 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_strict_right_mono
% 0.30/0.47 thf(fact_306_divide__strict__right__mono__neg,axiom,
% 0.30/0.47 ! [B: real,A: real,C: real] :
% 0.30/0.47 ( ( ord_less_real @ B @ A )
% 0.30/0.47 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.47 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_strict_right_mono_neg
% 0.30/0.47 thf(fact_307_zero__less__power,axiom,
% 0.30/0.47 ! [A: poly_real,N: nat] :
% 0.30/0.47 ( ( ord_less_poly_real @ zero_zero_poly_real @ A )
% 0.30/0.47 => ( ord_less_poly_real @ zero_zero_poly_real @ ( power_2108872382y_real @ A @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_less_power
% 0.30/0.47 thf(fact_308_zero__less__power,axiom,
% 0.30/0.47 ! [A: int,N: nat] :
% 0.30/0.47 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.47 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_less_power
% 0.30/0.47 thf(fact_309_zero__less__power,axiom,
% 0.30/0.47 ! [A: real,N: nat] :
% 0.30/0.47 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.47 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_less_power
% 0.30/0.47 thf(fact_310_zero__less__power,axiom,
% 0.30/0.47 ! [A: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.47 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_less_power
% 0.30/0.47 thf(fact_311_right__inverse__eq,axiom,
% 0.30/0.47 ! [B: real,A: real] :
% 0.30/0.47 ( ( B != zero_zero_real )
% 0.30/0.47 => ( ( ( divide_divide_real @ A @ B )
% 0.30/0.47 = one_one_real )
% 0.30/0.47 = ( A = B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % right_inverse_eq
% 0.30/0.47 thf(fact_312_power__0__left,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ( N = zero_zero_nat )
% 0.30/0.47 => ( ( power_2108872382y_real @ zero_zero_poly_real @ N )
% 0.30/0.47 = one_one_poly_real ) )
% 0.30/0.47 & ( ( N != zero_zero_nat )
% 0.30/0.47 => ( ( power_2108872382y_real @ zero_zero_poly_real @ N )
% 0.30/0.47 = zero_zero_poly_real ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_0_left
% 0.30/0.47 thf(fact_313_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_314_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_315_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_316_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_317_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_318_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_319_power__increasing,axiom,
% 0.30/0.47 ! [N: nat,N4: nat,A: real] :
% 0.30/0.47 ( ( ord_less_eq_nat @ N @ N4 )
% 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 @ N4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_increasing
% 0.30/0.47 thf(fact_320_power__increasing,axiom,
% 0.30/0.47 ! [N: nat,N4: nat,A: int] :
% 0.30/0.47 ( ( ord_less_eq_nat @ N @ N4 )
% 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 @ N4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_increasing
% 0.30/0.47 thf(fact_321_power__increasing,axiom,
% 0.30/0.47 ! [N: nat,N4: nat,A: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ N @ N4 )
% 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 @ N4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_increasing
% 0.30/0.47 thf(fact_322_power__less__imp__less__exp,axiom,
% 0.30/0.47 ! [A: int,M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.47 => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.30/0.47 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_less_imp_less_exp
% 0.30/0.47 thf(fact_323_power__less__imp__less__exp,axiom,
% 0.30/0.47 ! [A: real,M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.47 => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.30/0.47 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_less_imp_less_exp
% 0.30/0.47 thf(fact_324_power__less__imp__less__exp,axiom,
% 0.30/0.47 ! [A: nat,M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.47 => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.30/0.47 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_less_imp_less_exp
% 0.30/0.47 thf(fact_325_power__strict__increasing,axiom,
% 0.30/0.47 ! [N: nat,N4: nat,A: int] :
% 0.30/0.47 ( ( ord_less_nat @ N @ N4 )
% 0.30/0.47 => ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.47 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_strict_increasing
% 0.30/0.47 thf(fact_326_power__strict__increasing,axiom,
% 0.30/0.47 ! [N: nat,N4: nat,A: real] :
% 0.30/0.47 ( ( ord_less_nat @ N @ N4 )
% 0.30/0.47 => ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.47 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_strict_increasing
% 0.30/0.47 thf(fact_327_power__strict__increasing,axiom,
% 0.30/0.47 ! [N: nat,N4: nat,A: nat] :
% 0.30/0.47 ( ( ord_less_nat @ N @ N4 )
% 0.30/0.47 => ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.47 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_strict_increasing
% 0.30/0.47 thf(fact_328_power__one__over,axiom,
% 0.30/0.47 ! [A: real,N: nat] :
% 0.30/0.47 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 0.30/0.47 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_one_over
% 0.30/0.47 thf(fact_329_nonzero__abs__divide,axiom,
% 0.30/0.47 ! [B: real,A: real] :
% 0.30/0.47 ( ( B != zero_zero_real )
% 0.30/0.47 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.47 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nonzero_abs_divide
% 0.30/0.47 thf(fact_330_frac__le,axiom,
% 0.30/0.47 ! [Y2: real,X: real,W: real,Z: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
% 0.30/0.47 => ( ( ord_less_eq_real @ X @ Y2 )
% 0.30/0.47 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.30/0.47 => ( ( ord_less_eq_real @ W @ Z )
% 0.30/0.47 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % frac_le
% 0.30/0.47 thf(fact_331_frac__less,axiom,
% 0.30/0.47 ! [X: real,Y2: real,W: real,Z: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.47 => ( ( ord_less_real @ X @ Y2 )
% 0.30/0.47 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.30/0.47 => ( ( ord_less_eq_real @ W @ Z )
% 0.30/0.47 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % frac_less
% 0.30/0.47 thf(fact_332_frac__less2,axiom,
% 0.30/0.47 ! [X: real,Y2: real,W: real,Z: real] :
% 0.30/0.47 ( ( ord_less_real @ zero_zero_real @ X )
% 0.30/0.47 => ( ( ord_less_eq_real @ X @ Y2 )
% 0.30/0.47 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.30/0.47 => ( ( ord_less_real @ W @ Z )
% 0.30/0.47 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % frac_less2
% 0.30/0.47 thf(fact_333_divide__le__cancel,axiom,
% 0.30/0.47 ! [A: real,C: real,B: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.30/0.47 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.47 => ( ord_less_eq_real @ A @ B ) )
% 0.30/0.47 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.47 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_le_cancel
% 0.30/0.47 thf(fact_334_divide__nonneg__neg,axiom,
% 0.30/0.47 ! [X: real,Y2: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.47 => ( ( ord_less_real @ Y2 @ zero_zero_real )
% 0.30/0.47 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_nonneg_neg
% 0.30/0.47 thf(fact_335_divide__nonneg__pos,axiom,
% 0.30/0.47 ! [X: real,Y2: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.47 => ( ( ord_less_real @ zero_zero_real @ Y2 )
% 0.30/0.47 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_nonneg_pos
% 0.30/0.47 thf(fact_336_divide__nonpos__neg,axiom,
% 0.30/0.47 ! [X: real,Y2: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/0.47 => ( ( ord_less_real @ Y2 @ zero_zero_real )
% 0.30/0.47 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_nonpos_neg
% 0.30/0.47 thf(fact_337_divide__nonpos__pos,axiom,
% 0.30/0.47 ! [X: real,Y2: real] :
% 0.30/0.47 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/0.47 => ( ( ord_less_real @ zero_zero_real @ Y2 )
% 0.30/0.47 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % divide_nonpos_pos
% 0.30/0.47 thf(fact_338_power__less__imp__less__base,axiom,
% 0.30/0.47 ! [A: nat,N: nat,B: nat] :
% 0.30/0.47 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.30/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.47 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % power_less_imp_less_base
% 0.30/0.47 thf(fact_339_A__less_I2_J,axiom,
% 0.30/0.47 ord_less_real @ a @ ( divide_divide_real @ one_one_real @ m ) ).
% 0.30/0.47
% 0.30/0.47 % A_less(2)
% 0.30/0.47 thf(fact_340_ab,axiom,
% 0.30/0.47 ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ x @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) @ ( divide_divide_real @ a @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ n ) ) ).
% 0.30/0.47
% 0.30/0.47 % ab
% 0.30/0.47 thf(fact_341_ab_H,axiom,
% 0.30/0.47 ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ x @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) @ a ).
% 0.30/0.47
% 0.30/0.47 % ab'
% 0.30/0.47 thf(fact_342_div__pos__pos__trivial,axiom,
% 0.30/0.47 ! [K: int,L: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.47 => ( ( ord_less_int @ K @ L )
% 0.30/0.47 => ( ( divide_divide_int @ K @ L )
% 0.30/0.47 = zero_zero_int ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % div_pos_pos_trivial
% 0.30/0.47 thf(fact_343_div__neg__neg__trivial,axiom,
% 0.30/0.47 ! [K: int,L: int] :
% 0.30/0.47 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.30/0.47 => ( ( ord_less_int @ L @ K )
% 0.30/0.47 => ( ( divide_divide_int @ K @ L )
% 0.30/0.47 = zero_zero_int ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % div_neg_neg_trivial
% 0.30/0.47 thf(fact_344_M__pos,axiom,
% 0.30/0.47 ord_less_real @ zero_zero_real @ m ).
% 0.30/0.47
% 0.30/0.47 % M_pos
% 0.30/0.47 thf(fact_345_div__less,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ M @ N )
% 0.30/0.47 => ( ( divide_divide_nat @ M @ N )
% 0.30/0.47 = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % div_less
% 0.30/0.47 thf(fact_346_A__less_I3_J,axiom,
% 0.30/0.47 ! [X5: real] :
% 0.30/0.47 ( ( X5 != x )
% 0.30/0.47 => ( ( ( poly_real2 @ p @ X5 )
% 0.30/0.47 = zero_zero_real )
% 0.30/0.47 => ( ord_less_real @ a @ ( abs_abs_real @ ( minus_minus_real @ X5 @ x ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % A_less(3)
% 0.30/0.47 thf(fact_347_that,axiom,
% 0.30/0.47 ! [C: real,N: nat] :
% 0.30/0.47 ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.47 => ( ! [P2: int,Q3: int] :
% 0.30/0.47 ( ( ord_less_int @ zero_zero_int @ Q3 )
% 0.30/0.47 => ( ord_less_real @ ( divide_divide_real @ C @ ( power_power_real @ ( ring_1_of_int_real @ Q3 ) @ N ) ) @ ( abs_abs_real @ ( minus_minus_real @ x @ ( divide_divide_real @ ( ring_1_of_int_real @ P2 ) @ ( ring_1_of_int_real @ Q3 ) ) ) ) ) )
% 0.30/0.47 => thesis ) ) ).
% 0.30/0.47
% 0.30/0.47 % that
% 0.30/0.47 thf(fact_348_div__eq__dividend__iff,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.47 => ( ( ( divide_divide_nat @ M @ N )
% 0.30/0.47 = M )
% 0.30/0.47 = ( N = one_one_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % div_eq_dividend_iff
% 0.30/0.47 thf(fact_349_div__less__dividend,axiom,
% 0.30/0.47 ! [N: nat,M: nat] :
% 0.30/0.47 ( ( ord_less_nat @ one_one_nat @ N )
% 0.30/0.47 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.47 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % div_less_dividend
% 0.30/0.47 thf(fact_350_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ( divide_divide_nat @ M @ N )
% 0.30/0.47 = zero_zero_nat )
% 0.30/0.47 = ( ( ord_less_nat @ M @ N )
% 0.30/0.47 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Euclidean_Division.div_eq_0_iff
% 0.30/0.47 thf(fact_351_div__le__dividend,axiom,
% 0.30/0.47 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 0.30/0.47
% 0.30/0.47 % div_le_dividend
% 0.30/0.47 thf(fact_352_div__le__mono,axiom,
% 0.30/0.47 ! [M: nat,N: nat,K: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.47 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % div_le_mono
% 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_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 3.04/1.77 ---- Embedded in THF ---
% 3.04/1.77 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.04/1.77 %%% Generated on Tue Feb 24 08:26:28 EST 2026
% 3.04/1.77 %%% using '$$dhol' embedding, version 1.3.0.
% 3.04/1.77 %%% Logic specification used:
% 3.04/1.77 %%% thf(spec, logic, $$dhol).
% 3.04/1.77
% 3.04/1.77 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2DTF21775.p
% See solution above
% 3.04/1.77 ------------------------
% 3.04/1.79 ---- Cleaned THF ---
% 3.04/1.80 thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 3.04/1.80 poly_poly_real: $tType ).
% 3.04/1.80
% 3.04/1.80 thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 3.04/1.80 set_poly_real: $tType ).
% 3.04/1.80
% 3.04/1.80 thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 poly_real: $tType ).
% 3.04/1.80
% 3.04/1.80 thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 3.04/1.80 poly_nat: $tType ).
% 3.04/1.80
% 3.04/1.80 thf(ty_n_t__Polynomial__Opoly_It__Int__Oint_J,type,
% 3.04/1.80 poly_int: $tType ).
% 3.04/1.80
% 3.04/1.80 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 3.04/1.80 set_real: $tType ).
% 3.04/1.80
% 3.04/1.80 thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
% 3.04/1.80 set_int: $tType ).
% 3.04/1.80
% 3.04/1.80 thf(ty_n_t__Real__Oreal,type,
% 3.04/1.80 real: $tType ).
% 3.04/1.80
% 3.04/1.80 thf(ty_n_t__Nat__Onat,type,
% 3.04/1.80 nat: $tType ).
% 3.04/1.80
% 3.04/1.80 thf(ty_n_t__Int__Oint,type,
% 3.04/1.80 int: $tType ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
% 3.04/1.80 abs_abs_int: int > int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Oabs__class_Oabs_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 abs_abs_poly_real: poly_real > poly_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 3.04/1.80 abs_abs_real: real > real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 3.04/1.80 minus_minus_real: real > real > real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
% 3.04/1.80 one_one_int: int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 3.04/1.80 one_one_nat: nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 3.04/1.80 one_one_poly_int: poly_int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 3.04/1.80 one_one_poly_nat: poly_nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 one_one_poly_real: poly_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 3.04/1.80 one_one_real: real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 3.04/1.80 zero_zero_int: int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 3.04/1.80 zero_zero_nat: nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 3.04/1.80 zero_zero_poly_int: poly_int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 3.04/1.80 zero_zero_poly_nat: poly_nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
% 3.04/1.80 zero_z1423781445y_real: poly_poly_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 zero_zero_poly_real: poly_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 3.04/1.80 zero_zero_real: real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
% 3.04/1.80 ring_1_Ints_int: set_int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Int_Oring__1__class_OInts_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 ring_1690226883y_real: set_poly_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
% 3.04/1.80 ring_1_Ints_real: set_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 3.04/1.80 ring_1_of_int_int: int > int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 3.04/1.80 ring_12102921859ly_int: int > poly_int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 ring_11511526659y_real: int > poly_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 3.04/1.80 ring_1_of_int_real: int > real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 3.04/1.80 ord_less_int: int > int > $o ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 3.04/1.80 ord_less_nat: nat > nat > $o ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 ord_less_poly_real: poly_real > poly_real > $o ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 3.04/1.80 ord_less_real: real > real > $o ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 3.04/1.80 ord_less_eq_int: int > int > $o ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 3.04/1.80 ord_less_eq_nat: nat > nat > $o ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 ord_le1180086932y_real: poly_real > poly_real > $o ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 3.04/1.80 ord_less_eq_real: real > real > $o ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Polynomial_Oalgebraic_001t__Real__Oreal,type,
% 3.04/1.80 algebraic_real: real > $o ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Polynomial_Odegree_001t__Int__Oint,type,
% 3.04/1.80 degree_int: poly_int > nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Polynomial_Odegree_001t__Nat__Onat,type,
% 3.04/1.80 degree_nat: poly_nat > nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 degree_poly_real: poly_poly_real > nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Polynomial_Odegree_001t__Real__Oreal,type,
% 3.04/1.80 degree_real: poly_real > nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Polynomial_Opoly_001t__Int__Oint,type,
% 3.04/1.80 poly_int2: poly_int > int > int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Polynomial_Opoly_001t__Nat__Onat,type,
% 3.04/1.80 poly_nat2: poly_nat > nat > nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 poly_poly_real2: poly_poly_real > poly_real > poly_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Polynomial_Opoly_001t__Real__Oreal,type,
% 3.04/1.80 poly_real2: poly_real > real > real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Int__Oint,type,
% 3.04/1.80 coeff_int: poly_int > nat > int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat,type,
% 3.04/1.80 coeff_nat: poly_nat > nat > nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 coeff_poly_real: poly_poly_real > nat > poly_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Real__Oreal,type,
% 3.04/1.80 coeff_real: poly_real > nat > real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 3.04/1.80 power_power_int: int > nat > int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 3.04/1.80 power_power_nat: nat > nat > nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Int__Oint_J,type,
% 3.04/1.80 power_power_poly_int: poly_int > nat > poly_int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
% 3.04/1.80 power_power_poly_nat: poly_nat > nat > poly_nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 power_2108872382y_real: poly_real > nat > poly_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
% 3.04/1.80 power_power_real: real > nat > real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 3.04/1.80 field_1537545994s_real: set_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 3.04/1.80 divide_divide_int: int > int > int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 3.04/1.80 divide_divide_nat: nat > nat > nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Rings_Odivide__class_Odivide_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 divide1727078534y_real: poly_real > poly_real > poly_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 3.04/1.80 divide_divide_real: real > real > real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 3.04/1.80 collect_real: ( real > $o ) > set_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_member_001t__Int__Oint,type,
% 3.04/1.80 member_int: int > set_int > $o ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_member_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
% 3.04/1.80 member_poly_real: poly_real > set_poly_real > $o ).
% 3.04/1.80
% 3.04/1.80 thf(sy_c_member_001t__Real__Oreal,type,
% 3.04/1.80 member_real: real > set_real > $o ).
% 3.04/1.80
% 3.04/1.80 thf(sy_v_A____,type,
% 3.04/1.80 a: real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_v_M____,type,
% 3.04/1.80 m: real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_v_a____,type,
% 3.04/1.80 a2: int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_v_b____,type,
% 3.04/1.80 b: int ).
% 3.04/1.80
% 3.04/1.80 thf(sy_v_n____,type,
% 3.04/1.80 n: nat ).
% 3.04/1.80
% 3.04/1.80 thf(sy_v_p____,type,
% 3.04/1.80 p: poly_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_v_roots____,type,
% 3.04/1.80 roots: set_real ).
% 3.04/1.80
% 3.04/1.80 thf(sy_v_thesis,type,
% 3.04/1.80 thesis: $o ).
% 3.04/1.80
% 3.04/1.80 thf(sy_v_x,type,
% 3.04/1.80 x: real ).
% 3.04/1.80
% 3.04/1.80 thf(fact_0_assms_I2_J,axiom,
% 3.04/1.80 algebraic_real @ x ).
% 3.04/1.80
% 3.04/1.80 thf(fact_1_p_I2_J,axiom,
% 3.04/1.80 p != zero_zero_poly_real ).
% 3.04/1.80
% 3.04/1.80 thf(fact_2_p_I1_J,axiom,
% 3.04/1.80 ! [I: nat] : ( member_real @ ( coeff_real @ p @ I ) @ ring_1_Ints_real ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_3_b,axiom,
% 3.04/1.80 ord_less_int @ zero_zero_int @ b ).
% 3.04/1.80
% 3.04/1.80 thf(fact_4_p_I3_J,axiom,
% 3.04/1.80 ( ( poly_real2 @ p @ x )
% 3.04/1.80 = zero_zero_real ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_5_no__root,axiom,
% 3.04/1.80 ( ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) )
% 3.04/1.80 != zero_zero_real ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_6_n__def,axiom,
% 3.04/1.80 ( n
% 3.04/1.80 = ( degree_real @ p ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_7_of__int__le__of__int__power__cancel__iff,axiom,
% 3.04/1.80 ! [B: int,W: nat,X: int] :
% 3.04/1.80 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 3.04/1.80 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_8_of__int__le__of__int__power__cancel__iff,axiom,
% 3.04/1.80 ! [B: int,W: nat,X: int] :
% 3.04/1.80 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 3.04/1.80 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_9_of__int__power__le__of__int__cancel__iff,axiom,
% 3.04/1.80 ! [X: int,B: int,W: nat] :
% 3.04/1.80 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 3.04/1.80 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_10_of__int__power__le__of__int__cancel__iff,axiom,
% 3.04/1.80 ! [X: int,B: int,W: nat] :
% 3.04/1.80 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 3.04/1.80 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_11_of__int__1__le__iff,axiom,
% 3.04/1.80 ! [Z: int] :
% 3.04/1.80 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 3.04/1.80 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_12_of__int__1__le__iff,axiom,
% 3.04/1.80 ! [Z: int] :
% 3.04/1.80 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 3.04/1.80 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_13_of__int__le__1__iff,axiom,
% 3.04/1.80 ! [Z: int] :
% 3.04/1.80 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 3.04/1.80 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_14_of__int__le__1__iff,axiom,
% 3.04/1.80 ! [Z: int] :
% 3.04/1.80 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 3.04/1.80 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_15_poly__power,axiom,
% 3.04/1.80 ! [P: poly_real,N: nat,X: real] :
% 3.04/1.80 ( ( poly_real2 @ ( power_2108872382y_real @ P @ N ) @ X )
% 3.04/1.80 = ( power_power_real @ ( poly_real2 @ P @ X ) @ N ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_16_poly__power,axiom,
% 3.04/1.80 ! [P: poly_nat,N: nat,X: nat] :
% 3.04/1.80 ( ( poly_nat2 @ ( power_power_poly_nat @ P @ N ) @ X )
% 3.04/1.80 = ( power_power_nat @ ( poly_nat2 @ P @ X ) @ N ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_17_poly__power,axiom,
% 3.04/1.80 ! [P: poly_int,N: nat,X: int] :
% 3.04/1.80 ( ( poly_int2 @ ( power_power_poly_int @ P @ N ) @ X )
% 3.04/1.80 = ( power_power_int @ ( poly_int2 @ P @ X ) @ N ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_18_poly__1,axiom,
% 3.04/1.80 ! [X: real] :
% 3.04/1.80 ( ( poly_real2 @ one_one_poly_real @ X )
% 3.04/1.80 = one_one_real ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_19_poly__1,axiom,
% 3.04/1.80 ! [X: int] :
% 3.04/1.80 ( ( poly_int2 @ one_one_poly_int @ X )
% 3.04/1.80 = one_one_int ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_20_poly__1,axiom,
% 3.04/1.80 ! [X: nat] :
% 3.04/1.80 ( ( poly_nat2 @ one_one_poly_nat @ X )
% 3.04/1.80 = one_one_nat ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_21_of__int__abs,axiom,
% 3.04/1.80 ! [X: int] :
% 3.04/1.80 ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
% 3.04/1.80 = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_22_of__int__abs,axiom,
% 3.04/1.80 ! [X: int] :
% 3.04/1.80 ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
% 3.04/1.80 = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_23_of__int__power,axiom,
% 3.04/1.80 ! [Z: int,N: nat] :
% 3.04/1.80 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 3.04/1.80 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_24_of__int__power,axiom,
% 3.04/1.80 ! [Z: int,N: nat] :
% 3.04/1.80 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 3.04/1.80 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_25_of__int__eq__of__int__power__cancel__iff,axiom,
% 3.04/1.80 ! [B: int,W: nat,X: int] :
% 3.04/1.80 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 3.04/1.80 = ( ring_1_of_int_real @ X ) )
% 3.04/1.80 = ( ( power_power_int @ B @ W )
% 3.04/1.80 = X ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_26_of__int__eq__of__int__power__cancel__iff,axiom,
% 3.04/1.80 ! [B: int,W: nat,X: int] :
% 3.04/1.80 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 3.04/1.80 = ( ring_1_of_int_int @ X ) )
% 3.04/1.80 = ( ( power_power_int @ B @ W )
% 3.04/1.80 = X ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_27_of__int__power__eq__of__int__cancel__iff,axiom,
% 3.04/1.80 ! [X: int,B: int,W: nat] :
% 3.04/1.80 ( ( ( ring_1_of_int_real @ X )
% 3.04/1.80 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 3.04/1.80 = ( X
% 3.04/1.80 = ( power_power_int @ B @ W ) ) ) ).
% 3.04/1.80
% 3.04/1.80 thf(fact_28_of__int__power__eq__of__int__cancel__iff,axiom,
% 3.19/1.80 ! [X: int,B: int,W: nat] :
% 3.19/1.80 ( ( ( ring_1_of_int_int @ X )
% 3.19/1.80 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 3.19/1.80 = ( X
% 3.19/1.80 = ( power_power_int @ B @ W ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_29_of__int__1,axiom,
% 3.19/1.80 ( ( ring_1_of_int_real @ one_one_int )
% 3.19/1.80 = one_one_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_30_of__int__1,axiom,
% 3.19/1.80 ( ( ring_1_of_int_int @ one_one_int )
% 3.19/1.80 = one_one_int ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_31_of__int__eq__1__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ( ring_1_of_int_real @ Z )
% 3.19/1.80 = one_one_real )
% 3.19/1.80 = ( Z = one_one_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_32_of__int__eq__1__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ( ring_1_of_int_int @ Z )
% 3.19/1.80 = one_one_int )
% 3.19/1.80 = ( Z = one_one_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_33_irrationsl,axiom,
% 3.19/1.80 ~ ( member_real @ x @ field_1537545994s_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_34_of__int__eq__iff,axiom,
% 3.19/1.80 ! [W: int,Z: int] :
% 3.19/1.80 ( ( ( ring_1_of_int_real @ W )
% 3.19/1.80 = ( ring_1_of_int_real @ Z ) )
% 3.19/1.80 = ( W = Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_35_of__int__eq__iff,axiom,
% 3.19/1.80 ! [W: int,Z: int] :
% 3.19/1.80 ( ( ( ring_1_of_int_int @ W )
% 3.19/1.80 = ( ring_1_of_int_int @ Z ) )
% 3.19/1.80 = ( W = Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_36__092_060open_062_092_060And_062thesisa_O_A_I_092_060And_062p_O_A_092_060lbrakk_062_092_060And_062i_O_Acoeff_Ap_Ai_A_092_060in_062_A_092_060int_062_059_Ap_A_092_060noteq_062_A0_059_Apoly_Ap_Ax_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesisa_J_A_092_060Longrightarrow_062_Athesisa_092_060close_062,axiom,
% 3.19/1.80 ~ ! [P2: poly_real] :
% 3.19/1.80 ( ! [I2: nat] : ( member_real @ ( coeff_real @ P2 @ I2 ) @ ring_1_Ints_real )
% 3.19/1.80 => ( ( P2 != zero_zero_poly_real )
% 3.19/1.80 => ( ( poly_real2 @ P2 @ x )
% 3.19/1.80 != zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_37__092_060open_062x_A_092_060notin_062_Aroots_092_060close_062,axiom,
% 3.19/1.80 ~ ( member_real @ x @ roots ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_38_of__int__eq__0__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ( ring_11511526659y_real @ Z )
% 3.19/1.80 = zero_zero_poly_real )
% 3.19/1.80 = ( Z = zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_39_of__int__eq__0__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ( ring_1_of_int_real @ Z )
% 3.19/1.80 = zero_zero_real )
% 3.19/1.80 = ( Z = zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_40_of__int__eq__0__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ( ring_1_of_int_int @ Z )
% 3.19/1.80 = zero_zero_int )
% 3.19/1.80 = ( Z = zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_41_of__int__0__eq__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( zero_zero_poly_real
% 3.19/1.80 = ( ring_11511526659y_real @ Z ) )
% 3.19/1.80 = ( Z = zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_42_of__int__0__eq__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( zero_zero_real
% 3.19/1.80 = ( ring_1_of_int_real @ Z ) )
% 3.19/1.80 = ( Z = zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_43_of__int__0__eq__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( zero_zero_int
% 3.19/1.80 = ( ring_1_of_int_int @ Z ) )
% 3.19/1.80 = ( Z = zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_44_of__int__0,axiom,
% 3.19/1.80 ( ( ring_11511526659y_real @ zero_zero_int )
% 3.19/1.80 = zero_zero_poly_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_45_of__int__0,axiom,
% 3.19/1.80 ( ( ring_1_of_int_real @ zero_zero_int )
% 3.19/1.80 = zero_zero_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_46_of__int__0,axiom,
% 3.19/1.80 ( ( ring_1_of_int_int @ zero_zero_int )
% 3.19/1.80 = zero_zero_int ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_47_of__int__le__iff,axiom,
% 3.19/1.80 ! [W: int,Z: int] :
% 3.19/1.80 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 3.19/1.80 = ( ord_less_eq_int @ W @ Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_48_of__int__le__iff,axiom,
% 3.19/1.80 ! [W: int,Z: int] :
% 3.19/1.80 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 3.19/1.80 = ( ord_less_eq_int @ W @ Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_49_of__int__less__iff,axiom,
% 3.19/1.80 ! [W: int,Z: int] :
% 3.19/1.80 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 3.19/1.80 = ( ord_less_int @ W @ Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_50_of__int__less__iff,axiom,
% 3.19/1.80 ! [W: int,Z: int] :
% 3.19/1.80 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 3.19/1.80 = ( ord_less_int @ W @ Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_51_coeff__0,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( coeff_poly_real @ zero_z1423781445y_real @ N )
% 3.19/1.80 = zero_zero_poly_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_52_coeff__0,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( coeff_int @ zero_zero_poly_int @ N )
% 3.19/1.80 = zero_zero_int ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_53_coeff__0,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( coeff_nat @ zero_zero_poly_nat @ N )
% 3.19/1.80 = zero_zero_nat ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_54_coeff__0,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( coeff_real @ zero_zero_poly_real @ N )
% 3.19/1.80 = zero_zero_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_55_poly__0,axiom,
% 3.19/1.80 ! [X: poly_real] :
% 3.19/1.80 ( ( poly_poly_real2 @ zero_z1423781445y_real @ X )
% 3.19/1.80 = zero_zero_poly_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_56_poly__0,axiom,
% 3.19/1.80 ! [X: int] :
% 3.19/1.80 ( ( poly_int2 @ zero_zero_poly_int @ X )
% 3.19/1.80 = zero_zero_int ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_57_poly__0,axiom,
% 3.19/1.80 ! [X: nat] :
% 3.19/1.80 ( ( poly_nat2 @ zero_zero_poly_nat @ X )
% 3.19/1.80 = zero_zero_nat ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_58_poly__0,axiom,
% 3.19/1.80 ! [X: real] :
% 3.19/1.80 ( ( poly_real2 @ zero_zero_poly_real @ X )
% 3.19/1.80 = zero_zero_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_59_lead__coeff__of__int,axiom,
% 3.19/1.80 ! [K: int] :
% 3.19/1.80 ( ( coeff_real @ ( ring_11511526659y_real @ K ) @ ( degree_real @ ( ring_11511526659y_real @ K ) ) )
% 3.19/1.80 = ( ring_1_of_int_real @ K ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_60_lead__coeff__of__int,axiom,
% 3.19/1.80 ! [K: int] :
% 3.19/1.80 ( ( coeff_int @ ( ring_12102921859ly_int @ K ) @ ( degree_int @ ( ring_12102921859ly_int @ K ) ) )
% 3.19/1.80 = ( ring_1_of_int_int @ K ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_61_leading__coeff__0__iff,axiom,
% 3.19/1.80 ! [P: poly_poly_real] :
% 3.19/1.80 ( ( ( coeff_poly_real @ P @ ( degree_poly_real @ P ) )
% 3.19/1.80 = zero_zero_poly_real )
% 3.19/1.80 = ( P = zero_z1423781445y_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_62_leading__coeff__0__iff,axiom,
% 3.19/1.80 ! [P: poly_int] :
% 3.19/1.80 ( ( ( coeff_int @ P @ ( degree_int @ P ) )
% 3.19/1.80 = zero_zero_int )
% 3.19/1.80 = ( P = zero_zero_poly_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_63_leading__coeff__0__iff,axiom,
% 3.19/1.80 ! [P: poly_real] :
% 3.19/1.80 ( ( ( coeff_real @ P @ ( degree_real @ P ) )
% 3.19/1.80 = zero_zero_real )
% 3.19/1.80 = ( P = zero_zero_poly_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_64_leading__coeff__0__iff,axiom,
% 3.19/1.80 ! [P: poly_nat] :
% 3.19/1.80 ( ( ( coeff_nat @ P @ ( degree_nat @ P ) )
% 3.19/1.80 = zero_zero_nat )
% 3.19/1.80 = ( P = zero_zero_poly_nat ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_65_zabs__less__one__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 3.19/1.80 = ( Z = zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_66_lead__coeff__1,axiom,
% 3.19/1.80 ( ( coeff_int @ one_one_poly_int @ ( degree_int @ one_one_poly_int ) )
% 3.19/1.80 = one_one_int ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_67_lead__coeff__1,axiom,
% 3.19/1.80 ( ( coeff_nat @ one_one_poly_nat @ ( degree_nat @ one_one_poly_nat ) )
% 3.19/1.80 = one_one_nat ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_68_lead__coeff__1,axiom,
% 3.19/1.80 ( ( coeff_real @ one_one_poly_real @ ( degree_real @ one_one_poly_real ) )
% 3.19/1.80 = one_one_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_69_of__int__le__0__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_le1180086932y_real @ ( ring_11511526659y_real @ Z ) @ zero_zero_poly_real )
% 3.19/1.80 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_70_of__int__le__0__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 3.19/1.80 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_71_of__int__le__0__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 3.19/1.80 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_72_of__int__0__le__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_le1180086932y_real @ zero_zero_poly_real @ ( ring_11511526659y_real @ Z ) )
% 3.19/1.80 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_73_of__int__0__le__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 3.19/1.80 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_74_of__int__0__le__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 3.19/1.80 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_75_of__int__less__0__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_poly_real @ ( ring_11511526659y_real @ Z ) @ zero_zero_poly_real )
% 3.19/1.80 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_76_of__int__less__0__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 3.19/1.80 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_77_of__int__less__0__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 3.19/1.80 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_78_of__int__0__less__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_poly_real @ zero_zero_poly_real @ ( ring_11511526659y_real @ Z ) )
% 3.19/1.80 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_79_of__int__0__less__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 3.19/1.80 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_80_of__int__0__less__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 3.19/1.80 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_81_of__int__less__1__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 3.19/1.80 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_82_of__int__less__1__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 3.19/1.80 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_83_of__int__1__less__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 3.19/1.80 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_84_of__int__1__less__iff,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 3.19/1.80 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_85_of__int__power__less__of__int__cancel__iff,axiom,
% 3.19/1.80 ! [X: int,B: int,W: nat] :
% 3.19/1.80 ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 3.19/1.80 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_86_of__int__power__less__of__int__cancel__iff,axiom,
% 3.19/1.80 ! [X: int,B: int,W: nat] :
% 3.19/1.80 ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 3.19/1.80 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_87_of__int__less__of__int__power__cancel__iff,axiom,
% 3.19/1.80 ! [B: int,W: nat,X: int] :
% 3.19/1.80 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 3.19/1.80 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_88_of__int__less__of__int__power__cancel__iff,axiom,
% 3.19/1.80 ! [B: int,W: nat,X: int] :
% 3.19/1.80 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 3.19/1.80 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_89_less__int__code_I1_J,axiom,
% 3.19/1.80 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_90_less__eq__int__code_I1_J,axiom,
% 3.19/1.80 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 3.19/1.80
% 3.19/1.80 thf(fact_91_poly__eqI,axiom,
% 3.19/1.80 ! [P: poly_real,Q: poly_real] :
% 3.19/1.80 ( ! [N2: nat] :
% 3.19/1.80 ( ( coeff_real @ P @ N2 )
% 3.19/1.80 = ( coeff_real @ Q @ N2 ) )
% 3.19/1.80 => ( P = Q ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_92_mem__Collect__eq,axiom,
% 3.19/1.80 ! [A: real,P3: real > $o] :
% 3.19/1.80 ( ( member_real @ A @ ( collect_real @ P3 ) )
% 3.19/1.80 = ( P3 @ A ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_93_Collect__mem__eq,axiom,
% 3.19/1.80 ! [A2: set_real] :
% 3.19/1.80 ( ( collect_real
% 3.19/1.80 @ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
% 3.19/1.80 = A2 ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_94_poly__eq__iff,axiom,
% 3.19/1.80 ( ( ^ [Y: poly_real,Z2: poly_real] : ( Y = Z2 ) )
% 3.19/1.80 = ( ^ [P4: poly_real,Q2: poly_real] :
% 3.19/1.80 ! [N3: nat] :
% 3.19/1.80 ( ( coeff_real @ P4 @ N3 )
% 3.19/1.80 = ( coeff_real @ Q2 @ N3 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_95_Ints__0,axiom,
% 3.19/1.80 member_poly_real @ zero_zero_poly_real @ ring_1690226883y_real ).
% 3.19/1.80
% 3.19/1.80 thf(fact_96_Ints__0,axiom,
% 3.19/1.80 member_int @ zero_zero_int @ ring_1_Ints_int ).
% 3.19/1.80
% 3.19/1.80 thf(fact_97_Ints__0,axiom,
% 3.19/1.80 member_real @ zero_zero_real @ ring_1_Ints_real ).
% 3.19/1.80
% 3.19/1.80 thf(fact_98_zero__poly_Orep__eq,axiom,
% 3.19/1.80 ( ( coeff_poly_real @ zero_z1423781445y_real )
% 3.19/1.80 = ( ^ [Uu: nat] : zero_zero_poly_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_99_zero__poly_Orep__eq,axiom,
% 3.19/1.80 ( ( coeff_int @ zero_zero_poly_int )
% 3.19/1.80 = ( ^ [Uu: nat] : zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_100_zero__poly_Orep__eq,axiom,
% 3.19/1.80 ( ( coeff_nat @ zero_zero_poly_nat )
% 3.19/1.80 = ( ^ [Uu: nat] : zero_zero_nat ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_101_zero__poly_Orep__eq,axiom,
% 3.19/1.80 ( ( coeff_real @ zero_zero_poly_real )
% 3.19/1.80 = ( ^ [Uu: nat] : zero_zero_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_102_int__one__le__iff__zero__less,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_eq_int @ one_one_int @ Z )
% 3.19/1.80 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_103_coeff__inject,axiom,
% 3.19/1.80 ! [X: poly_real,Y2: poly_real] :
% 3.19/1.80 ( ( ( coeff_real @ X )
% 3.19/1.80 = ( coeff_real @ Y2 ) )
% 3.19/1.80 = ( X = Y2 ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_104_le__degree,axiom,
% 3.19/1.80 ! [P: poly_poly_real,N: nat] :
% 3.19/1.80 ( ( ( coeff_poly_real @ P @ N )
% 3.19/1.80 != zero_zero_poly_real )
% 3.19/1.80 => ( ord_less_eq_nat @ N @ ( degree_poly_real @ P ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_105_le__degree,axiom,
% 3.19/1.80 ! [P: poly_int,N: nat] :
% 3.19/1.80 ( ( ( coeff_int @ P @ N )
% 3.19/1.80 != zero_zero_int )
% 3.19/1.80 => ( ord_less_eq_nat @ N @ ( degree_int @ P ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_106_le__degree,axiom,
% 3.19/1.80 ! [P: poly_real,N: nat] :
% 3.19/1.80 ( ( ( coeff_real @ P @ N )
% 3.19/1.80 != zero_zero_real )
% 3.19/1.80 => ( ord_less_eq_nat @ N @ ( degree_real @ P ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_107_le__degree,axiom,
% 3.19/1.80 ! [P: poly_nat,N: nat] :
% 3.19/1.80 ( ( ( coeff_nat @ P @ N )
% 3.19/1.80 != zero_zero_nat )
% 3.19/1.80 => ( ord_less_eq_nat @ N @ ( degree_nat @ P ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_108_poly__0__coeff__0,axiom,
% 3.19/1.80 ! [P: poly_poly_real] :
% 3.19/1.80 ( ( poly_poly_real2 @ P @ zero_zero_poly_real )
% 3.19/1.80 = ( coeff_poly_real @ P @ zero_zero_nat ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_109_poly__0__coeff__0,axiom,
% 3.19/1.80 ! [P: poly_int] :
% 3.19/1.80 ( ( poly_int2 @ P @ zero_zero_int )
% 3.19/1.80 = ( coeff_int @ P @ zero_zero_nat ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_110_poly__0__coeff__0,axiom,
% 3.19/1.80 ! [P: poly_real] :
% 3.19/1.80 ( ( poly_real2 @ P @ zero_zero_real )
% 3.19/1.80 = ( coeff_real @ P @ zero_zero_nat ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_111_poly__0__coeff__0,axiom,
% 3.19/1.80 ! [P: poly_nat] :
% 3.19/1.80 ( ( poly_nat2 @ P @ zero_zero_nat )
% 3.19/1.80 = ( coeff_nat @ P @ zero_zero_nat ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_112_algebraic__def,axiom,
% 3.19/1.80 ( algebraic_real
% 3.19/1.80 = ( ^ [X2: real] :
% 3.19/1.80 ? [P4: poly_real] :
% 3.19/1.80 ( ! [I3: nat] : ( member_real @ ( coeff_real @ P4 @ I3 ) @ ring_1_Ints_real )
% 3.19/1.80 & ( P4 != zero_zero_poly_real )
% 3.19/1.80 & ( ( poly_real2 @ P4 @ X2 )
% 3.19/1.80 = zero_zero_real ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_113_algebraicI,axiom,
% 3.19/1.80 ! [P: poly_real,X: real] :
% 3.19/1.80 ( ! [I4: nat] : ( member_real @ ( coeff_real @ P @ I4 ) @ ring_1_Ints_real )
% 3.19/1.80 => ( ( P != zero_zero_poly_real )
% 3.19/1.80 => ( ( ( poly_real2 @ P @ X )
% 3.19/1.80 = zero_zero_real )
% 3.19/1.80 => ( algebraic_real @ X ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_114_algebraicE,axiom,
% 3.19/1.80 ! [X: real] :
% 3.19/1.80 ( ( algebraic_real @ X )
% 3.19/1.80 => ~ ! [P2: poly_real] :
% 3.19/1.80 ( ! [I2: nat] : ( member_real @ ( coeff_real @ P2 @ I2 ) @ ring_1_Ints_real )
% 3.19/1.80 => ( ( P2 != zero_zero_poly_real )
% 3.19/1.80 => ( ( poly_real2 @ P2 @ X )
% 3.19/1.80 != zero_zero_real ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_115_of__int__pos,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_int @ zero_zero_int @ Z )
% 3.19/1.80 => ( ord_less_poly_real @ zero_zero_poly_real @ ( ring_11511526659y_real @ Z ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_116_of__int__pos,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_int @ zero_zero_int @ Z )
% 3.19/1.80 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_117_of__int__pos,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_int @ zero_zero_int @ Z )
% 3.19/1.80 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_118_leading__coeff__neq__0,axiom,
% 3.19/1.80 ! [P: poly_poly_real] :
% 3.19/1.80 ( ( P != zero_z1423781445y_real )
% 3.19/1.80 => ( ( coeff_poly_real @ P @ ( degree_poly_real @ P ) )
% 3.19/1.80 != zero_zero_poly_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_119_leading__coeff__neq__0,axiom,
% 3.19/1.80 ! [P: poly_int] :
% 3.19/1.80 ( ( P != zero_zero_poly_int )
% 3.19/1.80 => ( ( coeff_int @ P @ ( degree_int @ P ) )
% 3.19/1.80 != zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_120_leading__coeff__neq__0,axiom,
% 3.19/1.80 ! [P: poly_nat] :
% 3.19/1.80 ( ( P != zero_zero_poly_nat )
% 3.19/1.80 => ( ( coeff_nat @ P @ ( degree_nat @ P ) )
% 3.19/1.80 != zero_zero_nat ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_121_leading__coeff__neq__0,axiom,
% 3.19/1.80 ! [P: poly_real] :
% 3.19/1.80 ( ( P != zero_zero_poly_real )
% 3.19/1.80 => ( ( coeff_real @ P @ ( degree_real @ P ) )
% 3.19/1.80 != zero_zero_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_122_poly__all__0__iff__0,axiom,
% 3.19/1.80 ! [P: poly_poly_real] :
% 3.19/1.80 ( ( ! [X2: poly_real] :
% 3.19/1.80 ( ( poly_poly_real2 @ P @ X2 )
% 3.19/1.80 = zero_zero_poly_real ) )
% 3.19/1.80 = ( P = zero_z1423781445y_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_123_poly__all__0__iff__0,axiom,
% 3.19/1.80 ! [P: poly_int] :
% 3.19/1.80 ( ( ! [X2: int] :
% 3.19/1.80 ( ( poly_int2 @ P @ X2 )
% 3.19/1.80 = zero_zero_int ) )
% 3.19/1.80 = ( P = zero_zero_poly_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_124_poly__all__0__iff__0,axiom,
% 3.19/1.80 ! [P: poly_real] :
% 3.19/1.80 ( ( ! [X2: real] :
% 3.19/1.80 ( ( poly_real2 @ P @ X2 )
% 3.19/1.80 = zero_zero_real ) )
% 3.19/1.80 = ( P = zero_zero_poly_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_125_Ints__1,axiom,
% 3.19/1.80 member_int @ one_one_int @ ring_1_Ints_int ).
% 3.19/1.80
% 3.19/1.80 thf(fact_126_Ints__1,axiom,
% 3.19/1.80 member_real @ one_one_real @ ring_1_Ints_real ).
% 3.19/1.80
% 3.19/1.80 thf(fact_127_Ints__power,axiom,
% 3.19/1.80 ! [A: real,N: nat] :
% 3.19/1.80 ( ( member_real @ A @ ring_1_Ints_real )
% 3.19/1.80 => ( member_real @ ( power_power_real @ A @ N ) @ ring_1_Ints_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_128_Ints__power,axiom,
% 3.19/1.80 ! [A: int,N: nat] :
% 3.19/1.80 ( ( member_int @ A @ ring_1_Ints_int )
% 3.19/1.80 => ( member_int @ ( power_power_int @ A @ N ) @ ring_1_Ints_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_129_Ints__abs,axiom,
% 3.19/1.80 ! [A: int] :
% 3.19/1.80 ( ( member_int @ A @ ring_1_Ints_int )
% 3.19/1.80 => ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_130_Ints__abs,axiom,
% 3.19/1.80 ! [A: real] :
% 3.19/1.80 ( ( member_real @ A @ ring_1_Ints_real )
% 3.19/1.80 => ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_131_Ints__of__int,axiom,
% 3.19/1.80 ! [Z: int] : ( member_real @ ( ring_1_of_int_real @ Z ) @ ring_1_Ints_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_132_Ints__of__int,axiom,
% 3.19/1.80 ! [Z: int] : ( member_int @ ( ring_1_of_int_int @ Z ) @ ring_1_Ints_int ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_133_Ints__induct,axiom,
% 3.19/1.80 ! [Q: real,P3: real > $o] :
% 3.19/1.80 ( ( member_real @ Q @ ring_1_Ints_real )
% 3.19/1.80 => ( ! [Z3: int] : ( P3 @ ( ring_1_of_int_real @ Z3 ) )
% 3.19/1.80 => ( P3 @ Q ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_134_Ints__induct,axiom,
% 3.19/1.80 ! [Q: int,P3: int > $o] :
% 3.19/1.80 ( ( member_int @ Q @ ring_1_Ints_int )
% 3.19/1.80 => ( ! [Z3: int] : ( P3 @ ( ring_1_of_int_int @ Z3 ) )
% 3.19/1.80 => ( P3 @ Q ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_135_Ints__cases,axiom,
% 3.19/1.80 ! [Q: real] :
% 3.19/1.80 ( ( member_real @ Q @ ring_1_Ints_real )
% 3.19/1.80 => ~ ! [Z3: int] :
% 3.19/1.80 ( Q
% 3.19/1.80 != ( ring_1_of_int_real @ Z3 ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_136_Ints__cases,axiom,
% 3.19/1.80 ! [Q: int] :
% 3.19/1.80 ( ( member_int @ Q @ ring_1_Ints_int )
% 3.19/1.80 => ~ ! [Z3: int] :
% 3.19/1.80 ( Q
% 3.19/1.80 != ( ring_1_of_int_int @ Z3 ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_137_Ints__nonzero__abs__less1,axiom,
% 3.19/1.80 ! [X: poly_real] :
% 3.19/1.80 ( ( member_poly_real @ X @ ring_1690226883y_real )
% 3.19/1.80 => ( ( ord_less_poly_real @ ( abs_abs_poly_real @ X ) @ one_one_poly_real )
% 3.19/1.80 => ( X = zero_zero_poly_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_138_Ints__nonzero__abs__less1,axiom,
% 3.19/1.80 ! [X: int] :
% 3.19/1.80 ( ( member_int @ X @ ring_1_Ints_int )
% 3.19/1.80 => ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
% 3.19/1.80 => ( X = zero_zero_int ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_139_Ints__nonzero__abs__less1,axiom,
% 3.19/1.80 ! [X: real] :
% 3.19/1.80 ( ( member_real @ X @ ring_1_Ints_real )
% 3.19/1.80 => ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 3.19/1.80 => ( X = zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_140_of__int__nonneg,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 3.19/1.80 => ( ord_le1180086932y_real @ zero_zero_poly_real @ ( ring_11511526659y_real @ Z ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_141_of__int__nonneg,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 3.19/1.80 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_142_of__int__nonneg,axiom,
% 3.19/1.80 ! [Z: int] :
% 3.19/1.80 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 3.19/1.80 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_143_lead__coeff__power,axiom,
% 3.19/1.80 ! [P: poly_real,N: nat] :
% 3.19/1.80 ( ( coeff_real @ ( power_2108872382y_real @ P @ N ) @ ( degree_real @ ( power_2108872382y_real @ P @ N ) ) )
% 3.19/1.80 = ( power_power_real @ ( coeff_real @ P @ ( degree_real @ P ) ) @ N ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_144_lead__coeff__power,axiom,
% 3.19/1.80 ! [P: poly_nat,N: nat] :
% 3.19/1.80 ( ( coeff_nat @ ( power_power_poly_nat @ P @ N ) @ ( degree_nat @ ( power_power_poly_nat @ P @ N ) ) )
% 3.19/1.80 = ( power_power_nat @ ( coeff_nat @ P @ ( degree_nat @ P ) ) @ N ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_145_lead__coeff__power,axiom,
% 3.19/1.80 ! [P: poly_int,N: nat] :
% 3.19/1.80 ( ( coeff_int @ ( power_power_poly_int @ P @ N ) @ ( degree_int @ ( power_power_poly_int @ P @ N ) ) )
% 3.19/1.80 = ( power_power_int @ ( coeff_int @ P @ ( degree_int @ P ) ) @ N ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_146_Ints__nonzero__abs__ge1,axiom,
% 3.19/1.80 ! [X: poly_real] :
% 3.19/1.80 ( ( member_poly_real @ X @ ring_1690226883y_real )
% 3.19/1.80 => ( ( X != zero_zero_poly_real )
% 3.19/1.80 => ( ord_le1180086932y_real @ one_one_poly_real @ ( abs_abs_poly_real @ X ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_147_Ints__nonzero__abs__ge1,axiom,
% 3.19/1.80 ! [X: real] :
% 3.19/1.80 ( ( member_real @ X @ ring_1_Ints_real )
% 3.19/1.80 => ( ( X != zero_zero_real )
% 3.19/1.80 => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_148_Ints__nonzero__abs__ge1,axiom,
% 3.19/1.80 ! [X: int] :
% 3.19/1.80 ( ( member_int @ X @ ring_1_Ints_int )
% 3.19/1.80 => ( ( X != zero_zero_int )
% 3.19/1.80 => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_149_poly__eq__poly__eq__iff,axiom,
% 3.19/1.80 ! [P: poly_real,Q: poly_real] :
% 3.19/1.80 ( ( ( poly_real2 @ P )
% 3.19/1.80 = ( poly_real2 @ Q ) )
% 3.19/1.80 = ( P = Q ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_150_of__int__lessD,axiom,
% 3.19/1.80 ! [N: int,X: int] :
% 3.19/1.80 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 3.19/1.80 => ( ( N = zero_zero_int )
% 3.19/1.80 | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_151_of__int__lessD,axiom,
% 3.19/1.80 ! [N: int,X: real] :
% 3.19/1.80 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 3.19/1.80 => ( ( N = zero_zero_int )
% 3.19/1.80 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_152_of__int__leD,axiom,
% 3.19/1.80 ! [N: int,X: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 3.19/1.80 => ( ( N = zero_zero_int )
% 3.19/1.80 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_153_of__int__leD,axiom,
% 3.19/1.80 ! [N: int,X: int] :
% 3.19/1.80 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 3.19/1.80 => ( ( N = zero_zero_int )
% 3.19/1.80 | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_154_int__poly__rat__no__root__ge,axiom,
% 3.19/1.80 ! [P: poly_real,B: int,A: int] :
% 3.19/1.80 ( ! [N2: nat] : ( member_real @ ( coeff_real @ P @ N2 ) @ ring_1_Ints_real )
% 3.19/1.80 => ( ( ord_less_int @ zero_zero_int @ B )
% 3.19/1.80 => ( ( ( poly_real2 @ P @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
% 3.19/1.80 != zero_zero_real )
% 3.19/1.80 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ ( degree_real @ P ) ) ) @ ( abs_abs_real @ ( poly_real2 @ P @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) ) ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_155_power__decreasing__iff,axiom,
% 3.19/1.80 ! [B: poly_real,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_poly_real @ zero_zero_poly_real @ B )
% 3.19/1.80 => ( ( ord_less_poly_real @ B @ one_one_poly_real )
% 3.19/1.80 => ( ( ord_le1180086932y_real @ ( power_2108872382y_real @ B @ M ) @ ( power_2108872382y_real @ B @ N ) )
% 3.19/1.80 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_156_power__decreasing__iff,axiom,
% 3.19/1.80 ! [B: real,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ B )
% 3.19/1.80 => ( ( ord_less_real @ B @ one_one_real )
% 3.19/1.80 => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 3.19/1.80 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_157_power__decreasing__iff,axiom,
% 3.19/1.80 ! [B: int,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_int @ zero_zero_int @ B )
% 3.19/1.80 => ( ( ord_less_int @ B @ one_one_int )
% 3.19/1.80 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 3.19/1.80 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_158_power__decreasing__iff,axiom,
% 3.19/1.80 ! [B: nat,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ B )
% 3.19/1.80 => ( ( ord_less_nat @ B @ one_one_nat )
% 3.19/1.80 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 3.19/1.80 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_159_le__divide__eq__1__pos,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ A )
% 3.19/1.80 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 3.19/1.80 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_160_le__divide__eq__1__neg,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_real @ A @ zero_zero_real )
% 3.19/1.80 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 3.19/1.80 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_161_divide__le__eq__1__pos,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ A )
% 3.19/1.80 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 3.19/1.80 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_162_divide__le__eq__1__neg,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_real @ A @ zero_zero_real )
% 3.19/1.80 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 3.19/1.80 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_163__092_060open_062A_A_P_Areal__of__int_Ab_A_094_An_A_092_060le_062_AA_A_P_A1_092_060close_062,axiom,
% 3.19/1.80 ord_less_eq_real @ ( divide_divide_real @ a @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ n ) ) @ ( divide_divide_real @ a @ one_one_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_164_zero__less__power__abs__iff,axiom,
% 3.19/1.80 ! [A: poly_real,N: nat] :
% 3.19/1.80 ( ( ord_less_poly_real @ zero_zero_poly_real @ ( power_2108872382y_real @ ( abs_abs_poly_real @ A ) @ N ) )
% 3.19/1.80 = ( ( A != zero_zero_poly_real )
% 3.19/1.80 | ( N = zero_zero_nat ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_165_zero__less__power__abs__iff,axiom,
% 3.19/1.80 ! [A: int,N: nat] :
% 3.19/1.80 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 3.19/1.80 = ( ( A != zero_zero_int )
% 3.19/1.80 | ( N = zero_zero_nat ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_166_zero__less__power__abs__iff,axiom,
% 3.19/1.80 ! [A: real,N: nat] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 3.19/1.80 = ( ( A != zero_zero_real )
% 3.19/1.80 | ( N = zero_zero_nat ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_167_zero__le__divide__abs__iff,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 3.19/1.80 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 3.19/1.80 | ( B = zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_168_divide__le__0__abs__iff,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 3.19/1.80 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 3.19/1.80 | ( B = zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_169_A__pos,axiom,
% 3.19/1.80 ord_less_real @ zero_zero_real @ a ).
% 3.19/1.80
% 3.19/1.80 thf(fact_170_A__less_I1_J,axiom,
% 3.19/1.80 ord_less_real @ a @ one_one_real ).
% 3.19/1.80
% 3.19/1.80 thf(fact_171__092_060open_062A_A_092_060le_062_A1_092_060close_062,axiom,
% 3.19/1.80 ord_less_eq_real @ a @ one_one_real ).
% 3.19/1.80
% 3.19/1.80 thf(fact_172_power__one__right,axiom,
% 3.19/1.80 ! [A: real] :
% 3.19/1.80 ( ( power_power_real @ A @ one_one_nat )
% 3.19/1.80 = A ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_173_power__one__right,axiom,
% 3.19/1.80 ! [A: nat] :
% 3.19/1.80 ( ( power_power_nat @ A @ one_one_nat )
% 3.19/1.80 = A ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_174_power__one__right,axiom,
% 3.19/1.80 ! [A: int] :
% 3.19/1.80 ( ( power_power_int @ A @ one_one_nat )
% 3.19/1.80 = A ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_175_nat__zero__less__power__iff,axiom,
% 3.19/1.80 ! [X: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 3.19/1.80 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 3.19/1.80 | ( N = zero_zero_nat ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_176_divide__eq__0__iff,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ( divide_divide_real @ A @ B )
% 3.19/1.80 = zero_zero_real )
% 3.19/1.80 = ( ( A = zero_zero_real )
% 3.19/1.80 | ( B = zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_177_divide__cancel__left,axiom,
% 3.19/1.80 ! [C: real,A: real,B: real] :
% 3.19/1.80 ( ( ( divide_divide_real @ C @ A )
% 3.19/1.80 = ( divide_divide_real @ C @ B ) )
% 3.19/1.80 = ( ( C = zero_zero_real )
% 3.19/1.80 | ( A = B ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_178_divide__cancel__right,axiom,
% 3.19/1.80 ! [A: real,C: real,B: real] :
% 3.19/1.80 ( ( ( divide_divide_real @ A @ C )
% 3.19/1.80 = ( divide_divide_real @ B @ C ) )
% 3.19/1.80 = ( ( C = zero_zero_real )
% 3.19/1.80 | ( A = B ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_179_division__ring__divide__zero,axiom,
% 3.19/1.80 ! [A: real] :
% 3.19/1.80 ( ( divide_divide_real @ A @ zero_zero_real )
% 3.19/1.80 = zero_zero_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_180_power__one,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( power_power_real @ one_one_real @ N )
% 3.19/1.80 = one_one_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_181_power__one,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( power_power_nat @ one_one_nat @ N )
% 3.19/1.80 = one_one_nat ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_182_power__one,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( power_power_int @ one_one_int @ N )
% 3.19/1.80 = one_one_int ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_183_abs__divide,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 3.19/1.80 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_184_divide__eq__1__iff,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ( divide_divide_real @ A @ B )
% 3.19/1.80 = one_one_real )
% 3.19/1.80 = ( ( B != zero_zero_real )
% 3.19/1.80 & ( A = B ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_185_one__eq__divide__iff,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( one_one_real
% 3.19/1.80 = ( divide_divide_real @ A @ B ) )
% 3.19/1.80 = ( ( B != zero_zero_real )
% 3.19/1.80 & ( A = B ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_186_divide__self,axiom,
% 3.19/1.80 ! [A: real] :
% 3.19/1.80 ( ( A != zero_zero_real )
% 3.19/1.80 => ( ( divide_divide_real @ A @ A )
% 3.19/1.80 = one_one_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_187_divide__self__if,axiom,
% 3.19/1.80 ! [A: real] :
% 3.19/1.80 ( ( ( A = zero_zero_real )
% 3.19/1.80 => ( ( divide_divide_real @ A @ A )
% 3.19/1.80 = zero_zero_real ) )
% 3.19/1.80 & ( ( A != zero_zero_real )
% 3.19/1.80 => ( ( divide_divide_real @ A @ A )
% 3.19/1.80 = one_one_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_188_divide__eq__eq__1,axiom,
% 3.19/1.80 ! [B: real,A: real] :
% 3.19/1.80 ( ( ( divide_divide_real @ B @ A )
% 3.19/1.80 = one_one_real )
% 3.19/1.80 = ( ( A != zero_zero_real )
% 3.19/1.80 & ( A = B ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_189_eq__divide__eq__1,axiom,
% 3.19/1.80 ! [B: real,A: real] :
% 3.19/1.80 ( ( one_one_real
% 3.19/1.80 = ( divide_divide_real @ B @ A ) )
% 3.19/1.80 = ( ( A != zero_zero_real )
% 3.19/1.80 & ( A = B ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_190_one__divide__eq__0__iff,axiom,
% 3.19/1.80 ! [A: real] :
% 3.19/1.80 ( ( ( divide_divide_real @ one_one_real @ A )
% 3.19/1.80 = zero_zero_real )
% 3.19/1.80 = ( A = zero_zero_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_191_zero__eq__1__divide__iff,axiom,
% 3.19/1.80 ! [A: real] :
% 3.19/1.80 ( ( zero_zero_real
% 3.19/1.80 = ( divide_divide_real @ one_one_real @ A ) )
% 3.19/1.80 = ( A = zero_zero_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_192_power__inject__exp,axiom,
% 3.19/1.80 ! [A: int,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_int @ one_one_int @ A )
% 3.19/1.80 => ( ( ( power_power_int @ A @ M )
% 3.19/1.80 = ( power_power_int @ A @ N ) )
% 3.19/1.80 = ( M = N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_193_power__inject__exp,axiom,
% 3.19/1.80 ! [A: real,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_real @ one_one_real @ A )
% 3.19/1.80 => ( ( ( power_power_real @ A @ M )
% 3.19/1.80 = ( power_power_real @ A @ N ) )
% 3.19/1.80 = ( M = N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_194_power__inject__exp,axiom,
% 3.19/1.80 ! [A: nat,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ one_one_nat @ A )
% 3.19/1.80 => ( ( ( power_power_nat @ A @ M )
% 3.19/1.80 = ( power_power_nat @ A @ N ) )
% 3.19/1.80 = ( M = N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_195_power__strict__increasing__iff,axiom,
% 3.19/1.80 ! [B: int,X: nat,Y2: nat] :
% 3.19/1.80 ( ( ord_less_int @ one_one_int @ B )
% 3.19/1.80 => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
% 3.19/1.80 = ( ord_less_nat @ X @ Y2 ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_196_power__strict__increasing__iff,axiom,
% 3.19/1.80 ! [B: real,X: nat,Y2: nat] :
% 3.19/1.80 ( ( ord_less_real @ one_one_real @ B )
% 3.19/1.80 => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y2 ) )
% 3.19/1.80 = ( ord_less_nat @ X @ Y2 ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_197_power__strict__increasing__iff,axiom,
% 3.19/1.80 ! [B: nat,X: nat,Y2: nat] :
% 3.19/1.80 ( ( ord_less_nat @ one_one_nat @ B )
% 3.19/1.80 => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
% 3.19/1.80 = ( ord_less_nat @ X @ Y2 ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_198_power__eq__0__iff,axiom,
% 3.19/1.80 ! [A: poly_real,N: nat] :
% 3.19/1.80 ( ( ( power_2108872382y_real @ A @ N )
% 3.19/1.80 = zero_zero_poly_real )
% 3.19/1.80 = ( ( A = zero_zero_poly_real )
% 3.19/1.80 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_199_power__eq__0__iff,axiom,
% 3.19/1.80 ! [A: real,N: nat] :
% 3.19/1.80 ( ( ( power_power_real @ A @ N )
% 3.19/1.80 = zero_zero_real )
% 3.19/1.80 = ( ( A = zero_zero_real )
% 3.19/1.80 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_200_power__eq__0__iff,axiom,
% 3.19/1.80 ! [A: nat,N: nat] :
% 3.19/1.80 ( ( ( power_power_nat @ A @ N )
% 3.19/1.80 = zero_zero_nat )
% 3.19/1.80 = ( ( A = zero_zero_nat )
% 3.19/1.80 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_201_power__eq__0__iff,axiom,
% 3.19/1.80 ! [A: int,N: nat] :
% 3.19/1.80 ( ( ( power_power_int @ A @ N )
% 3.19/1.80 = zero_zero_int )
% 3.19/1.80 = ( ( A = zero_zero_int )
% 3.19/1.80 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_202_degree__0,axiom,
% 3.19/1.80 ( ( degree_real @ zero_zero_poly_real )
% 3.19/1.80 = zero_zero_nat ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_203_degree__1,axiom,
% 3.19/1.80 ( ( degree_real @ one_one_poly_real )
% 3.19/1.80 = zero_zero_nat ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_204_degree__of__int,axiom,
% 3.19/1.80 ! [K: int] :
% 3.19/1.80 ( ( degree_real @ ( ring_11511526659y_real @ K ) )
% 3.19/1.80 = zero_zero_nat ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_205_divide__le__0__1__iff,axiom,
% 3.19/1.80 ! [A: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 3.19/1.80 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_206_zero__le__divide__1__iff,axiom,
% 3.19/1.80 ! [A: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 3.19/1.80 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_207_divide__less__0__1__iff,axiom,
% 3.19/1.80 ! [A: real] :
% 3.19/1.80 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 3.19/1.80 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_208_divide__less__eq__1__neg,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_real @ A @ zero_zero_real )
% 3.19/1.80 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 3.19/1.80 = ( ord_less_real @ A @ B ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_209_divide__less__eq__1__pos,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ A )
% 3.19/1.80 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 3.19/1.80 = ( ord_less_real @ B @ A ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_210_less__divide__eq__1__neg,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_real @ A @ zero_zero_real )
% 3.19/1.80 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 3.19/1.80 = ( ord_less_real @ B @ A ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_211_less__divide__eq__1__pos,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ A )
% 3.19/1.80 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 3.19/1.80 = ( ord_less_real @ A @ B ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_212_zero__less__divide__1__iff,axiom,
% 3.19/1.80 ! [A: real] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 3.19/1.80 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_213_power__strict__decreasing__iff,axiom,
% 3.19/1.80 ! [B: poly_real,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_poly_real @ zero_zero_poly_real @ B )
% 3.19/1.80 => ( ( ord_less_poly_real @ B @ one_one_poly_real )
% 3.19/1.80 => ( ( ord_less_poly_real @ ( power_2108872382y_real @ B @ M ) @ ( power_2108872382y_real @ B @ N ) )
% 3.19/1.80 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_214_power__strict__decreasing__iff,axiom,
% 3.19/1.80 ! [B: int,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_int @ zero_zero_int @ B )
% 3.19/1.80 => ( ( ord_less_int @ B @ one_one_int )
% 3.19/1.80 => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 3.19/1.80 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_215_power__strict__decreasing__iff,axiom,
% 3.19/1.80 ! [B: real,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ B )
% 3.19/1.80 => ( ( ord_less_real @ B @ one_one_real )
% 3.19/1.80 => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 3.19/1.80 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_216_power__strict__decreasing__iff,axiom,
% 3.19/1.80 ! [B: nat,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ B )
% 3.19/1.80 => ( ( ord_less_nat @ B @ one_one_nat )
% 3.19/1.80 => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 3.19/1.80 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_217_power__mono__iff,axiom,
% 3.19/1.80 ! [A: poly_real,B: poly_real,N: nat] :
% 3.19/1.80 ( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
% 3.19/1.80 => ( ( ord_le1180086932y_real @ zero_zero_poly_real @ B )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ( ord_le1180086932y_real @ ( power_2108872382y_real @ A @ N ) @ ( power_2108872382y_real @ B @ N ) )
% 3.19/1.80 = ( ord_le1180086932y_real @ A @ B ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_218_power__mono__iff,axiom,
% 3.19/1.80 ! [A: real,B: real,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 3.19/1.80 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 3.19/1.80 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_219_power__mono__iff,axiom,
% 3.19/1.80 ! [A: int,B: int,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 3.19/1.80 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 3.19/1.80 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_220_power__mono__iff,axiom,
% 3.19/1.80 ! [A: nat,B: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 3.19/1.80 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 3.19/1.80 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_221_power__increasing__iff,axiom,
% 3.19/1.80 ! [B: real,X: nat,Y2: nat] :
% 3.19/1.80 ( ( ord_less_real @ one_one_real @ B )
% 3.19/1.80 => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y2 ) )
% 3.19/1.80 = ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_222_power__increasing__iff,axiom,
% 3.19/1.80 ! [B: int,X: nat,Y2: nat] :
% 3.19/1.80 ( ( ord_less_int @ one_one_int @ B )
% 3.19/1.80 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
% 3.19/1.80 = ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_223_power__increasing__iff,axiom,
% 3.19/1.80 ! [B: nat,X: nat,Y2: nat] :
% 3.19/1.80 ( ( ord_less_nat @ one_one_nat @ B )
% 3.19/1.80 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
% 3.19/1.80 = ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_224__092_060open_062_092_060not_062_AA_A_P_Areal__of__int_Ab_A_094_An_A_060_A_092_060bar_062x_A_N_Areal__of__int_Aa_A_P_Areal__of__int_Ab_092_060bar_062_092_060close_062,axiom,
% 3.19/1.80 ~ ( ord_less_real @ ( divide_divide_real @ a @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ n ) ) @ ( abs_abs_real @ ( minus_minus_real @ x @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_225_nat__power__less__imp__less,axiom,
% 3.19/1.80 ! [I: nat,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ I )
% 3.19/1.80 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 3.19/1.80 => ( ord_less_nat @ M @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_226_div__poly__less,axiom,
% 3.19/1.80 ! [X: poly_real,Y2: poly_real] :
% 3.19/1.80 ( ( ord_less_nat @ ( degree_real @ X ) @ ( degree_real @ Y2 ) )
% 3.19/1.80 => ( ( divide1727078534y_real @ X @ Y2 )
% 3.19/1.80 = zero_zero_poly_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_227_self__le__power,axiom,
% 3.19/1.80 ! [A: real,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_real @ one_one_real @ A )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_228_self__le__power,axiom,
% 3.19/1.80 ! [A: int,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_int @ one_one_int @ A )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_229_self__le__power,axiom,
% 3.19/1.80 ! [A: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_230_divide__poly__0,axiom,
% 3.19/1.80 ! [F: poly_real] :
% 3.19/1.80 ( ( divide1727078534y_real @ F @ zero_zero_poly_real )
% 3.19/1.80 = zero_zero_poly_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_231_zero__power,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ( power_2108872382y_real @ zero_zero_poly_real @ N )
% 3.19/1.80 = zero_zero_poly_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_232_zero__power,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ( power_power_real @ zero_zero_real @ N )
% 3.19/1.80 = zero_zero_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_233_zero__power,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ( power_power_nat @ zero_zero_nat @ N )
% 3.19/1.80 = zero_zero_nat ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_234_zero__power,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ( power_power_int @ zero_zero_int @ N )
% 3.19/1.80 = zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_235_power__0,axiom,
% 3.19/1.80 ! [A: real] :
% 3.19/1.80 ( ( power_power_real @ A @ zero_zero_nat )
% 3.19/1.80 = one_one_real ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_236_power__0,axiom,
% 3.19/1.80 ! [A: nat] :
% 3.19/1.80 ( ( power_power_nat @ A @ zero_zero_nat )
% 3.19/1.80 = one_one_nat ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_237_power__0,axiom,
% 3.19/1.80 ! [A: int] :
% 3.19/1.80 ( ( power_power_int @ A @ zero_zero_nat )
% 3.19/1.80 = one_one_int ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_238_power__eq__iff__eq__base,axiom,
% 3.19/1.80 ! [N: nat,A: poly_real,B: poly_real] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
% 3.19/1.80 => ( ( ord_le1180086932y_real @ zero_zero_poly_real @ B )
% 3.19/1.80 => ( ( ( power_2108872382y_real @ A @ N )
% 3.19/1.80 = ( power_2108872382y_real @ B @ N ) )
% 3.19/1.80 = ( A = B ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_239_power__eq__iff__eq__base,axiom,
% 3.19/1.80 ! [N: nat,A: real,B: real] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 3.19/1.80 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 3.19/1.80 => ( ( ( power_power_real @ A @ N )
% 3.19/1.80 = ( power_power_real @ B @ N ) )
% 3.19/1.80 = ( A = B ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_240_power__eq__iff__eq__base,axiom,
% 3.19/1.80 ! [N: nat,A: int,B: int] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 3.19/1.80 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 3.19/1.80 => ( ( ( power_power_int @ A @ N )
% 3.19/1.80 = ( power_power_int @ B @ N ) )
% 3.19/1.80 = ( A = B ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_241_power__eq__iff__eq__base,axiom,
% 3.19/1.80 ! [N: nat,A: nat,B: nat] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 3.19/1.80 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 3.19/1.80 => ( ( ( power_power_nat @ A @ N )
% 3.19/1.80 = ( power_power_nat @ B @ N ) )
% 3.19/1.80 = ( A = B ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_242_power__eq__imp__eq__base,axiom,
% 3.19/1.80 ! [A: poly_real,N: nat,B: poly_real] :
% 3.19/1.80 ( ( ( power_2108872382y_real @ A @ N )
% 3.19/1.80 = ( power_2108872382y_real @ B @ N ) )
% 3.19/1.80 => ( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
% 3.19/1.80 => ( ( ord_le1180086932y_real @ zero_zero_poly_real @ B )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( A = B ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_243_power__eq__imp__eq__base,axiom,
% 3.19/1.80 ! [A: real,N: nat,B: real] :
% 3.19/1.80 ( ( ( power_power_real @ A @ N )
% 3.19/1.80 = ( power_power_real @ B @ N ) )
% 3.19/1.80 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 3.19/1.80 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( A = B ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_244_power__eq__imp__eq__base,axiom,
% 3.19/1.80 ! [A: int,N: nat,B: int] :
% 3.19/1.80 ( ( ( power_power_int @ A @ N )
% 3.19/1.80 = ( power_power_int @ B @ N ) )
% 3.19/1.80 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 3.19/1.80 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( A = B ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_245_power__eq__imp__eq__base,axiom,
% 3.19/1.80 ! [A: nat,N: nat,B: nat] :
% 3.19/1.80 ( ( ( power_power_nat @ A @ N )
% 3.19/1.80 = ( power_power_nat @ B @ N ) )
% 3.19/1.80 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 3.19/1.80 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( A = B ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_246_one__less__power,axiom,
% 3.19/1.80 ! [A: int,N: nat] :
% 3.19/1.80 ( ( ord_less_int @ one_one_int @ A )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_247_one__less__power,axiom,
% 3.19/1.80 ! [A: real,N: nat] :
% 3.19/1.80 ( ( ord_less_real @ one_one_real @ A )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_248_one__less__power,axiom,
% 3.19/1.80 ! [A: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ one_one_nat @ A )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.19/1.80 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_249_poly__IVT__neg,axiom,
% 3.19/1.80 ! [A: real,B: real,P: poly_real] :
% 3.19/1.80 ( ( ord_less_real @ A @ B )
% 3.19/1.80 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ A ) )
% 3.19/1.80 => ( ( ord_less_real @ ( poly_real2 @ P @ B ) @ zero_zero_real )
% 3.19/1.80 => ? [X3: real] :
% 3.19/1.80 ( ( ord_less_real @ A @ X3 )
% 3.19/1.80 & ( ord_less_real @ X3 @ B )
% 3.19/1.80 & ( ( poly_real2 @ P @ X3 )
% 3.19/1.80 = zero_zero_real ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_250_poly__IVT__pos,axiom,
% 3.19/1.80 ! [A: real,B: real,P: poly_real] :
% 3.19/1.80 ( ( ord_less_real @ A @ B )
% 3.19/1.80 => ( ( ord_less_real @ ( poly_real2 @ P @ A ) @ zero_zero_real )
% 3.19/1.80 => ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ B ) )
% 3.19/1.80 => ? [X3: real] :
% 3.19/1.80 ( ( ord_less_real @ A @ X3 )
% 3.19/1.80 & ( ord_less_real @ X3 @ B )
% 3.19/1.80 & ( ( poly_real2 @ P @ X3 )
% 3.19/1.80 = zero_zero_real ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_251_linordered__field__no__ub,axiom,
% 3.19/1.80 ! [X4: real] :
% 3.19/1.80 ? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_252_linordered__field__no__lb,axiom,
% 3.19/1.80 ! [X4: real] :
% 3.19/1.80 ? [Y3: real] : ( ord_less_real @ Y3 @ X4 ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_253_coeff__eq__0,axiom,
% 3.19/1.80 ! [P: poly_poly_real,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ ( degree_poly_real @ P ) @ N )
% 3.19/1.80 => ( ( coeff_poly_real @ P @ N )
% 3.19/1.80 = zero_zero_poly_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_254_coeff__eq__0,axiom,
% 3.19/1.80 ! [P: poly_int,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ ( degree_int @ P ) @ N )
% 3.19/1.80 => ( ( coeff_int @ P @ N )
% 3.19/1.80 = zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_255_coeff__eq__0,axiom,
% 3.19/1.80 ! [P: poly_real,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ ( degree_real @ P ) @ N )
% 3.19/1.80 => ( ( coeff_real @ P @ N )
% 3.19/1.80 = zero_zero_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_256_coeff__eq__0,axiom,
% 3.19/1.80 ! [P: poly_nat,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ ( degree_nat @ P ) @ N )
% 3.19/1.80 => ( ( coeff_nat @ P @ N )
% 3.19/1.80 = zero_zero_nat ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_257_less__degree__imp,axiom,
% 3.19/1.80 ! [N: nat,P: poly_poly_real] :
% 3.19/1.80 ( ( ord_less_nat @ N @ ( degree_poly_real @ P ) )
% 3.19/1.80 => ? [I4: nat] :
% 3.19/1.80 ( ( ord_less_nat @ N @ I4 )
% 3.19/1.80 & ( ( coeff_poly_real @ P @ I4 )
% 3.19/1.80 != zero_zero_poly_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_258_less__degree__imp,axiom,
% 3.19/1.80 ! [N: nat,P: poly_int] :
% 3.19/1.80 ( ( ord_less_nat @ N @ ( degree_int @ P ) )
% 3.19/1.80 => ? [I4: nat] :
% 3.19/1.80 ( ( ord_less_nat @ N @ I4 )
% 3.19/1.80 & ( ( coeff_int @ P @ I4 )
% 3.19/1.80 != zero_zero_int ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_259_less__degree__imp,axiom,
% 3.19/1.80 ! [N: nat,P: poly_real] :
% 3.19/1.80 ( ( ord_less_nat @ N @ ( degree_real @ P ) )
% 3.19/1.80 => ? [I4: nat] :
% 3.19/1.80 ( ( ord_less_nat @ N @ I4 )
% 3.19/1.80 & ( ( coeff_real @ P @ I4 )
% 3.19/1.80 != zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_260_less__degree__imp,axiom,
% 3.19/1.80 ! [N: nat,P: poly_nat] :
% 3.19/1.80 ( ( ord_less_nat @ N @ ( degree_nat @ P ) )
% 3.19/1.80 => ? [I4: nat] :
% 3.19/1.80 ( ( ord_less_nat @ N @ I4 )
% 3.19/1.80 & ( ( coeff_nat @ P @ I4 )
% 3.19/1.80 != zero_zero_nat ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_261_degree__le,axiom,
% 3.19/1.80 ! [N: nat,P: poly_poly_real] :
% 3.19/1.80 ( ! [I4: nat] :
% 3.19/1.80 ( ( ord_less_nat @ N @ I4 )
% 3.19/1.80 => ( ( coeff_poly_real @ P @ I4 )
% 3.19/1.80 = zero_zero_poly_real ) )
% 3.19/1.80 => ( ord_less_eq_nat @ ( degree_poly_real @ P ) @ N ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_262_degree__le,axiom,
% 3.19/1.80 ! [N: nat,P: poly_int] :
% 3.19/1.80 ( ! [I4: nat] :
% 3.19/1.80 ( ( ord_less_nat @ N @ I4 )
% 3.19/1.80 => ( ( coeff_int @ P @ I4 )
% 3.19/1.80 = zero_zero_int ) )
% 3.19/1.80 => ( ord_less_eq_nat @ ( degree_int @ P ) @ N ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_263_degree__le,axiom,
% 3.19/1.80 ! [N: nat,P: poly_real] :
% 3.19/1.80 ( ! [I4: nat] :
% 3.19/1.80 ( ( ord_less_nat @ N @ I4 )
% 3.19/1.80 => ( ( coeff_real @ P @ I4 )
% 3.19/1.80 = zero_zero_real ) )
% 3.19/1.80 => ( ord_less_eq_nat @ ( degree_real @ P ) @ N ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_264_degree__le,axiom,
% 3.19/1.80 ! [N: nat,P: poly_nat] :
% 3.19/1.80 ( ! [I4: nat] :
% 3.19/1.80 ( ( ord_less_nat @ N @ I4 )
% 3.19/1.80 => ( ( coeff_nat @ P @ I4 )
% 3.19/1.80 = zero_zero_nat ) )
% 3.19/1.80 => ( ord_less_eq_nat @ ( degree_nat @ P ) @ N ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_265_Liouville__Numbers__Misc_ORats__cases_H,axiom,
% 3.19/1.80 ! [X: real] :
% 3.19/1.80 ( ( member_real @ X @ field_1537545994s_real )
% 3.19/1.80 => ~ ! [P2: int,Q3: int] :
% 3.19/1.80 ( ( ord_less_int @ zero_zero_int @ Q3 )
% 3.19/1.80 => ( X
% 3.19/1.80 != ( divide_divide_real @ ( ring_1_of_int_real @ P2 ) @ ( ring_1_of_int_real @ Q3 ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_266_coeff__0__power,axiom,
% 3.19/1.80 ! [P: poly_real,N: nat] :
% 3.19/1.80 ( ( coeff_real @ ( power_2108872382y_real @ P @ N ) @ zero_zero_nat )
% 3.19/1.80 = ( power_power_real @ ( coeff_real @ P @ zero_zero_nat ) @ N ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_267_coeff__0__power,axiom,
% 3.19/1.80 ! [P: poly_nat,N: nat] :
% 3.19/1.80 ( ( coeff_nat @ ( power_power_poly_nat @ P @ N ) @ zero_zero_nat )
% 3.19/1.80 = ( power_power_nat @ ( coeff_nat @ P @ zero_zero_nat ) @ N ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_268_coeff__0__power,axiom,
% 3.19/1.80 ! [P: poly_int,N: nat] :
% 3.19/1.80 ( ( coeff_int @ ( power_power_poly_int @ P @ N ) @ zero_zero_nat )
% 3.19/1.80 = ( power_power_int @ ( coeff_int @ P @ zero_zero_nat ) @ N ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_269_eq__zero__or__degree__less,axiom,
% 3.19/1.80 ! [P: poly_poly_real,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_nat @ ( degree_poly_real @ P ) @ N )
% 3.19/1.80 => ( ( ( coeff_poly_real @ P @ N )
% 3.19/1.80 = zero_zero_poly_real )
% 3.19/1.80 => ( ( P = zero_z1423781445y_real )
% 3.19/1.80 | ( ord_less_nat @ ( degree_poly_real @ P ) @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_270_eq__zero__or__degree__less,axiom,
% 3.19/1.80 ! [P: poly_int,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_nat @ ( degree_int @ P ) @ N )
% 3.19/1.80 => ( ( ( coeff_int @ P @ N )
% 3.19/1.80 = zero_zero_int )
% 3.19/1.80 => ( ( P = zero_zero_poly_int )
% 3.19/1.80 | ( ord_less_nat @ ( degree_int @ P ) @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_271_eq__zero__or__degree__less,axiom,
% 3.19/1.80 ! [P: poly_real,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_nat @ ( degree_real @ P ) @ N )
% 3.19/1.80 => ( ( ( coeff_real @ P @ N )
% 3.19/1.80 = zero_zero_real )
% 3.19/1.80 => ( ( P = zero_zero_poly_real )
% 3.19/1.80 | ( ord_less_nat @ ( degree_real @ P ) @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_272_eq__zero__or__degree__less,axiom,
% 3.19/1.80 ! [P: poly_nat,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_nat @ ( degree_nat @ P ) @ N )
% 3.19/1.80 => ( ( ( coeff_nat @ P @ N )
% 3.19/1.80 = zero_zero_nat )
% 3.19/1.80 => ( ( P = zero_zero_poly_nat )
% 3.19/1.80 | ( ord_less_nat @ ( degree_nat @ P ) @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_273_power__not__zero,axiom,
% 3.19/1.80 ! [A: poly_real,N: nat] :
% 3.19/1.80 ( ( A != zero_zero_poly_real )
% 3.19/1.80 => ( ( power_2108872382y_real @ A @ N )
% 3.19/1.80 != zero_zero_poly_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_274_power__not__zero,axiom,
% 3.19/1.80 ! [A: real,N: nat] :
% 3.19/1.80 ( ( A != zero_zero_real )
% 3.19/1.80 => ( ( power_power_real @ A @ N )
% 3.19/1.80 != zero_zero_real ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_275_power__not__zero,axiom,
% 3.19/1.80 ! [A: nat,N: nat] :
% 3.19/1.80 ( ( A != zero_zero_nat )
% 3.19/1.80 => ( ( power_power_nat @ A @ N )
% 3.19/1.80 != zero_zero_nat ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_276_power__not__zero,axiom,
% 3.19/1.80 ! [A: int,N: nat] :
% 3.19/1.80 ( ( A != zero_zero_int )
% 3.19/1.80 => ( ( power_power_int @ A @ N )
% 3.19/1.80 != zero_zero_int ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_277_power__divide,axiom,
% 3.19/1.80 ! [A: real,B: real,N: nat] :
% 3.19/1.80 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 3.19/1.80 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_278_poly__pinfty__gt__lc,axiom,
% 3.19/1.80 ! [P: poly_real] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ ( coeff_real @ P @ ( degree_real @ P ) ) )
% 3.19/1.80 => ? [N2: real] :
% 3.19/1.80 ! [X4: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ N2 @ X4 )
% 3.19/1.80 => ( ord_less_eq_real @ ( coeff_real @ P @ ( degree_real @ P ) ) @ ( poly_real2 @ P @ X4 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_279_power__abs,axiom,
% 3.19/1.80 ! [A: real,N: nat] :
% 3.19/1.80 ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 3.19/1.80 = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_280_power__abs,axiom,
% 3.19/1.80 ! [A: int,N: nat] :
% 3.19/1.80 ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 3.19/1.80 = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_281_algebraic__altdef,axiom,
% 3.19/1.80 ( algebraic_real
% 3.19/1.80 = ( ^ [X2: real] :
% 3.19/1.80 ? [P4: poly_real] :
% 3.19/1.80 ( ! [I3: nat] : ( member_real @ ( coeff_real @ P4 @ I3 ) @ field_1537545994s_real )
% 3.19/1.80 & ( P4 != zero_zero_poly_real )
% 3.19/1.80 & ( ( poly_real2 @ P4 @ X2 )
% 3.19/1.80 = zero_zero_real ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_282_divide__le__0__iff,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 3.19/1.80 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 3.19/1.80 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 3.19/1.80 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 3.19/1.80 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_283_divide__right__mono,axiom,
% 3.19/1.80 ! [A: real,B: real,C: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ A @ B )
% 3.19/1.80 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 3.19/1.80 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_284_zero__le__divide__iff,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 3.19/1.80 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 3.19/1.80 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 3.19/1.80 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 3.19/1.80 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_285_divide__nonneg__nonneg,axiom,
% 3.19/1.80 ! [X: real,Y2: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 3.19/1.80 => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
% 3.19/1.80 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_286_divide__nonneg__nonpos,axiom,
% 3.19/1.80 ! [X: real,Y2: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 3.19/1.80 => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
% 3.19/1.80 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_287_divide__nonpos__nonneg,axiom,
% 3.19/1.80 ! [X: real,Y2: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 3.19/1.80 => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
% 3.19/1.80 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_288_divide__nonpos__nonpos,axiom,
% 3.19/1.80 ! [X: real,Y2: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 3.19/1.80 => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
% 3.19/1.80 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_289_divide__right__mono__neg,axiom,
% 3.19/1.80 ! [A: real,B: real,C: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ A @ B )
% 3.19/1.80 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 3.19/1.80 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_290_power__mono,axiom,
% 3.19/1.80 ! [A: poly_real,B: poly_real,N: nat] :
% 3.19/1.80 ( ( ord_le1180086932y_real @ A @ B )
% 3.19/1.80 => ( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
% 3.19/1.80 => ( ord_le1180086932y_real @ ( power_2108872382y_real @ A @ N ) @ ( power_2108872382y_real @ B @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_291_power__mono,axiom,
% 3.19/1.80 ! [A: real,B: real,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_real @ A @ B )
% 3.19/1.80 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 3.19/1.80 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_292_power__mono,axiom,
% 3.19/1.80 ! [A: int,B: int,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_int @ A @ B )
% 3.19/1.80 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 3.19/1.80 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_293_power__mono,axiom,
% 3.19/1.80 ! [A: nat,B: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_nat @ A @ B )
% 3.19/1.80 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 3.19/1.80 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_294_zero__le__power,axiom,
% 3.19/1.80 ! [A: poly_real,N: nat] :
% 3.19/1.80 ( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
% 3.19/1.80 => ( ord_le1180086932y_real @ zero_zero_poly_real @ ( power_2108872382y_real @ A @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_295_zero__le__power,axiom,
% 3.19/1.80 ! [A: real,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 3.19/1.80 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_296_zero__le__power,axiom,
% 3.19/1.80 ! [A: int,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 3.19/1.80 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_297_zero__le__power,axiom,
% 3.19/1.80 ! [A: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 3.19/1.80 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_298_divide__neg__neg,axiom,
% 3.19/1.80 ! [X: real,Y2: real] :
% 3.19/1.80 ( ( ord_less_real @ X @ zero_zero_real )
% 3.19/1.80 => ( ( ord_less_real @ Y2 @ zero_zero_real )
% 3.19/1.80 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_299_divide__neg__pos,axiom,
% 3.19/1.80 ! [X: real,Y2: real] :
% 3.19/1.80 ( ( ord_less_real @ X @ zero_zero_real )
% 3.19/1.80 => ( ( ord_less_real @ zero_zero_real @ Y2 )
% 3.19/1.80 => ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_300_divide__pos__neg,axiom,
% 3.19/1.80 ! [X: real,Y2: real] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ X )
% 3.19/1.80 => ( ( ord_less_real @ Y2 @ zero_zero_real )
% 3.19/1.80 => ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_301_divide__pos__pos,axiom,
% 3.19/1.80 ! [X: real,Y2: real] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ X )
% 3.19/1.80 => ( ( ord_less_real @ zero_zero_real @ Y2 )
% 3.19/1.80 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_302_divide__less__0__iff,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 3.19/1.80 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 3.19/1.80 & ( ord_less_real @ B @ zero_zero_real ) )
% 3.19/1.80 | ( ( ord_less_real @ A @ zero_zero_real )
% 3.19/1.80 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_303_divide__less__cancel,axiom,
% 3.19/1.80 ! [A: real,C: real,B: real] :
% 3.19/1.80 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 3.19/1.80 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 3.19/1.80 => ( ord_less_real @ A @ B ) )
% 3.19/1.80 & ( ( ord_less_real @ C @ zero_zero_real )
% 3.19/1.80 => ( ord_less_real @ B @ A ) )
% 3.19/1.80 & ( C != zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_304_zero__less__divide__iff,axiom,
% 3.19/1.80 ! [A: real,B: real] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 3.19/1.80 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 3.19/1.80 & ( ord_less_real @ zero_zero_real @ B ) )
% 3.19/1.80 | ( ( ord_less_real @ A @ zero_zero_real )
% 3.19/1.80 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_305_divide__strict__right__mono,axiom,
% 3.19/1.80 ! [A: real,B: real,C: real] :
% 3.19/1.80 ( ( ord_less_real @ A @ B )
% 3.19/1.80 => ( ( ord_less_real @ zero_zero_real @ C )
% 3.19/1.80 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_306_divide__strict__right__mono__neg,axiom,
% 3.19/1.80 ! [B: real,A: real,C: real] :
% 3.19/1.80 ( ( ord_less_real @ B @ A )
% 3.19/1.80 => ( ( ord_less_real @ C @ zero_zero_real )
% 3.19/1.80 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_307_zero__less__power,axiom,
% 3.19/1.80 ! [A: poly_real,N: nat] :
% 3.19/1.80 ( ( ord_less_poly_real @ zero_zero_poly_real @ A )
% 3.19/1.80 => ( ord_less_poly_real @ zero_zero_poly_real @ ( power_2108872382y_real @ A @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_308_zero__less__power,axiom,
% 3.19/1.80 ! [A: int,N: nat] :
% 3.19/1.80 ( ( ord_less_int @ zero_zero_int @ A )
% 3.19/1.80 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_309_zero__less__power,axiom,
% 3.19/1.80 ! [A: real,N: nat] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ A )
% 3.19/1.80 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_310_zero__less__power,axiom,
% 3.19/1.80 ! [A: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ A )
% 3.19/1.80 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_311_right__inverse__eq,axiom,
% 3.19/1.80 ! [B: real,A: real] :
% 3.19/1.80 ( ( B != zero_zero_real )
% 3.19/1.80 => ( ( ( divide_divide_real @ A @ B )
% 3.19/1.80 = one_one_real )
% 3.19/1.80 = ( A = B ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_312_power__0__left,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( ( N = zero_zero_nat )
% 3.19/1.80 => ( ( power_2108872382y_real @ zero_zero_poly_real @ N )
% 3.19/1.80 = one_one_poly_real ) )
% 3.19/1.80 & ( ( N != zero_zero_nat )
% 3.19/1.80 => ( ( power_2108872382y_real @ zero_zero_poly_real @ N )
% 3.19/1.80 = zero_zero_poly_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_313_power__0__left,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( ( N = zero_zero_nat )
% 3.19/1.80 => ( ( power_power_real @ zero_zero_real @ N )
% 3.19/1.80 = one_one_real ) )
% 3.19/1.80 & ( ( N != zero_zero_nat )
% 3.19/1.80 => ( ( power_power_real @ zero_zero_real @ N )
% 3.19/1.80 = zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_314_power__0__left,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( ( N = zero_zero_nat )
% 3.19/1.80 => ( ( power_power_nat @ zero_zero_nat @ N )
% 3.19/1.80 = one_one_nat ) )
% 3.19/1.80 & ( ( N != zero_zero_nat )
% 3.19/1.80 => ( ( power_power_nat @ zero_zero_nat @ N )
% 3.19/1.80 = zero_zero_nat ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_315_power__0__left,axiom,
% 3.19/1.80 ! [N: nat] :
% 3.19/1.80 ( ( ( N = zero_zero_nat )
% 3.19/1.80 => ( ( power_power_int @ zero_zero_int @ N )
% 3.19/1.80 = one_one_int ) )
% 3.19/1.80 & ( ( N != zero_zero_nat )
% 3.19/1.80 => ( ( power_power_int @ zero_zero_int @ N )
% 3.19/1.80 = zero_zero_int ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_316_one__le__power,axiom,
% 3.19/1.80 ! [A: real,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_real @ one_one_real @ A )
% 3.19/1.80 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_317_one__le__power,axiom,
% 3.19/1.80 ! [A: int,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_int @ one_one_int @ A )
% 3.19/1.80 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_318_one__le__power,axiom,
% 3.19/1.80 ! [A: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 3.19/1.80 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_319_power__increasing,axiom,
% 3.19/1.80 ! [N: nat,N4: nat,A: real] :
% 3.19/1.80 ( ( ord_less_eq_nat @ N @ N4 )
% 3.19/1.80 => ( ( ord_less_eq_real @ one_one_real @ A )
% 3.19/1.80 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_320_power__increasing,axiom,
% 3.19/1.80 ! [N: nat,N4: nat,A: int] :
% 3.19/1.80 ( ( ord_less_eq_nat @ N @ N4 )
% 3.19/1.80 => ( ( ord_less_eq_int @ one_one_int @ A )
% 3.19/1.80 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_321_power__increasing,axiom,
% 3.19/1.80 ! [N: nat,N4: nat,A: nat] :
% 3.19/1.80 ( ( ord_less_eq_nat @ N @ N4 )
% 3.19/1.80 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 3.19/1.80 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_322_power__less__imp__less__exp,axiom,
% 3.19/1.80 ! [A: int,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_int @ one_one_int @ A )
% 3.19/1.80 => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 3.19/1.80 => ( ord_less_nat @ M @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_323_power__less__imp__less__exp,axiom,
% 3.19/1.80 ! [A: real,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_real @ one_one_real @ A )
% 3.19/1.80 => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 3.19/1.80 => ( ord_less_nat @ M @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_324_power__less__imp__less__exp,axiom,
% 3.19/1.80 ! [A: nat,M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ one_one_nat @ A )
% 3.19/1.80 => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 3.19/1.80 => ( ord_less_nat @ M @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_325_power__strict__increasing,axiom,
% 3.19/1.80 ! [N: nat,N4: nat,A: int] :
% 3.19/1.80 ( ( ord_less_nat @ N @ N4 )
% 3.19/1.80 => ( ( ord_less_int @ one_one_int @ A )
% 3.19/1.80 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_326_power__strict__increasing,axiom,
% 3.19/1.80 ! [N: nat,N4: nat,A: real] :
% 3.19/1.80 ( ( ord_less_nat @ N @ N4 )
% 3.19/1.80 => ( ( ord_less_real @ one_one_real @ A )
% 3.19/1.80 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_327_power__strict__increasing,axiom,
% 3.19/1.80 ! [N: nat,N4: nat,A: nat] :
% 3.19/1.80 ( ( ord_less_nat @ N @ N4 )
% 3.19/1.80 => ( ( ord_less_nat @ one_one_nat @ A )
% 3.19/1.80 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_328_power__one__over,axiom,
% 3.19/1.80 ! [A: real,N: nat] :
% 3.19/1.80 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 3.19/1.80 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_329_nonzero__abs__divide,axiom,
% 3.19/1.80 ! [B: real,A: real] :
% 3.19/1.80 ( ( B != zero_zero_real )
% 3.19/1.80 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 3.19/1.80 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_330_frac__le,axiom,
% 3.19/1.80 ! [Y2: real,X: real,W: real,Z: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
% 3.19/1.80 => ( ( ord_less_eq_real @ X @ Y2 )
% 3.19/1.80 => ( ( ord_less_real @ zero_zero_real @ W )
% 3.19/1.80 => ( ( ord_less_eq_real @ W @ Z )
% 3.19/1.80 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_331_frac__less,axiom,
% 3.19/1.80 ! [X: real,Y2: real,W: real,Z: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 3.19/1.80 => ( ( ord_less_real @ X @ Y2 )
% 3.19/1.80 => ( ( ord_less_real @ zero_zero_real @ W )
% 3.19/1.80 => ( ( ord_less_eq_real @ W @ Z )
% 3.19/1.80 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_332_frac__less2,axiom,
% 3.19/1.80 ! [X: real,Y2: real,W: real,Z: real] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ X )
% 3.19/1.80 => ( ( ord_less_eq_real @ X @ Y2 )
% 3.19/1.80 => ( ( ord_less_real @ zero_zero_real @ W )
% 3.19/1.80 => ( ( ord_less_real @ W @ Z )
% 3.19/1.80 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_333_divide__le__cancel,axiom,
% 3.19/1.80 ! [A: real,C: real,B: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 3.19/1.80 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 3.19/1.80 => ( ord_less_eq_real @ A @ B ) )
% 3.19/1.80 & ( ( ord_less_real @ C @ zero_zero_real )
% 3.19/1.80 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_334_divide__nonneg__neg,axiom,
% 3.19/1.80 ! [X: real,Y2: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 3.19/1.80 => ( ( ord_less_real @ Y2 @ zero_zero_real )
% 3.19/1.80 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_335_divide__nonneg__pos,axiom,
% 3.19/1.80 ! [X: real,Y2: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 3.19/1.80 => ( ( ord_less_real @ zero_zero_real @ Y2 )
% 3.19/1.80 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_336_divide__nonpos__neg,axiom,
% 3.19/1.80 ! [X: real,Y2: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 3.19/1.80 => ( ( ord_less_real @ Y2 @ zero_zero_real )
% 3.19/1.80 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_337_divide__nonpos__pos,axiom,
% 3.19/1.80 ! [X: real,Y2: real] :
% 3.19/1.80 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 3.19/1.80 => ( ( ord_less_real @ zero_zero_real @ Y2 )
% 3.19/1.80 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_338_power__less__imp__less__base,axiom,
% 3.19/1.80 ! [A: nat,N: nat,B: nat] :
% 3.19/1.80 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 3.19/1.80 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 3.19/1.80 => ( ord_less_nat @ A @ B ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_339_A__less_I2_J,axiom,
% 3.19/1.80 ord_less_real @ a @ ( divide_divide_real @ one_one_real @ m ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_340_ab,axiom,
% 3.19/1.80 ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ x @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) @ ( divide_divide_real @ a @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ n ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_341_ab_H,axiom,
% 3.19/1.80 ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ x @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) @ a ).
% 3.19/1.80
% 3.19/1.80 thf(fact_342_div__pos__pos__trivial,axiom,
% 3.19/1.80 ! [K: int,L: int] :
% 3.19/1.80 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 3.19/1.80 => ( ( ord_less_int @ K @ L )
% 3.19/1.80 => ( ( divide_divide_int @ K @ L )
% 3.19/1.80 = zero_zero_int ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_343_div__neg__neg__trivial,axiom,
% 3.19/1.80 ! [K: int,L: int] :
% 3.19/1.80 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 3.19/1.80 => ( ( ord_less_int @ L @ K )
% 3.19/1.80 => ( ( divide_divide_int @ K @ L )
% 3.19/1.80 = zero_zero_int ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_344_M__pos,axiom,
% 3.19/1.80 ord_less_real @ zero_zero_real @ m ).
% 3.19/1.80
% 3.19/1.80 thf(fact_345_div__less,axiom,
% 3.19/1.80 ! [M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ M @ N )
% 3.19/1.80 => ( ( divide_divide_nat @ M @ N )
% 3.19/1.80 = zero_zero_nat ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_346_A__less_I3_J,axiom,
% 3.19/1.80 ! [X5: real] :
% 3.19/1.80 ( ( X5 != x )
% 3.19/1.80 => ( ( ( poly_real2 @ p @ X5 )
% 3.19/1.80 = zero_zero_real )
% 3.19/1.80 => ( ord_less_real @ a @ ( abs_abs_real @ ( minus_minus_real @ X5 @ x ) ) ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_347_that,axiom,
% 3.19/1.80 ! [C: real,N: nat] :
% 3.19/1.80 ( ( ord_less_real @ zero_zero_real @ C )
% 3.19/1.80 => ( ! [P2: int,Q3: int] :
% 3.19/1.80 ( ( ord_less_int @ zero_zero_int @ Q3 )
% 3.19/1.80 => ( ord_less_real @ ( divide_divide_real @ C @ ( power_power_real @ ( ring_1_of_int_real @ Q3 ) @ N ) ) @ ( abs_abs_real @ ( minus_minus_real @ x @ ( divide_divide_real @ ( ring_1_of_int_real @ P2 ) @ ( ring_1_of_int_real @ Q3 ) ) ) ) ) )
% 3.19/1.80 => thesis ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_348_div__eq__dividend__iff,axiom,
% 3.19/1.80 ! [M: nat,N: nat] :
% 3.19/1.80 ( ( ord_less_nat @ zero_zero_nat @ M )
% 3.19/1.80 => ( ( ( divide_divide_nat @ M @ N )
% 3.19/1.80 = M )
% 3.19/1.80 = ( N = one_one_nat ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_349_div__less__dividend,axiom,
% 3.19/1.80 ! [N: nat,M: nat] :
% 3.19/1.80 ( ( ord_less_nat @ one_one_nat @ N )
% 3.19/1.80 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 3.19/1.80 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_350_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 3.19/1.80 ! [M: nat,N: nat] :
% 3.19/1.80 ( ( ( divide_divide_nat @ M @ N )
% 3.19/1.80 = zero_zero_nat )
% 3.19/1.80 = ( ( ord_less_nat @ M @ N )
% 3.19/1.80 | ( N = zero_zero_nat ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_351_div__le__dividend,axiom,
% 3.19/1.80 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 3.19/1.80
% 3.19/1.80 thf(fact_352_div__le__mono,axiom,
% 3.19/1.80 ! [M: nat,N: nat,K: nat] :
% 3.19/1.80 ( ( ord_less_eq_nat @ M @ N )
% 3.19/1.80 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 3.19/1.80
% 3.19/1.80 thf(conj_0,conjecture,
% 3.19/1.80 ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) ).
% 3.19/1.80 ------------------------
% 3.19/1.82 % (21881)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/183Mi)
% 3.19/1.82 % (21883)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_21775 for (2999ds/27Mi)
% 3.19/1.83 % (21885)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_21775 for (2999ds/2Mi)
% 3.19/1.83 % (21884)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/2Mi)
% 3.19/1.83 % (21883)Instruction limit reached!
% 3.19/1.83 % (21883)------------------------------
% 3.19/1.83 % (21883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.83 % (21883)Termination reason: Unknown
% 3.19/1.83 % (21883)Termination phase: Property scanning
% 3.19/1.83
% 3.19/1.83 % (21883)Memory used [KB]: 1663
% 3.19/1.83 % (21883)Time elapsed: 0.010 s
% 3.19/1.83 % (21883)Instructions burned: 30 (million)
% 3.19/1.83 % (21883)------------------------------
% 3.19/1.83 % (21883)------------------------------
% 3.19/1.83 % (21887)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/18Mi)
% 3.19/1.83 % (21886)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_21775 for (2999ds/275Mi)
% 3.19/1.83 % (21884)Instruction limit reached!
% 3.19/1.83 % (21884)------------------------------
% 3.19/1.83 % (21884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.83 % (21884)Termination reason: Unknown
% 3.19/1.83 % (21885)Instruction limit reached!
% 3.19/1.83 % (21885)------------------------------
% 3.19/1.83 % (21885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.83 % (21884)Termination phase: shuffling
% 3.19/1.83
% 3.19/1.83 % (21884)Memory used [KB]: 1407
% 3.19/1.83 % (21884)Time elapsed: 0.003 s
% 3.19/1.83 % (21884)Instructions burned: 2 (million)
% 3.19/1.83 % (21884)------------------------------
% 3.19/1.83 % (21884)------------------------------
% 3.19/1.83 % (21885)Termination reason: Unknown
% 3.19/1.83 % (21885)Termination phase: shuffling
% 3.19/1.83
% 3.19/1.83 % (21885)Memory used [KB]: 1407
% 3.19/1.83 % (21885)Time elapsed: 0.003 s
% 3.19/1.83 % (21885)Instructions burned: 2 (million)
% 3.19/1.83 % (21885)------------------------------
% 3.19/1.83 % (21885)------------------------------
% 3.19/1.83 % (21882)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_21775 for (2999ds/4Mi)
% 3.19/1.83 % (21882)Instruction limit reached!
% 3.19/1.83 % (21882)------------------------------
% 3.19/1.83 % (21882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.83 % (21882)Termination reason: Unknown
% 3.19/1.83 % (21882)Termination phase: shuffling
% 3.19/1.83
% 3.19/1.83 % (21882)Memory used [KB]: 1407
% 3.19/1.83 % (21882)Time elapsed: 0.003 s
% 3.19/1.83 % (21882)Instructions burned: 7 (million)
% 3.19/1.83 % (21882)------------------------------
% 3.19/1.83 % (21882)------------------------------
% 3.19/1.84 % (21888)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/3Mi)
% 3.19/1.84 % (21887)Instruction limit reached!
% 3.19/1.84 % (21887)------------------------------
% 3.19/1.84 % (21887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.84 % (21887)Termination reason: Unknown
% 3.19/1.84 % (21887)Termination phase: shuffling
% 3.19/1.84
% 3.19/1.84 % (21887)Memory used [KB]: 1663
% 3.19/1.84 % (21887)Time elapsed: 0.012 s
% 3.19/1.84 % (21887)Instructions burned: 18 (million)
% 3.19/1.84 % (21887)------------------------------
% 3.19/1.84 % (21887)------------------------------
% 3.19/1.84 % (21888)Instruction limit reached!
% 3.19/1.84 % (21888)------------------------------
% 3.19/1.84 % (21888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.84 % (21888)Termination reason: Unknown
% 3.19/1.84 % (21888)Termination phase: shuffling
% 3.19/1.84
% 3.19/1.84 % (21888)Memory used [KB]: 1407
% 3.19/1.84 % (21888)Time elapsed: 0.002 s
% 3.19/1.84 % (21888)Instructions burned: 3 (million)
% 3.19/1.84 % (21888)------------------------------
% 3.19/1.84 % (21888)------------------------------
% 3.19/1.84 % (21891)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_21775 for (2999ds/3Mi)
% 3.19/1.84 % (21891)Instruction limit reached!
% 3.19/1.84 % (21891)------------------------------
% 3.19/1.84 % (21891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.84 % (21891)Termination reason: Unknown
% 3.19/1.84 % (21891)Termination phase: shuffling
% 3.19/1.84
% 3.19/1.84 % (21891)Memory used [KB]: 1407
% 3.19/1.84 % (21891)Time elapsed: 0.002 s
% 3.19/1.84 % (21891)Instructions burned: 3 (million)
% 3.19/1.84 % (21891)------------------------------
% 3.19/1.84 % (21891)------------------------------
% 3.19/1.85 % (21889)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_21775 for (2999ds/37Mi)
% 3.19/1.85 % (21890)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_21775 for (2999ds/15Mi)
% 3.19/1.85 % (21893)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/7Mi)
% 3.19/1.85 % (21893)Instruction limit reached!
% 3.19/1.85 % (21893)------------------------------
% 3.19/1.85 % (21893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.85 % (21893)Termination reason: Unknown
% 3.19/1.85 % (21893)Termination phase: shuffling
% 3.19/1.85
% 3.19/1.85 % (21893)Memory used [KB]: 1407
% 3.19/1.85 % (21893)Time elapsed: 0.003 s
% 3.19/1.85 % (21893)Instructions burned: 7 (million)
% 3.19/1.85 % (21893)------------------------------
% 3.19/1.85 % (21893)------------------------------
% 3.19/1.85 % (21894)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/16Mi)
% 3.19/1.86 % (21890)Instruction limit reached!
% 3.19/1.86 % (21890)------------------------------
% 3.19/1.86 % (21890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.86 % (21890)Termination reason: Unknown
% 3.19/1.86 % (21890)Termination phase: shuffling
% 3.19/1.86
% 3.19/1.86 % (21890)Memory used [KB]: 1535
% 3.19/1.86 % (21890)Time elapsed: 0.010 s
% 3.19/1.86 % (21890)Instructions burned: 15 (million)
% 3.19/1.86 % (21890)------------------------------
% 3.19/1.86 % (21890)------------------------------
% 3.19/1.86 % (21892)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_21775 for (2999ds/1041Mi)
% 3.19/1.86 % (21894)Instruction limit reached!
% 3.19/1.86 % (21894)------------------------------
% 3.19/1.86 % (21894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.86 % (21894)Termination reason: Unknown
% 3.19/1.86 % (21894)Termination phase: shuffling
% 3.19/1.86
% 3.19/1.86 % (21894)Memory used [KB]: 1535
% 3.19/1.86 % (21894)Time elapsed: 0.006 s
% 3.19/1.86 % (21894)Instructions burned: 16 (million)
% 3.19/1.86 % (21894)------------------------------
% 3.19/1.86 % (21894)------------------------------
% 3.19/1.86 % (21895)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_21775 for (2999ds/3Mi)
% 3.19/1.86 % (21895)Instruction limit reached!
% 3.19/1.86 % (21895)------------------------------
% 3.19/1.86 % (21895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.86 % (21895)Termination reason: Unknown
% 3.19/1.86 % (21895)Termination phase: shuffling
% 3.19/1.86
% 3.19/1.86 % (21895)Memory used [KB]: 1407
% 3.19/1.86 % (21895)Time elapsed: 0.002 s
% 3.19/1.86 % (21895)Instructions burned: 3 (million)
% 3.19/1.86 % (21895)------------------------------
% 3.19/1.86 % (21895)------------------------------
% 3.19/1.87 % (21889)Instruction limit reached!
% 3.19/1.87 % (21889)------------------------------
% 3.19/1.87 % (21889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.87 % (21889)Termination reason: Unknown
% 3.19/1.87 % (21889)Termination phase: shuffling
% 3.19/1.87
% 3.19/1.87 % (21889)Memory used [KB]: 2046
% 3.19/1.87 % (21889)Time elapsed: 0.021 s
% 3.19/1.87 % (21889)Instructions burned: 37 (million)
% 3.19/1.87 % (21889)------------------------------
% 3.19/1.87 % (21889)------------------------------
% 3.19/1.87 % (21897)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/7Mi)
% 3.19/1.87 % (21897)Instruction limit reached!
% 3.19/1.87 % (21897)------------------------------
% 3.19/1.87 % (21897)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.87 % (21897)Termination reason: Unknown
% 3.19/1.87 % (21897)Termination phase: shuffling
% 3.19/1.87
% 3.19/1.87 % (21897)Memory used [KB]: 1407
% 3.19/1.87 % (21897)Time elapsed: 0.003 s
% 3.19/1.87 % (21897)Instructions burned: 7 (million)
% 3.19/1.87 % (21897)------------------------------
% 3.19/1.87 % (21897)------------------------------
% 3.19/1.87 % (21898)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/3Mi)
% 3.19/1.87 % (21898)Instruction limit reached!
% 3.19/1.87 % (21898)------------------------------
% 3.19/1.87 % (21898)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.87 % (21898)Termination reason: Unknown
% 3.19/1.87 % (21898)Termination phase: shuffling
% 3.19/1.87
% 3.19/1.87 % (21898)Memory used [KB]: 1407
% 3.19/1.87 % (21898)Time elapsed: 0.002 s
% 3.19/1.87 % (21898)Instructions burned: 4 (million)
% 3.19/1.87 % (21898)------------------------------
% 3.19/1.87 % (21898)------------------------------
% 3.19/1.87 % (21896)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_21775 for (2999ds/3Mi)
% 3.19/1.87 % (21886)Refutation not found, incomplete strategy
% 3.19/1.87 % (21886)------------------------------
% 3.19/1.87 % (21886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.87 % (21886)Termination reason: Refutation not found, incomplete strategy
% 3.19/1.87
% 3.19/1.87
% 3.19/1.87 % (21886)Memory used [KB]: 6652
% 3.19/1.87 % (21886)Time elapsed: 0.048 s
% 3.19/1.87 % (21886)Instructions burned: 84 (million)
% 3.19/1.87 % (21886)------------------------------
% 3.19/1.87 % (21886)------------------------------
% 3.19/1.88 % (21896)Instruction limit reached!
% 3.19/1.88 % (21896)------------------------------
% 3.19/1.88 % (21896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.88 % (21896)Termination reason: Unknown
% 3.19/1.88 % (21896)Termination phase: shuffling
% 3.19/1.88
% 3.19/1.88 % (21896)Memory used [KB]: 1407
% 3.19/1.88 % (21896)Time elapsed: 0.004 s
% 3.19/1.88 % (21896)Instructions burned: 4 (million)
% 3.19/1.88 % (21896)------------------------------
% 3.19/1.88 % (21896)------------------------------
% 3.19/1.88 % (21900)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/18Mi)
% 3.19/1.88 % (21901)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/710Mi)
% 3.19/1.88 % (21900)Instruction limit reached!
% 3.19/1.88 % (21900)------------------------------
% 3.19/1.88 % (21900)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.88 % (21900)Termination reason: Unknown
% 3.19/1.88 % (21900)Termination phase: shuffling
% 3.19/1.88
% 3.19/1.88 % (21900)Memory used [KB]: 1663
% 3.19/1.88 % (21900)Time elapsed: 0.007 s
% 3.19/1.88 % (21900)Instructions burned: 20 (million)
% 3.19/1.88 % (21900)------------------------------
% 3.19/1.88 % (21900)------------------------------
% 3.19/1.89 % (21899)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/4Mi)
% 3.19/1.89 % (21899)Instruction limit reached!
% 3.19/1.89 % (21899)------------------------------
% 3.19/1.89 % (21899)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.89 % (21899)Termination reason: Unknown
% 3.19/1.89 % (21899)Termination phase: shuffling
% 3.19/1.89
% 3.19/1.89 % (21899)Memory used [KB]: 1407
% 3.19/1.89 % (21899)Time elapsed: 0.004 s
% 3.19/1.89 % (21899)Instructions burned: 4 (million)
% 3.19/1.89 % (21899)------------------------------
% 3.19/1.89 % (21899)------------------------------
% 3.19/1.89 % (21902)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/6Mi)
% 3.19/1.89 % (21903)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_21775 for (2999ds/902Mi)
% 3.19/1.90 % (21902)Instruction limit reached!
% 3.19/1.90 % (21902)------------------------------
% 3.19/1.90 % (21902)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.90 % (21902)Termination reason: Unknown
% 3.19/1.90 % (21902)Termination phase: shuffling
% 3.19/1.90
% 3.19/1.90 % (21902)Memory used [KB]: 1407
% 3.19/1.90 % (21902)Time elapsed: 0.003 s
% 3.19/1.90 % (21902)Instructions burned: 7 (million)
% 3.19/1.90 % (21902)------------------------------
% 3.19/1.90 % (21902)------------------------------
% 3.19/1.90 % (21906)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_21775 for (2999ds/6Mi)
% 3.19/1.91 % (21905)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/5Mi)
% 3.19/1.91 % (21906)Instruction limit reached!
% 3.19/1.91 % (21906)------------------------------
% 3.19/1.91 % (21906)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.91 % (21906)Termination reason: Unknown
% 3.19/1.91 % (21906)Termination phase: shuffling
% 3.19/1.91
% 3.19/1.91 % (21906)Memory used [KB]: 1407
% 3.19/1.91 % (21906)Time elapsed: 0.003 s
% 3.19/1.91 % (21906)Instructions burned: 8 (million)
% 3.19/1.91 % (21906)------------------------------
% 3.19/1.91 % (21906)------------------------------
% 3.19/1.91 % (21904)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_21775 for (2999ds/21Mi)
% 3.19/1.91 % (21905)Instruction limit reached!
% 3.19/1.91 % (21905)------------------------------
% 3.19/1.91 % (21905)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.91 % (21905)Termination reason: Unknown
% 3.19/1.91 % (21905)Termination phase: shuffling
% 3.19/1.91
% 3.19/1.91 % (21905)Memory used [KB]: 1407
% 3.19/1.91 % (21905)Time elapsed: 0.004 s
% 3.19/1.91 % (21905)Instructions burned: 5 (million)
% 3.19/1.91 % (21905)------------------------------
% 3.19/1.91 % (21905)------------------------------
% 3.19/1.92 % (21907)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_21775 for (2999ds/377Mi)
% 3.19/1.92 % (21904)Instruction limit reached!
% 3.19/1.92 % (21904)------------------------------
% 3.19/1.92 % (21904)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.92 % (21904)Termination reason: Unknown
% 3.19/1.92 % (21904)Termination phase: shuffling
% 3.19/1.92
% 3.19/1.92 % (21904)Memory used [KB]: 1663
% 3.19/1.92 % (21904)Time elapsed: 0.013 s
% 3.19/1.92 % (21904)Instructions burned: 21 (million)
% 3.19/1.92 % (21904)------------------------------
% 3.19/1.92 % (21904)------------------------------
% 3.19/1.92 % (21881)Instruction limit reached!
% 3.19/1.92 % (21881)------------------------------
% 3.19/1.92 % (21881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.93 % (21881)Termination reason: Unknown
% 3.19/1.93 % (21881)Termination phase: Saturation
% 3.19/1.93
% 3.19/1.93 % (21881)Memory used [KB]: 7036
% 3.19/1.93 % (21881)Time elapsed: 0.101 s
% 3.19/1.93 % (21881)Instructions burned: 183 (million)
% 3.19/1.93 % (21881)------------------------------
% 3.19/1.93 % (21881)------------------------------
% 3.19/1.93 % (21908)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_21775 for (2999ds/779Mi)
% 3.19/1.94 % (21909)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_21775 for (2998ds/19Mi)
% 3.19/1.95 % (21909)Instruction limit reached!
% 3.19/1.95 % (21909)------------------------------
% 3.19/1.95 % (21909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.95 % (21909)Termination reason: Unknown
% 3.19/1.95 % (21909)Termination phase: shuffling
% 3.19/1.95
% 3.19/1.95 % (21909)Memory used [KB]: 1663
% 3.19/1.95 % (21909)Time elapsed: 0.011 s
% 3.19/1.95 % (21909)Instructions burned: 19 (million)
% 3.19/1.95 % (21909)------------------------------
% 3.19/1.95 % (21909)------------------------------
% 3.19/1.95 % (21903)Refutation not found, incomplete strategy
% 3.19/1.95 % (21903)------------------------------
% 3.19/1.95 % (21903)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.95 % (21903)Termination reason: Refutation not found, incomplete strategy
% 3.19/1.95
% 3.19/1.95
% 3.19/1.95 % (21903)Memory used [KB]: 6908
% 3.19/1.95 % (21903)Time elapsed: 0.059 s
% 3.19/1.95 % (21903)Instructions burned: 103 (million)
% 3.19/1.95 % (21903)------------------------------
% 3.19/1.95 % (21903)------------------------------
% 3.19/1.96 % (21910)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_21775 for (2998ds/879Mi)
% 3.19/1.97 % (21911)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_21775 for (2998ds/17Mi)
% 3.19/1.97 % (21912)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_21775 for (2998ds/3Mi)
% 3.19/1.97 % (21912)Instruction limit reached!
% 3.19/1.97 % (21912)------------------------------
% 3.19/1.97 % (21912)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.97 % (21912)Termination reason: Unknown
% 3.19/1.97 % (21912)Termination phase: shuffling
% 3.19/1.97
% 3.19/1.97 % (21912)Memory used [KB]: 1407
% 3.19/1.97 % (21912)Time elapsed: 0.003 s
% 3.19/1.97 % (21912)Instructions burned: 3 (million)
% 3.19/1.97 % (21912)------------------------------
% 3.19/1.97 % (21912)------------------------------
% 3.19/1.98 % (21911)Instruction limit reached!
% 3.19/1.98 % (21911)------------------------------
% 3.19/1.98 % (21911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/1.98 % (21911)Termination reason: Unknown
% 3.19/1.98 % (21911)Termination phase: shuffling
% 3.19/1.98
% 3.19/1.98 % (21911)Memory used [KB]: 1535
% 3.19/1.98 % (21911)Time elapsed: 0.011 s
% 3.19/1.98 % (21911)Instructions burned: 18 (million)
% 3.19/1.98 % (21911)------------------------------
% 3.19/1.98 % (21911)------------------------------
% 3.19/1.99 % (21913)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_21775 for (2998ds/30Mi)
% 3.19/1.99 % (21914)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_21775 for (2998ds/127Mi)
% 3.19/2.01 % (21913)Instruction limit reached!
% 3.19/2.01 % (21913)------------------------------
% 3.19/2.01 % (21913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.19/2.01 % (21913)Termination reason: Unknown
% 3.19/2.01 % (21913)Termination phase: Property scanning
% 3.19/2.01
% 3.19/2.01 % (21913)Memory used [KB]: 1791
% 3.19/2.01 % (21913)Time elapsed: 0.019 s
% 3.19/2.01 % (21913)Instructions burned: 31 (million)
% 3.19/2.01 % (21913)------------------------------
% 3.19/2.01 % (21913)------------------------------
% 4.47/2.03 % (21915)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_21775 for (2997ds/100Mi)
% 4.47/2.06 % (21914)Instruction limit reached!
% 4.47/2.06 % (21914)------------------------------
% 4.47/2.06 % (21914)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.47/2.06 % (21914)Termination reason: Unknown
% 4.47/2.06 % (21914)Termination phase: Saturation
% 4.47/2.06
% 4.47/2.06 % (21914)Memory used [KB]: 6780
% 4.47/2.06 % (21914)Time elapsed: 0.069 s
% 4.47/2.06 % (21914)Instructions burned: 127 (million)
% 4.47/2.06 % (21914)------------------------------
% 4.47/2.06 % (21914)------------------------------
% 4.47/2.08 % (21915)Instruction limit reached!
% 4.47/2.08 % (21915)------------------------------
% 4.47/2.08 % (21915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.47/2.08 % (21915)Termination reason: Unknown
% 4.47/2.08 % (21915)Termination phase: Function definition elimination
% 4.47/2.08
% 4.47/2.08 % (21915)Memory used [KB]: 2302
% 4.47/2.08 % (21915)Time elapsed: 0.053 s
% 4.47/2.08 % (21915)Instructions burned: 101 (million)
% 4.47/2.08 % (21915)------------------------------
% 4.47/2.08 % (21915)------------------------------
% 4.47/2.08 % (21916)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_21775 for (2997ds/3Mi)
% 4.47/2.08 % (21916)Instruction limit reached!
% 4.47/2.08 % (21916)------------------------------
% 4.47/2.08 % (21916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.47/2.08 % (21916)Termination reason: Unknown
% 4.47/2.08 % (21916)Termination phase: shuffling
% 4.47/2.08
% 4.47/2.08 % (21916)Memory used [KB]: 1407
% 4.47/2.08 % (21916)Time elapsed: 0.004 s
% 4.47/2.08 % (21916)Instructions burned: 4 (million)
% 4.47/2.08 % (21916)------------------------------
% 4.47/2.08 % (21916)------------------------------
% 4.47/2.10 % (21917)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_21775 for (2997ds/20Mi)
% 4.47/2.10 % (21907)Instruction limit reached!
% 4.47/2.10 % (21907)------------------------------
% 4.47/2.10 % (21907)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.47/2.10 % (21907)Termination reason: Unknown
% 4.47/2.10 % (21907)Termination phase: Saturation
% 4.47/2.10
% 4.47/2.10 % (21907)Memory used [KB]: 7675
% 4.47/2.10 % (21907)Time elapsed: 0.180 s
% 4.47/2.10 % (21907)Instructions burned: 377 (million)
% 4.47/2.10 % (21907)------------------------------
% 4.47/2.10 % (21907)------------------------------
% 4.47/2.10 % (21917)Instruction limit reached!
% 4.47/2.10 % (21917)------------------------------
% 4.47/2.10 % (21917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.47/2.10 % (21917)Termination reason: Unknown
% 4.47/2.10 % (21917)Termination phase: Property scanning
% 4.47/2.10
% 4.47/2.10 % (21917)Memory used [KB]: 1663
% 4.47/2.10 % (21917)Time elapsed: 0.011 s
% 4.47/2.10 % (21917)Instructions burned: 20 (million)
% 4.47/2.10 % (21917)------------------------------
% 4.47/2.10 % (21917)------------------------------
% 4.47/2.11 % (21918)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_21775 for (2997ds/86Mi)
% 4.47/2.12 % (21919)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_21775 for (2996ds/107Mi)
% 5.16/2.12 % (21901)Instruction limit reached!
% 5.16/2.12 % (21901)------------------------------
% 5.16/2.12 % (21901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.12 % (21901)Termination reason: Unknown
% 5.16/2.12 % (21901)Termination phase: Saturation
% 5.16/2.12
% 5.16/2.12 % (21901)Memory used [KB]: 7675
% 5.16/2.12 % (21901)Time elapsed: 0.258 s
% 5.16/2.12 % (21901)Instructions burned: 710 (million)
% 5.16/2.12 % (21901)------------------------------
% 5.16/2.12 % (21901)------------------------------
% 5.16/2.12 % (21920)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_21775 for (2996ds/39Mi)
% 5.16/2.13 % (21918)Instruction limit reached!
% 5.16/2.13 % (21918)------------------------------
% 5.16/2.13 % (21918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.13 % (21918)Termination reason: Unknown
% 5.16/2.13 % (21918)Termination phase: Saturation
% 5.16/2.13
% 5.16/2.13 % (21918)Memory used [KB]: 6524
% 5.16/2.13 % (21918)Time elapsed: 0.025 s
% 5.16/2.13 % (21918)Instructions burned: 87 (million)
% 5.16/2.13 % (21918)------------------------------
% 5.16/2.13 % (21918)------------------------------
% 5.16/2.13 % (21921)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_21775 for (2996ds/448Mi)
% 5.16/2.14 % (21922)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_21775 for (2996ds/46Mi)
% 5.16/2.14 % (21920)Instruction limit reached!
% 5.16/2.14 % (21920)------------------------------
% 5.16/2.14 % (21920)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.14 % (21920)Termination reason: Unknown
% 5.16/2.14 % (21920)Termination phase: shuffling
% 5.16/2.14
% 5.16/2.14 % (21920)Memory used [KB]: 2174
% 5.16/2.14 % (21920)Time elapsed: 0.022 s
% 5.16/2.14 % (21920)Instructions burned: 40 (million)
% 5.16/2.14 % (21920)------------------------------
% 5.16/2.14 % (21920)------------------------------
% 5.16/2.15 % (21922)Instruction limit reached!
% 5.16/2.15 % (21922)------------------------------
% 5.16/2.15 % (21922)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.15 % (21922)Termination reason: Unknown
% 5.16/2.15 % (21922)Termination phase: Naming
% 5.16/2.15
% 5.16/2.15 % (21922)Memory used [KB]: 1791
% 5.16/2.15 % (21922)Time elapsed: 0.015 s
% 5.16/2.15 % (21922)Instructions burned: 48 (million)
% 5.16/2.15 % (21922)------------------------------
% 5.16/2.15 % (21922)------------------------------
% 5.16/2.16 % (21923)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_21775 for (2996ds/98Mi)
% 5.16/2.16 % (21924)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_21775 for (2996ds/507Mi)
% 5.16/2.17 % (21919)Instruction limit reached!
% 5.16/2.17 % (21919)------------------------------
% 5.16/2.17 % (21919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.17 % (21919)Termination reason: Unknown
% 5.16/2.17 % (21919)Termination phase: Saturation
% 5.16/2.17
% 5.16/2.17 % (21919)Memory used [KB]: 6652
% 5.16/2.17 % (21919)Time elapsed: 0.057 s
% 5.16/2.17 % (21919)Instructions burned: 107 (million)
% 5.16/2.17 % (21919)------------------------------
% 5.16/2.17 % (21919)------------------------------
% 5.16/2.18 % (21923)Instruction limit reached!
% 5.16/2.18 % (21923)------------------------------
% 5.16/2.18 % (21923)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.18 % (21923)Termination reason: Unknown
% 5.16/2.18 % (21923)Termination phase: Function definition elimination
% 5.16/2.18
% 5.16/2.18 % (21923)Memory used [KB]: 2302
% 5.16/2.18 % (21923)Time elapsed: 0.026 s
% 5.16/2.18 % (21923)Instructions burned: 98 (million)
% 5.16/2.18 % (21923)------------------------------
% 5.16/2.18 % (21923)------------------------------
% 5.16/2.19 % (21925)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_21775 for (2996ds/149Mi)
% 5.16/2.20 % (21926)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_21775 for (2996ds/39Mi)
% 5.62/2.20 % (21924)First to succeed.
% 5.62/2.20 % (21924)Refutation found. Thanks to Tanya!
% 5.62/2.20 % SZS status Theorem for DTF2THF_21775
% 5.62/2.20 % SZS output start Proof for DTF2THF_21775
% 5.62/2.20 thf(type_def_5, type, int: $tType).
% 5.62/2.20 thf(type_def_7, type, poly_real: $tType).
% 5.62/2.20 thf(type_def_8, type, real: $tType).
% 5.62/2.20 thf(type_def_9, type, nat: $tType).
% 5.62/2.20 thf(type_def_10, type, poly_int: $tType).
% 5.62/2.20 thf(type_def_11, type, poly_nat: $tType).
% 5.62/2.20 thf(type_def_12, type, poly_poly_real: $tType).
% 5.62/2.20 thf(type_def_13, type, set_int: $tType).
% 5.62/2.20 thf(type_def_14, type, set_poly_real: $tType).
% 5.62/2.20 thf(type_def_15, type, set_real: $tType).
% 5.62/2.20 thf(func_def_0, type, poly_poly_real: $tType).
% 5.62/2.20 thf(func_def_1, type, set_poly_real: $tType).
% 5.62/2.20 thf(func_def_2, type, poly_real: $tType).
% 5.62/2.20 thf(func_def_3, type, poly_nat: $tType).
% 5.62/2.20 thf(func_def_4, type, poly_int: $tType).
% 5.62/2.20 thf(func_def_5, type, set_real: $tType).
% 5.62/2.20 thf(func_def_6, type, set_int: $tType).
% 5.62/2.20 thf(func_def_7, type, real: $tType).
% 5.62/2.20 thf(func_def_8, type, nat: $tType).
% 5.62/2.20 thf(func_def_9, type, int: $tType).
% 5.62/2.20 thf(func_def_10, type, abs_abs_int: int > int).
% 5.62/2.20 thf(func_def_11, type, abs_abs_poly_real: poly_real > poly_real).
% 5.62/2.20 thf(func_def_12, type, abs_abs_real: real > real).
% 5.62/2.20 thf(func_def_13, type, minus_minus_real: real > real > real).
% 5.62/2.20 thf(func_def_14, type, one_one_int: int).
% 5.62/2.20 thf(func_def_15, type, one_one_nat: nat).
% 5.62/2.20 thf(func_def_16, type, one_one_poly_int: poly_int).
% 5.62/2.20 thf(func_def_17, type, one_one_poly_nat: poly_nat).
% 5.62/2.20 thf(func_def_18, type, one_one_poly_real: poly_real).
% 5.62/2.20 thf(func_def_19, type, one_one_real: real).
% 5.62/2.20 thf(func_def_20, type, zero_zero_int: int).
% 5.62/2.20 thf(func_def_21, type, zero_zero_nat: nat).
% 5.62/2.20 thf(func_def_22, type, zero_zero_poly_int: poly_int).
% 5.62/2.20 thf(func_def_23, type, zero_zero_poly_nat: poly_nat).
% 5.62/2.20 thf(func_def_24, type, zero_z1423781445y_real: poly_poly_real).
% 5.62/2.20 thf(func_def_25, type, zero_zero_poly_real: poly_real).
% 5.62/2.20 thf(func_def_26, type, zero_zero_real: real).
% 5.62/2.20 thf(func_def_27, type, ring_1_Ints_int: set_int).
% 5.62/2.20 thf(func_def_28, type, ring_1690226883y_real: set_poly_real).
% 5.62/2.20 thf(func_def_29, type, ring_1_Ints_real: set_real).
% 5.62/2.20 thf(func_def_30, type, ring_1_of_int_int: int > int).
% 5.62/2.20 thf(func_def_31, type, ring_12102921859ly_int: int > poly_int).
% 5.62/2.20 thf(func_def_32, type, ring_11511526659y_real: int > poly_real).
% 5.62/2.20 thf(func_def_33, type, ring_1_of_int_real: int > real).
% 5.62/2.20 thf(func_def_34, type, ord_less_int: int > int > $o).
% 5.62/2.20 thf(func_def_35, type, ord_less_nat: nat > nat > $o).
% 5.62/2.20 thf(func_def_36, type, ord_less_poly_real: poly_real > poly_real > $o).
% 5.62/2.20 thf(func_def_37, type, ord_less_real: real > real > $o).
% 5.62/2.20 thf(func_def_38, type, ord_less_eq_int: int > int > $o).
% 5.62/2.20 thf(func_def_39, type, ord_less_eq_nat: nat > nat > $o).
% 5.62/2.20 thf(func_def_40, type, ord_le1180086932y_real: poly_real > poly_real > $o).
% 5.62/2.20 thf(func_def_41, type, ord_less_eq_real: real > real > $o).
% 5.62/2.20 thf(func_def_42, type, algebraic_real: real > $o).
% 5.62/2.20 thf(func_def_43, type, degree_int: poly_int > nat).
% 5.62/2.20 thf(func_def_44, type, degree_nat: poly_nat > nat).
% 5.62/2.20 thf(func_def_45, type, degree_poly_real: poly_poly_real > nat).
% 5.62/2.20 thf(func_def_46, type, degree_real: poly_real > nat).
% 5.62/2.20 thf(func_def_47, type, poly_int2: poly_int > int > int).
% 5.62/2.20 thf(func_def_48, type, poly_nat2: poly_nat > nat > nat).
% 5.62/2.20 thf(func_def_49, type, poly_poly_real2: poly_poly_real > poly_real > poly_real).
% 5.62/2.20 thf(func_def_50, type, poly_real2: poly_real > real > real).
% 5.62/2.20 thf(func_def_51, type, coeff_int: poly_int > nat > int).
% 5.62/2.20 thf(func_def_52, type, coeff_nat: poly_nat > nat > nat).
% 5.62/2.20 thf(func_def_53, type, coeff_poly_real: poly_poly_real > nat > poly_real).
% 5.62/2.20 thf(func_def_54, type, coeff_real: poly_real > nat > real).
% 5.62/2.20 thf(func_def_55, type, power_power_int: int > nat > int).
% 5.62/2.20 thf(func_def_56, type, power_power_nat: nat > nat > nat).
% 5.62/2.20 thf(func_def_57, type, power_power_poly_int: poly_int > nat > poly_int).
% 5.62/2.20 thf(func_def_58, type, power_power_poly_nat: poly_nat > nat > poly_nat).
% 5.62/2.20 thf(func_def_59, type, power_2108872382y_real: poly_real > nat > poly_real).
% 5.62/2.20 thf(func_def_60, type, power_power_real: real > nat > real).
% 5.62/2.20 thf(func_def_61, type, field_1537545994s_real: set_real).
% 5.62/2.20 thf(func_def_62, type, divide_divide_int: int > int > int).
% 5.62/2.20 thf(func_def_63, type, divide_divide_nat: nat > nat > nat).
% 5.62/2.20 thf(func_def_64, type, divide1727078534y_real: poly_real > poly_real > poly_real).
% 5.62/2.20 thf(func_def_65, type, divide_divide_real: real > real > real).
% 5.62/2.20 thf(func_def_66, type, collect_real: (real > $o) > set_real).
% 5.62/2.20 thf(func_def_67, type, member_int: int > set_int > $o).
% 5.62/2.20 thf(func_def_68, type, member_poly_real: poly_real > set_poly_real > $o).
% 5.62/2.20 thf(func_def_69, type, member_real: real > set_real > $o).
% 5.62/2.20 thf(func_def_70, type, a: real).
% 5.62/2.20 thf(func_def_71, type, m: real).
% 5.62/2.20 thf(func_def_72, type, a2: int).
% 5.62/2.20 thf(func_def_73, type, b: int).
% 5.62/2.20 thf(func_def_74, type, n: nat).
% 5.62/2.20 thf(func_def_75, type, p: poly_real).
% 5.62/2.20 thf(func_def_76, type, roots: set_real).
% 5.62/2.20 thf(func_def_78, type, x: real).
% 5.62/2.20 thf(func_def_93, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.62/2.20 thf(func_def_94, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.62/2.20 thf(func_def_95, type, sK2: poly_real > poly_real > nat).
% 5.62/2.20 thf(func_def_96, type, sK3: poly_real > nat).
% 5.62/2.20 thf(func_def_97, type, sK4: real > poly_real > real > real).
% 5.62/2.20 thf(func_def_98, type, sK5: poly_poly_real > nat > nat).
% 5.62/2.20 thf(func_def_99, type, sK6: nat > poly_nat > nat).
% 5.62/2.20 thf(func_def_100, type, sK7: real > nat > int).
% 5.62/2.20 thf(func_def_101, type, sK8: real > nat > int).
% 5.62/2.20 thf(func_def_102, type, sK9: real > poly_real).
% 5.62/2.20 thf(func_def_103, type, sK10: poly_real > nat > nat).
% 5.62/2.20 thf(func_def_104, type, sK11: real > int).
% 5.62/2.20 thf(func_def_105, type, sK12: nat > poly_real > nat).
% 5.62/2.20 thf(func_def_106, type, sK13: real > poly_real > real > real).
% 5.62/2.20 thf(func_def_107, type, sK14: real > real).
% 5.62/2.20 thf(func_def_108, type, sK15: poly_real).
% 5.62/2.20 thf(func_def_109, type, sK16: nat > poly_int > nat).
% 5.62/2.20 thf(func_def_110, type, sK17: poly_real > real).
% 5.62/2.20 thf(func_def_111, type, sK18: nat > poly_int > nat).
% 5.62/2.20 thf(func_def_112, type, sK19: (real > $o) > int).
% 5.62/2.20 thf(func_def_113, type, sK20: real > real).
% 5.62/2.20 thf(func_def_114, type, sK21: poly_poly_real > nat > nat).
% 5.62/2.20 thf(func_def_115, type, sK22: (int > $o) > int).
% 5.62/2.20 thf(func_def_116, type, sK23: int > int).
% 5.62/2.20 thf(func_def_117, type, sK24: real > int).
% 5.62/2.20 thf(func_def_118, type, sK25: real > int).
% 5.62/2.20 thf(func_def_119, type, sK26: poly_real > nat).
% 5.62/2.20 thf(func_def_120, type, sK27: poly_nat > nat > nat).
% 5.62/2.20 thf(func_def_122, type, ph29: !>[X0: $tType]:(X0)).
% 5.62/2.20 thf(f2062,plain,(
% 5.62/2.20 $false),
% 5.62/2.20 inference(subsumption_resolution,[status(thm)],[f2061,f1486])).
% 5.62/2.20 thf(f1486,plain,(
% 5.62/2.20 ( ! [X0 : nat] : (((member_real @ (coeff_real @ p @ X0) @ ring_1_Ints_real) = $true)) )),
% 5.62/2.20 inference(cnf_transformation,[status(thm)],[f403])).
% 5.62/2.20 thf(f403,plain,(
% 5.62/2.20 ! [X0 : nat] : ((member_real @ (coeff_real @ p @ X0) @ ring_1_Ints_real) = $true)),
% 5.62/2.20 inference(fool_elimination,[status(thm)],[f402])).
% 5.62/2.20 thf(f402,plain,(
% 5.62/2.20 ! [X0 : nat] : (member_real @ (coeff_real @ p @ X0) @ ring_1_Ints_real)),
% 5.62/2.20 inference(rectify,[status(thm)],[f3])).
% 5.62/2.20 thf(f3,axiom,(
% 5.62/2.20 ! [X0 : nat] : (member_real @ (coeff_real @ p @ X0) @ ring_1_Ints_real)),
% 5.62/2.20 file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',fact_2_p_I1_J)).
% 5.62/2.20 thf(f2061,plain,(
% 5.62/2.20 ($true != (member_real @ (coeff_real @ p @ (sK26 @ p)) @ ring_1_Ints_real))),
% 5.62/2.20 inference(subsumption_resolution,[status(thm)],[f2060,f1536])).
% 5.62/2.20 thf(f1536,plain,(
% 5.62/2.20 ((ord_less_int @ zero_zero_int @ b) = $true)),
% 5.62/2.20 inference(cnf_transformation,[status(thm)],[f1024])).
% 5.62/2.20 thf(f1024,plain,(
% 5.62/2.20 ((ord_less_int @ zero_zero_int @ b) = $true)),
% 5.62/2.20 inference(fool_elimination,[status(thm)],[f1023])).
% 5.62/2.20 thf(f1023,plain,(
% 5.62/2.20 (ord_less_int @ zero_zero_int @ b)),
% 5.62/2.20 inference(rectify,[status(thm)],[f4])).
% 5.62/2.20 thf(f4,axiom,(
% 5.62/2.20 (ord_less_int @ zero_zero_int @ b)),
% 5.62/2.20 file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',fact_3_b)).
% 5.62/2.20 thf(f2060,plain,(
% 5.62/2.20 ((ord_less_int @ zero_zero_int @ b) != $true) | ($true != (member_real @ (coeff_real @ p @ (sK26 @ p)) @ ring_1_Ints_real))),
% 5.62/2.20 inference(subsumption_resolution,[status(thm)],[f2045,f1804])).
% 5.62/2.20 thf(f1804,plain,(
% 5.62/2.20 ((poly_real2 @ p @ x) != (poly_real2 @ p @ (divide_divide_real @ (ring_1_of_int_real @ a2) @ (ring_1_of_int_real @ b))))),
% 5.62/2.20 inference(definition_unfolding,[status(thm)],[f1445,f1400])).
% 5.62/2.20 thf(f1400,plain,(
% 5.62/2.20 ((poly_real2 @ p @ x) = zero_zero_real)),
% 5.62/2.20 inference(cnf_transformation,[status(thm)],[f776])).
% 5.62/2.20 thf(f776,plain,(
% 5.62/2.20 ((poly_real2 @ p @ x) = zero_zero_real)),
% 5.62/2.20 inference(fool_elimination,[status(thm)],[f5])).
% 5.62/2.20 thf(f5,axiom,(
% 5.62/2.20 ((poly_real2 @ p @ x) = zero_zero_real)),
% 5.62/2.20 file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',fact_4_p_I3_J)).
% 5.62/2.20 thf(f1445,plain,(
% 5.62/2.20 (zero_zero_real != (poly_real2 @ p @ (divide_divide_real @ (ring_1_of_int_real @ a2) @ (ring_1_of_int_real @ b))))),
% 5.62/2.20 inference(cnf_transformation,[status(thm)],[f699])).
% 5.62/2.20 thf(f699,plain,(
% 5.62/2.20 (zero_zero_real != (poly_real2 @ p @ (divide_divide_real @ (ring_1_of_int_real @ a2) @ (ring_1_of_int_real @ b))))),
% 5.62/2.20 inference(fool_elimination,[status(thm)],[f6])).
% 5.62/2.20 thf(f6,axiom,(
% 5.62/2.20 (zero_zero_real != (poly_real2 @ p @ (divide_divide_real @ (ring_1_of_int_real @ a2) @ (ring_1_of_int_real @ b))))),
% 5.62/2.20 file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',fact_5_no__root)).
% 5.62/2.20 thf(f2045,plain,(
% 5.62/2.20 ((poly_real2 @ p @ x) = (poly_real2 @ p @ (divide_divide_real @ (ring_1_of_int_real @ a2) @ (ring_1_of_int_real @ b)))) | ((ord_less_int @ zero_zero_int @ b) != $true) | ($true != (member_real @ (coeff_real @ p @ (sK26 @ p)) @ ring_1_Ints_real))),
% 5.62/2.20 inference(trivial_inequality_removal,[status(thm)],[f2044])).
% 5.62/2.20 thf(f2044,plain,(
% 5.62/2.20 ((poly_real2 @ p @ x) = (poly_real2 @ p @ (divide_divide_real @ (ring_1_of_int_real @ a2) @ (ring_1_of_int_real @ b)))) | ($true != $true) | ($true != (member_real @ (coeff_real @ p @ (sK26 @ p)) @ ring_1_Ints_real)) | ((ord_less_int @ zero_zero_int @ b) != $true)),
% 5.62/2.20 inference(superposition,[status(thm)],[f1951,f2007])).
% 5.62/2.20 thf(f2007,plain,(
% 5.62/2.20 ( ! [X2 : int,X0 : poly_real,X1 : int] : (($true = (ord_less_eq_real @ (divide_divide_real @ (coeff_real @ one_one_poly_real @ (degree_real @ one_one_poly_real)) @ (power_power_real @ (ring_1_of_int_real @ X1) @ (degree_real @ X0))) @ (abs_abs_real @ (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1)))))) | ((poly_real2 @ p @ x) = (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1)))) | ($true != (member_real @ (coeff_real @ X0 @ (sK26 @ X0)) @ ring_1_Ints_real)) | ((ord_less_int @ zero_zero_int @ X1) != $true)) )),
% 5.62/2.20 inference(definition_unfolding,[status(thm)],[f1729,f1400,f1437])).
% 5.62/2.20 thf(f1437,plain,(
% 5.62/2.20 (one_one_real = (coeff_real @ one_one_poly_real @ (degree_real @ one_one_poly_real)))),
% 5.62/2.20 inference(cnf_transformation,[status(thm)],[f418])).
% 5.62/2.20 thf(f418,plain,(
% 5.62/2.20 (one_one_real = (coeff_real @ one_one_poly_real @ (degree_real @ one_one_poly_real)))),
% 5.62/2.20 inference(fool_elimination,[status(thm)],[f69])).
% 5.62/2.20 thf(f69,axiom,(
% 5.62/2.20 (one_one_real = (coeff_real @ one_one_poly_real @ (degree_real @ one_one_poly_real)))),
% 5.62/2.20 file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',fact_68_lead__coeff__1)).
% 5.62/2.20 thf(f1729,plain,(
% 5.62/2.20 ( ! [X2 : int,X0 : poly_real,X1 : int] : (((ord_less_int @ zero_zero_int @ X1) != $true) | ($true != (member_real @ (coeff_real @ X0 @ (sK26 @ X0)) @ ring_1_Ints_real)) | (zero_zero_real = (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1)))) | ($true = (ord_less_eq_real @ (divide_divide_real @ one_one_real @ (power_power_real @ (ring_1_of_int_real @ X1) @ (degree_real @ X0))) @ (abs_abs_real @ (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1))))))) )),
% 5.62/2.20 inference(cnf_transformation,[status(thm)],[f1375])).
% 5.62/2.20 thf(f1375,plain,(
% 5.62/2.20 ! [X0 : poly_real,X1 : int,X2 : int] : (((ord_less_int @ zero_zero_int @ X1) != $true) | ($true != (member_real @ (coeff_real @ X0 @ (sK26 @ X0)) @ ring_1_Ints_real)) | (zero_zero_real = (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1)))) | ($true = (ord_less_eq_real @ (divide_divide_real @ one_one_real @ (power_power_real @ (ring_1_of_int_real @ X1) @ (degree_real @ X0))) @ (abs_abs_real @ (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1)))))))),
% 5.62/2.20 inference(skolemisation,[status(esa),new_symbols(skolem,[sK26])],[f1309,f1374])).
% 5.62/2.20 thf(f1374,plain,(
% 5.62/2.20 ! [X0 : poly_real] : (? [X3 : nat] : ($true != (member_real @ (coeff_real @ X0 @ X3) @ ring_1_Ints_real)) => ($true != (member_real @ (coeff_real @ X0 @ (sK26 @ X0)) @ ring_1_Ints_real)))),
% 5.62/2.20 introduced(definition,[],[choice_axiom])).
% 5.62/2.20 thf(f1309,plain,(
% 5.62/2.20 ! [X0 : poly_real,X1 : int,X2 : int] : (((ord_less_int @ zero_zero_int @ X1) != $true) | ? [X3 : nat] : ($true != (member_real @ (coeff_real @ X0 @ X3) @ ring_1_Ints_real)) | (zero_zero_real = (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1)))) | ($true = (ord_less_eq_real @ (divide_divide_real @ one_one_real @ (power_power_real @ (ring_1_of_int_real @ X1) @ (degree_real @ X0))) @ (abs_abs_real @ (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1)))))))),
% 5.62/2.20 inference(flattening,[status(thm)],[f1308])).
% 5.62/2.20 thf(f1308,plain,(
% 5.62/2.20 ! [X0 : poly_real,X1 : int,X2 : int] : (((($true = (ord_less_eq_real @ (divide_divide_real @ one_one_real @ (power_power_real @ (ring_1_of_int_real @ X1) @ (degree_real @ X0))) @ (abs_abs_real @ (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1)))))) | (zero_zero_real = (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1))))) | ((ord_less_int @ zero_zero_int @ X1) != $true)) | ? [X3 : nat] : ($true != (member_real @ (coeff_real @ X0 @ X3) @ ring_1_Ints_real)))),
% 5.62/2.20 inference(ennf_transformation,[status(thm)],[f890])).
% 5.62/2.20 thf(f890,plain,(
% 5.62/2.20 ! [X0 : poly_real,X1 : int,X2 : int] : (! [X3 : nat] : ($true = (member_real @ (coeff_real @ X0 @ X3) @ ring_1_Ints_real)) => (((ord_less_int @ zero_zero_int @ X1) = $true) => ((zero_zero_real != (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1)))) => ($true = (ord_less_eq_real @ (divide_divide_real @ one_one_real @ (power_power_real @ (ring_1_of_int_real @ X1) @ (degree_real @ X0))) @ (abs_abs_real @ (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1)))))))))),
% 5.62/2.20 inference(fool_elimination,[status(thm)],[f889])).
% 5.62/2.20 thf(f889,plain,(
% 5.62/2.20 ! [X0 : poly_real,X1 : int,X2 : int] : (! [X3 : nat] : (member_real @ (coeff_real @ X0 @ X3) @ ring_1_Ints_real) => ((ord_less_int @ zero_zero_int @ X1) => ((zero_zero_real != (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1)))) => (ord_less_eq_real @ (divide_divide_real @ one_one_real @ (power_power_real @ (ring_1_of_int_real @ X1) @ (degree_real @ X0))) @ (abs_abs_real @ (poly_real2 @ X0 @ (divide_divide_real @ (ring_1_of_int_real @ X2) @ (ring_1_of_int_real @ X1))))))))),
% 5.62/2.20 inference(rectify,[status(thm)],[f155])).
% 5.62/2.20 thf(f155,axiom,(
% 5.62/2.20 ! [X5 : poly_real,X1 : int,X12 : int] : (! [X11 : nat] : (member_real @ (coeff_real @ X5 @ X11) @ ring_1_Ints_real) => ((ord_less_int @ zero_zero_int @ X1) => ((zero_zero_real != (poly_real2 @ X5 @ (divide_divide_real @ (ring_1_of_int_real @ X12) @ (ring_1_of_int_real @ X1)))) => (ord_less_eq_real @ (divide_divide_real @ one_one_real @ (power_power_real @ (ring_1_of_int_real @ X1) @ (degree_real @ X5))) @ (abs_abs_real @ (poly_real2 @ X5 @ (divide_divide_real @ (ring_1_of_int_real @ X12) @ (ring_1_of_int_real @ X1))))))))),
% 5.62/2.20 file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',fact_154_int__poly__rat__no__root__ge)).
% 5.62/2.20 thf(f1951,plain,(
% 5.62/2.20 ($true != (ord_less_eq_real @ (divide_divide_real @ (coeff_real @ one_one_poly_real @ (degree_real @ one_one_poly_real)) @ (power_power_real @ (ring_1_of_int_real @ b) @ (degree_real @ p))) @ (abs_abs_real @ (poly_real2 @ p @ (divide_divide_real @ (ring_1_of_int_real @ a2) @ (ring_1_of_int_real @ b))))))),
% 5.62/2.20 inference(definition_unfolding,[status(thm)],[f1641,f1437])).
% 5.62/2.20 thf(f1641,plain,(
% 5.62/2.20 ((ord_less_eq_real @ (divide_divide_real @ one_one_real @ (power_power_real @ (ring_1_of_int_real @ b) @ (degree_real @ p))) @ (abs_abs_real @ (poly_real2 @ p @ (divide_divide_real @ (ring_1_of_int_real @ a2) @ (ring_1_of_int_real @ b))))) != $true)),
% 5.62/2.20 inference(cnf_transformation,[status(thm)],[f1071])).
% 5.62/2.20 thf(f1071,plain,(
% 5.62/2.20 ((ord_less_eq_real @ (divide_divide_real @ one_one_real @ (power_power_real @ (ring_1_of_int_real @ b) @ (degree_real @ p))) @ (abs_abs_real @ (poly_real2 @ p @ (divide_divide_real @ (ring_1_of_int_real @ a2) @ (ring_1_of_int_real @ b))))) != $true)),
% 5.62/2.20 inference(flattening,[status(thm)],[f863])).
% 5.62/2.20 thf(f863,plain,(
% 5.62/2.20 ~((ord_less_eq_real @ (divide_divide_real @ one_one_real @ (power_power_real @ (ring_1_of_int_real @ b) @ (degree_real @ p))) @ (abs_abs_real @ (poly_real2 @ p @ (divide_divide_real @ (ring_1_of_int_real @ a2) @ (ring_1_of_int_real @ b))))) = $true)),
% 5.62/2.20 inference(fool_elimination,[status(thm)],[f862])).
% 5.62/2.20 thf(f862,plain,(
% 5.62/2.20 ~(ord_less_eq_real @ (divide_divide_real @ one_one_real @ (power_power_real @ (ring_1_of_int_real @ b) @ (degree_real @ p))) @ (abs_abs_real @ (poly_real2 @ p @ (divide_divide_real @ (ring_1_of_int_real @ a2) @ (ring_1_of_int_real @ b)))))),
% 5.62/2.20 inference(rectify,[status(thm)],[f355])).
% 5.62/2.20 thf(f355,negated_conjecture,(
% 5.62/2.20 ~(ord_less_eq_real @ (divide_divide_real @ one_one_real @ (power_power_real @ (ring_1_of_int_real @ b) @ (degree_real @ p))) @ (abs_abs_real @ (poly_real2 @ p @ (divide_divide_real @ (ring_1_of_int_real @ a2) @ (ring_1_of_int_real @ b)))))),
% 5.62/2.20 inference(negated_conjecture,[status(cth)],[f354])).
% 5.62/2.20 thf(f354,conjecture,(
% 5.62/2.20 (ord_less_eq_real @ (divide_divide_real @ one_one_real @ (power_power_real @ (ring_1_of_int_real @ b) @ (degree_real @ p))) @ (abs_abs_real @ (poly_real2 @ p @ (divide_divide_real @ (ring_1_of_int_real @ a2) @ (ring_1_of_int_real @ b)))))),
% 5.62/2.20 file('/export/starexec/sandbox/tmp/tmp.Rh8XhsTbSL/DTF2THF_21775.p',conj_0)).
% 5.62/2.20 % SZS output end Proof for DTF2THF_21775
% 5.62/2.20 % (21924)------------------------------
% 5.62/2.20 % (21924)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.62/2.20 % (21924)Termination reason: Refutation
% 5.62/2.20
% 5.62/2.20 % (21924)Memory used [KB]: 7291
% 5.62/2.20 % (21924)Time elapsed: 0.039 s
% 5.62/2.20 % (21924)Instructions burned: 126 (million)
% 5.62/2.20 % (21924)------------------------------
% 5.62/2.20 % (21924)------------------------------
% 5.62/2.20 % (21880)Success in time 0.402 s
%------------------------------------------------------------------------------